:: MATRIX13 semantic presentation

REAL is set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal V103() Element of bool REAL
bool REAL is non empty set
RAT is set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -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 V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal V103() set
bool NAT is non empty non trivial non finite V103() set
bool NAT is non empty non trivial non finite V103() set
Fin NAT is preBoolean set
INT is set
COMPLEX is set
1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[:NAT,NAT:],NAT:] is Relation-like non empty non trivial non finite V103() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite V103() set
3 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -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 V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() Element of NAT
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -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 V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() set
Seg 1 is non empty trivial finite 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V37() 1 -element without_zero V103() set
Seg 2 is non empty finite 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V37() without_zero V103() set
[1,1] is set
{1,1} is non empty finite V37() without_zero V103() set
{{1,1},{1}} is non empty finite V37() without_zero V103() set
6 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
4 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
24 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular Matrix of m,m, the carrier of n
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
Indices (K @) is set
dom (K @) is finite Element of bool NAT
width (K @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (K @)) is finite width (K @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (K @) ) } is set
[:(dom (K @)),(Seg (width (K @))):] is Relation-like finite set
[R,M] is set
{R,M} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,M},{R}} is non empty finite V37() without_zero V103() set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
K * (R,M) is Element of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(K @) * (M,R) is Element of the carrier of n
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
[R,M] is set
{R,M} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,M},{R}} is non empty finite V37() without_zero V103() set
Indices (K @) is set
dom (K @) is finite Element of bool NAT
width (K @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (K @)) is finite width (K @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (K @) ) } is set
[:(dom (K @)),(Seg (width (K @))):] is Relation-like finite set
(K @) * (R,M) is Element of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K * (M,R) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix (K @) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (diagonal_of_Matrix K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(diagonal_of_Matrix (K @)) . P is set
(K @) * (P,P) is Element of the carrier of n
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
[:(Seg m),(Seg m):] is Relation-like finite set
[P,P] is set
{P,P} is non empty finite V37() set
{P} is non empty trivial finite V37() 1 -element set
{{P,P},{P}} is non empty finite V37() without_zero V103() set
(diagonal_of_Matrix K) . P is set
K * (P,P) is Element of the carrier of n
len (diagonal_of_Matrix (K @)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
n is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
K is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
dom K is finite Element of bool (Seg m)
bool (Seg m) is non empty finite V37() set
M is set
n . M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R - 1 is V105() complex ext-real set
Seg R is finite R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
K .: (Seg R) is finite set
Q is finite set
card (Seg R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n .: (Seg R) is finite set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P is finite P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
S is set
x1 is set
n . x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
n . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
rng K is finite set
rng n is finite set
n . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
rng K is finite set
rng n is finite set
card (Seg P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
nat_interval (R,m) is finite Element of bool NAT
K .: (nat_interval (R,m)) is finite set
Q is finite set
S is set
rng K is finite set
x1 is set
n . x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (nat_interval (R,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is set
n . S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
M is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
(Path_product K) . M is Element of the carrier of n
Path_matrix (M,K) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
R is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
rng R is finite set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M . S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom R is finite Element of bool (Seg m)
bool (Seg m) is non empty finite V37() set
R . S is set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[S,x1] is set
{S,x1} is non empty finite V37() set
{S} is non empty trivial finite V37() 1 -element set
{{S,x1},{S}} is non empty finite V37() without_zero V103() set
K * (S,x1) is Element of the carrier of n
len (Path_matrix (M,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Path_matrix (M,K)) is finite Element of bool NAT
(Path_matrix (M,K)) . S is set
R is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
rng R is finite set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M . S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom R is finite Element of bool (Seg m)
bool (Seg m) is non empty finite V37() set
R . S is set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[S,x1] is set
{S,x1} is non empty finite V37() set
{S} is non empty trivial finite V37() 1 -element set
{{S,x1},{S}} is non empty finite V37() without_zero V103() set
K * (S,x1) is Element of the carrier of n
len (Path_matrix (M,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Path_matrix (M,K)) is finite Element of bool NAT
(Path_matrix (M,K)) . S is set
dom (Path_matrix (M,K)) is finite Element of bool NAT
dom (Path_matrix (M,K)) is finite Element of bool NAT
Product (Path_matrix (M,K)) is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n $$ (Path_matrix (M,K)) is Element of the carrier of n
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Path_matrix (M,K)) . S is set
- ((0. n),M) is Element of the carrier of n
- (0. n) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
M is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
Path_matrix (M,K) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (Path_matrix (M,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom (Path_matrix (M,K)) is finite Element of bool NAT
(Path_matrix (M,K)) . Q is set
K * (Q,Q) is Element of the carrier of n
(diagonal_of_Matrix K) . Q is set
len (diagonal_of_Matrix K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular upper_triangular Matrix of m,m, the carrier of n
Det K is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product K)) is Element of the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (diagonal_of_Matrix K) is Element of the carrier of n
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
Group_of_Perm m is non empty strict Group-like associative multMagma
the carrier of (Group_of_Perm m) is non empty set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
x1 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
{x1} is functional non empty trivial finite V37() 1 -element Element of bool (Permutations m)
bool (Permutations m) is non empty set
(Permutations m) \ {x1} is Element of bool (Permutations m)
i1 is Element of Fin (Permutations m)
x2 is Element of Fin (Permutations m)
the addF of n $$ (x2,(Path_product K)) is Element of the carrier of n
i1 is Element of Fin (Permutations m)
(Path_product K) .: i1 is Element of Fin the carrier of n
Fin the carrier of n is preBoolean set
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty trivial finite 1 -element Element of bool the carrier of n
bool the carrier of n is non empty set
i2 is set
dom (Path_product K) is non empty Element of bool (Permutations m)
y1 is set
(Path_product K) . y1 is set
y2 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
(Path_product K) . y2 is Element of the carrier of n
the_unity_wrt the addF of n is Element of the carrier of n
dom (Path_product K) is non empty Element of bool (Permutations m)
the addF of n $$ (i1,(Path_product K)) is Element of the carrier of n
x2 is Element of Fin (Permutations m)
i1 \/ x2 is Element of Fin (Permutations m)
x2 \/ (Permutations m) is non empty set
the addF of n $$ (x2,(Path_product K)) is Element of the carrier of n
( the addF of n $$ (x2,(Path_product K))) + (0. n) is Element of the carrier of n
the addF of n . (( the addF of n $$ (x2,(Path_product K))),(0. n)) is Element of the carrier of n
i1 is Element of Fin (Permutations m)
x2 is Element of Fin (Permutations m)
the addF of n $$ (x2,(Path_product K)) is Element of the carrier of n
x2 is Element of Fin (Permutations m)
the addF of n $$ (x2,(Path_product K)) is Element of the carrier of n
(Path_product K) . x1 is Element of the carrier of n
Path_matrix (x1,K) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (Path_matrix (x1,K)) is Element of the carrier of n
- (( the multF of n $$ (Path_matrix (x1,K))),x1) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular lower_triangular Matrix of m,m, the carrier of n
Det K is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product K)) is Element of the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (diagonal_of_Matrix K) is Element of the carrier of n
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det (K @) is Element of the carrier of n
Path_product (K @) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (K @))) is Element of the carrier of n
diagonal_of_Matrix (K @) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (diagonal_of_Matrix (K @)) is Element of the carrier of n
m is finite set
bool m is non empty finite V37() set
card m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{ b₁ where b₁ is finite Element of bool m : card b₁ = n } is set
card { b₁ where b₁ is finite Element of bool m : card b₁ = n } is V26() V27() V28() cardinal set
(card m) choose n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Choose (m,K,1,0) is functional Element of bool (Funcs (m,{1,0}))
{1,0} is non empty finite V37() set
Funcs (m,{1,0}) is functional non empty FUNCTION_DOMAIN of m,{1,0}
bool (Funcs (m,{1,0})) is non empty set
m --> 0 is Relation-like m -defined NAT -valued RAT -valued Function-like constant total quasi_total finite Function-yielding V147() V202() V203() V204() V205() Element of bool [:m,NAT:]
[:m,NAT:] is Relation-like set
bool [:m,NAT:] is non empty set
{0} is functional non empty trivial finite V37() 1 -element set
[:m,{0}:] is Relation-like finite set
P is Relation-like Function-like set
dom P is set
rng P is set
Q is set
dom (m --> 0) is finite Element of bool m
{0,1} is non empty finite V37() Element of bool NAT
[:m,{0,1}:] is Relation-like finite set
bool [:m,{0,1}:] is non empty finite V37() set
{1} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
S is Relation-like m -defined {0,1} -valued Function-like total quasi_total finite Element of bool [:m,{0,1}:]
S " {1} is finite set
card (S " {1}) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom S is finite Element of bool m
x1 is finite Element of bool m
x2 is set
S . x2 is set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
(x1 --> 1) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
dom (x1 --> 1) is finite Element of bool x1
bool x1 is non empty finite V37() set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x1) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
S . x2 is set
rng S is finite set
(m --> 0) . x2 is Relation-like Function-like V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
dom (x1 --> 1) is finite Element of bool x1
bool x1 is non empty finite V37() set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x1) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
S . x2 is set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x1) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
S . x2 is set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x1) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
dom (x1 --> 1) is finite Element of bool x1
bool x1 is non empty finite V37() set
dom H₁(x1) is set
m \/ x1 is finite set
P . x1 is set
Q is set
S is set
P . Q is set
P . S is set
x1 is finite Element of bool m
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 is finite Element of bool m
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 --> 1 is Relation-like non-empty x2 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x2,NAT:]
[:x2,NAT:] is Relation-like set
bool [:x2,NAT:] is non empty set
[:x2,{1}:] is Relation-like finite set
dom (x2 --> 1) is finite Element of bool x2
bool x2 is non empty finite V37() set
i1 is set
(x2 --> 1) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
(m --> 0) +* (x2 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x2) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
dom (x1 --> 1) is finite Element of bool x1
bool x1 is non empty finite V37() set
(m --> 0) . i1 is Relation-like Function-like V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
H₁(x1) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative V201() set
Q is set
S is set
P . S is set
x1 is finite Element of bool m
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 \/ m is finite set
x1 --> 1 is Relation-like non-empty x1 -defined NAT -valued RAT -valued Function-like constant total quasi_total finite V202() V203() V204() V205() Element of bool [:x1,NAT:]
[:x1,NAT:] is Relation-like set
bool [:x1,NAT:] is non empty set
[:x1,{1}:] is Relation-like finite set
(m --> 0) +* (x1 --> 1) is Relation-like RAT -valued Function-like finite V202() V203() V204() V205() set
card (Choose (m,K,1,0)) is V26() V27() V28() cardinal set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
TWOELEMENTSETS (Seg m) is set
card (TWOELEMENTSETS (Seg m)) is V26() V27() V28() cardinal set
m choose 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
bool (Seg m) is non empty finite V37() set
{ b₁ where b₁ is finite Element of bool (Seg m) : card b₁ = 2 } is set
card (Seg m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card (Seg m)) choose 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
Rev (idseq m) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m choose 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m choose 2) mod 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
m - 1 is V105() complex ext-real set
m * (m - 1) is V105() complex ext-real set
(m * (m - 1)) / 2 is V105() complex ext-real set
m - 2 is V105() complex ext-real set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M + 2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Permutations (M + 2) is non empty permutational set
len (Permutations (M + 2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations (M + 2))) is finite len (Permutations (M + 2)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (M + 2)) ) } is set
the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
Seg (M + 2) is non empty finite M + 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M + 2 ) } is set
TWOELEMENTSETS (Seg (M + 2)) is non empty finite set
FinOmega (TWOELEMENTSETS (Seg (M + 2))) is Element of Fin (TWOELEMENTSETS (Seg (M + 2)))
Fin (TWOELEMENTSETS (Seg (M + 2))) is preBoolean set
idseq (M + 2) is Relation-like NAT -defined Function-like finite M + 2 -element FinSequence-like FinSubsequence-like set
id (Seg (M + 2)) is Relation-like Seg (M + 2) -defined Seg (M + 2) -valued Function-like one-to-one non empty total quasi_total finite Element of bool [:(Seg (M + 2)),(Seg (M + 2)):]
[:(Seg (M + 2)),(Seg (M + 2)):] is Relation-like non empty finite set
bool [:(Seg (M + 2)),(Seg (M + 2)):] is non empty finite V37() set
Group_of_Perm (M + 2) is non empty strict Group-like associative multMagma
the carrier of (Group_of_Perm (M + 2)) is non empty set
the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is non empty non trivial V103() set
S is Relation-like Seg (len (Permutations (M + 2))) -defined Seg (len (Permutations (M + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (M + 2)
Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Relation-like TWOELEMENTSETS (Seg (M + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (M + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :]
[:(TWOELEMENTSETS (Seg (M + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (M + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is non empty set
R is Relation-like Seg (len (Permutations (M + 2))) -defined Seg (len (Permutations (M + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (M + 2)
Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Relation-like TWOELEMENTSETS (Seg (M + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (M + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :]
{ b₁ where b₁ is Element of TWOELEMENTSETS (Seg (M + 2)) : ( b₁ in TWOELEMENTSETS (Seg (M + 2)) & not (Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . b₁ = (Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . b₁ ) } is set
x2 is set
i1 is Element of TWOELEMENTSETS (Seg (M + 2))
(Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . i1 is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
(Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . i1 is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
x2 is finite set
i1 is set
i2 is Element of TWOELEMENTSETS (Seg (M + 2))
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{y1,y2} is non empty finite V37() set
S . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 + 2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(Q1 + 2) - Q is V105() complex ext-real set
((Q1 + 2) - Q) + 1 is V105() complex ext-real set
(Q1 + 2) - P1 is V105() complex ext-real set
((Q1 + 2) - P1) + 1 is V105() complex ext-real set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S . P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
len (idseq (M + 2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom S is finite Element of bool (Seg (len (Permutations (M + 2))))
bool (Seg (len (Permutations (M + 2)))) is non empty finite V37() set
R . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . i2 is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1. the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is non zero Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the OneF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
(Q1 + 2) + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
((Q1 + 2) + 1) - Q is V105() complex ext-real set
((Q1 + 2) + 1) - P1 is V105() complex ext-real set
(Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) . i2 is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
- (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
card (TWOELEMENTSETS (Seg (M + 2))) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Relation-like [: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :]
[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
[:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is non empty set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (M + 2)))),(Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ))) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1. the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is non zero Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the OneF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (M + 2)))),(Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ))) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Relation-like [: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :]
[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
[:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr :] is non empty set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (M + 2)))),(Part_sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ))) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
1. the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is non zero Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the OneF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (M + 2)))),(Part_sgn (R, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr ))) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
- (sgn (S, the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr )) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital Fanoian doubleLoopStr
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
Rev (idseq m) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
M is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
K * M is Relation-like NAT -defined Seg m -defined the carrier of n * -valued the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[Q,S] is set
{Q,S} is non empty finite V37() set
{Q} is non empty trivial finite V37() 1 -element set
{{Q,S},{Q}} is non empty finite V37() without_zero V103() set
Indices (K * M) is set
dom (K * M) is finite Element of bool NAT
width (K * M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (K * M)) is finite width (K * M) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (K * M) ) } is set
[:(dom (K * M)),(Seg (width (K * M))):] is Relation-like finite set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m - x1 is V105() complex ext-real set
(m - x1) + 1 is V105() complex ext-real set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(idseq m) . x2 is set
m + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(m + 1) - S is V105() complex ext-real set
(m + 1) - Q is V105() complex ext-real set
((m + 1) - S) + S is V105() complex ext-real set
x2 + S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K * (x2,S) is Element of the carrier of n
len (idseq m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
dom (idseq m) is finite m -element Element of bool NAT
Seg (len (idseq m)) is finite len (idseq m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (idseq m) ) } is set
M . Q is set
(K * M) * (Q,S) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
Rev (idseq m) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
M is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
K * M is Relation-like NAT -defined Seg m -defined the carrier of n * -valued the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[Q,S] is set
{Q,S} is non empty finite V37() set
{Q} is non empty trivial finite V37() 1 -element set
{{Q,S},{Q}} is non empty finite V37() without_zero V103() set
Indices (K * M) is set
dom (K * M) is finite Element of bool NAT
width (K * M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (K * M)) is finite width (K * M) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (K * M) ) } is set
[:(dom (K * M)),(Seg (width (K * M))):] is Relation-like finite set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m - x1 is V105() complex ext-real set
(m - x1) + 1 is V105() complex ext-real set
(m + 1) - Q is V105() complex ext-real set
(m + 1) - S is V105() complex ext-real set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 + S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((m + 1) - S) + S is V105() complex ext-real set
len (idseq m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
(idseq m) . x2 is set
K * (x2,S) is Element of the carrier of n
dom (idseq m) is finite m -element Element of bool NAT
Seg (len (idseq m)) is finite len (idseq m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (idseq m) ) } is set
M . Q is set
(K * M) * (Q,S) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
Rev (idseq m) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det K is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product K)) is Element of the carrier of n
M is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
Path_matrix (M,K) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (Path_matrix (M,K)) is Element of the carrier of n
- (( the multF of n $$ (Path_matrix (M,K))),M) is Element of the carrier of n
Group_of_Perm m is non empty strict Group-like associative multMagma
the carrier of (Group_of_Perm m) is non empty set
S is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
K * S is Relation-like NAT -defined Seg m -defined the carrier of n * -valued the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
P is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
Path_matrix (P,(K * S)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (Path_matrix (M,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Path_matrix (P,(K * S))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
the multF of n $$ (Path_matrix (P,(K * S))) is Element of the carrier of n
rng P is finite set
rng (Rev (idseq m)) is finite set
(Path_matrix (M,K)) * M is Relation-like Seg (len (Permutations m)) -defined the carrier of n -valued Function-like finite Element of bool [:(Seg (len (Permutations m))), the carrier of n:]
[:(Seg (len (Permutations m))), the carrier of n:] is Relation-like set
bool [:(Seg (len (Permutations m))), the carrier of n:] is non empty set
dom S is finite Element of bool (Seg m)
bool (Seg m) is non empty finite V37() set
dom (Path_matrix (M,K)) is finite Element of bool NAT
i2 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom i2 is finite Element of bool NAT
dom M is finite Element of bool (Seg (len (Permutations m)))
bool (Seg (len (Permutations m))) is non empty finite V37() set
dom (Path_matrix (P,(K * S))) is finite Element of bool NAT
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m - y1 is V105() complex ext-real set
(m - y1) + 1 is V105() complex ext-real set
P . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Path_matrix (P,(K * S))) . y1 is set
(K * S) * (y1,y1) is Element of the carrier of n
len (idseq m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S . y1 is set
P . ((m - y1) + 1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
[y1,y1] is set
{y1,y1} is non empty finite V37() set
{y1} is non empty trivial finite V37() 1 -element set
{{y1,y1},{y1}} is non empty finite V37() without_zero V103() set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[y2,y1] is set
{y2,y1} is non empty finite V37() set
{y2} is non empty trivial finite V37() 1 -element set
{{y2,y1},{y2}} is non empty finite V37() without_zero V103() set
K * (y2,y1) is Element of the carrier of n
m - y2 is V105() complex ext-real set
(m - y2) + 1 is V105() complex ext-real set
S . y2 is set
P . ((m - y2) + 1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Path_matrix (M,K)) . y2 is set
i2 . y1 is set
len i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
the multF of n $$ (Path_matrix (P,(K * S))) is Element of the carrier of n
the multF of n $$ (Path_matrix (P,(K * S))) is Element of the carrier of n
the multF of n $$ (Path_matrix (P,(K * S))) is Element of the carrier of n
diagonal_of_Matrix (K * S) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n $$ (diagonal_of_Matrix (K * S)) is Element of the carrier of n
Det (K * S) is Element of the carrier of n
Path_product (K * S) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (K * S))) is Element of the carrier of n
- ((Det K),M) is Element of the carrier of n
- ( the multF of n $$ (Path_matrix (M,K))) is Element of the carrier of n
- (Det K) is Element of the carrier of n
(- (( the multF of n $$ (Path_matrix (M,K))),M)) + (- (Det K)) is Element of the carrier of n
the addF of n . ((- (( the multF of n $$ (Path_matrix (M,K))),M)),(- (Det K))) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular upper_triangular Matrix of m,m, the carrier of n
P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular upper_triangular Matrix of m,m, the carrier of n
R * P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix R is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
diagonal_of_Matrix P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
mlt ((diagonal_of_Matrix R),(diagonal_of_Matrix P)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(diagonal_of_Matrix R),(diagonal_of_Matrix P)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
m |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
m -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 = m } is set
(Seg m) --> (0. n) is Relation-like Seg m -defined Seg m -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. n)}:] is Relation-like finite set
bool [:(Seg m),{(0. n)}:] is non empty finite V37() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[x2,i1] is set
{x2,i1} is non empty finite V37() set
{x2} is non empty trivial finite V37() 1 -element set
{{x2,i1},{x2}} is non empty finite V37() without_zero V103() set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
Col (P,i1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len P -element FinSequence-like FinSubsequence-like Element of (len P) -tuples_on the carrier of n
(len P) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len P } is set
Line (R,x2) is Relation-like NAT -defined the carrier of n -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of n
(width R) -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 R } is set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M -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 = M } is set
y2 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
Q is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
mlt (y2,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,y2,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (mlt (y2,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m |-> (0. n)) . Q1 is set
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
Indices R is set
dom R is finite Element of bool NAT
Seg (width R) is finite width R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width R ) } is set
[:(dom R),(Seg (width R)):] is Relation-like finite set
(Line (R,x2)) . Q1 is set
R * (x2,Q1) is Element of the carrier of n
[x2,Q1] is set
{x2,Q1} is non empty finite V37() set
{{x2,Q1},{x2}} is non empty finite V37() without_zero V103() set
[Q1,i1] is set
{Q1,i1} is non empty finite V37() set
{Q1} is non empty trivial finite V37() 1 -element set
{{Q1,i1},{Q1}} is non empty finite V37() without_zero V103() set
dom (mlt (y2,Q)) is finite M -element Element of bool NAT
(Col (P,i1)) . Q1 is set
P * (Q1,i1) is Element of the carrier of n
(mlt (y2,Q)) . Q1 is set
(R * (x2,Q1)) * (P * (Q1,i1)) is Element of the carrier of n
the multF of n . ((R * (x2,Q1)),(P * (Q1,i1))) is Element of the carrier of n
(mlt (y2,Q)) . Q1 is set
(R * (x2,Q1)) * (P * (Q1,i1)) is Element of the carrier of n
the multF of n . ((R * (x2,Q1)),(P * (Q1,i1))) is Element of the carrier of n
len (m |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (R,x2)) "*" (Col (P,i1)) is Element of the carrier of n
mlt ((Line (R,x2)),(Col (P,i1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line (R,x2)),(Col (P,i1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line (R,x2)),(Col (P,i1)))) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the addF of n $$ (mlt ((Line (R,x2)),(Col (P,i1)))) is Element of the carrier of n
K * (x2,i1) is Element of the carrier of n
len (diagonal_of_Matrix P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (diagonal_of_Matrix R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
y1 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
mlt (i2,y1) is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,i2,y1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (mlt (i2,y1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(diagonal_of_Matrix K) . P1 is set
K * (P1,P1) is Element of the carrier of n
Col (P,P1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len P -element FinSequence-like FinSubsequence-like Element of (len P) -tuples_on the carrier of n
Line (R,P1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of n
Q2 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
i is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
mlt (Q2,i) is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,Q2,i) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
[P1,P1] is set
{P1,P1} is non empty finite V37() set
{P1} is non empty trivial finite V37() 1 -element set
{{P1,P1},{P1}} is non empty finite V37() without_zero V103() set
(Line (R,P1)) "*" (Col (P,P1)) is Element of the carrier of n
mlt ((Line (R,P1)),(Col (P,P1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line (R,P1)),(Col (P,P1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line (R,P1)),(Col (P,P1)))) is Element of the carrier of n
the addF of n $$ (mlt ((Line (R,P1)),(Col (P,P1)))) is Element of the carrier of n
(diagonal_of_Matrix P) . P1 is set
P * (P1,P1) is Element of the carrier of n
(diagonal_of_Matrix R) . P1 is set
R * (P1,P1) is Element of the carrier of n
len (mlt (Q2,i)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
(mlt (Q2,i)) . 1 is set
Q2i is Relation-like NAT -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of n:]
Q2i . 1 is Element of the carrier of n
Q2i . m is set
(mlt (i2,y1)) . P1 is set
dom (mlt (i2,y1)) is finite M -element Element of bool NAT
(R * (P1,P1)) * (P * (P1,P1)) is Element of the carrier of n
the multF of n . ((R * (P1,P1)),(P * (P1,P1))) is Element of the carrier of n
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(mlt (Q2,i)) . SQ2i is set
(Line (R,P1)) . SQ2i is set
R * (P1,SQ2i) is Element of the carrier of n
[P1,SQ2i] is set
{P1,SQ2i} is non empty finite V37() set
{{P1,SQ2i},{P1}} is non empty finite V37() without_zero V103() set
(Col (P,P1)) . SQ2i is set
P * (SQ2i,P1) is Element of the carrier of n
[SQ2i,P1] is set
{SQ2i,P1} is non empty finite V37() set
{SQ2i} is non empty trivial finite V37() 1 -element set
{{SQ2i,P1},{SQ2i}} is non empty finite V37() without_zero V103() set
(R * (P1,SQ2i)) * (P * (SQ2i,P1)) is Element of the carrier of n
the multF of n . ((R * (P1,SQ2i)),(P * (SQ2i,P1))) is Element of the carrier of n
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2i . SQ2i is set
SQ2i + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q2i . (SQ2i + 1) is set
Q2i . (SQ2i + 1) is Element of the carrier of n
(mlt (Q2,i)) . (SQ2i + 1) is set
Q2i . (SQ2i + 1) is Element of the carrier of n
(mlt (Q2,i)) . (SQ2i + 1) is set
the addF of n . ((Q2i . SQ2i),((mlt (Q2,i)) . (SQ2i + 1))) is set
(0. n) + (0. n) is Element of the carrier of n
the addF of n . ((0. n),(0. n)) is Element of the carrier of n
(0. n) + ((R * (P1,P1)) * (P * (P1,P1))) is Element of the carrier of n
the addF of n . ((0. n),((R * (P1,P1)) * (P * (P1,P1)))) is Element of the carrier of n
((R * (P1,P1)) * (P * (P1,P1))) + (0. n) is Element of the carrier of n
the addF of n . (((R * (P1,P1)) * (P * (P1,P1))),(0. n)) is Element of the carrier of n
Q2i . 0 is set
len (diagonal_of_Matrix K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular lower_triangular Matrix of m,m, the carrier of n
P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular lower_triangular Matrix of m,m, the carrier of n
R * P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix R is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
diagonal_of_Matrix P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
mlt ((diagonal_of_Matrix R),(diagonal_of_Matrix P)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(diagonal_of_Matrix R),(diagonal_of_Matrix P)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
R @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(P @) * (R @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
len ((P @) * (R @)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
len (K @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
P @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
R @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(P @) * (R @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
P @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
R @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(P @) * (R @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
P @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
R @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(P @) * (R @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
diagonal_of_Matrix (P @) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
diagonal_of_Matrix (R @) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (diagonal_of_Matrix P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (diagonal_of_Matrix R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M -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 = M } is set
diagonal_of_Matrix (K @) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
mlt ((diagonal_of_Matrix (P @)),(diagonal_of_Matrix (R @))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(diagonal_of_Matrix (P @)),(diagonal_of_Matrix (R @))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
mlt ((diagonal_of_Matrix P),(diagonal_of_Matrix (R @))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,(diagonal_of_Matrix P),(diagonal_of_Matrix (R @))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
i2 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
i1 is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
mlt (i2,i1) is Relation-like NAT -defined the carrier of n -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of n
K388( the carrier of n, the carrier of n, the carrier of n, the multF of n,i2,i1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
P is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of S,Q,m
Indices x1 is set
dom x1 is finite Element of bool NAT
width x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width x1) is finite width x1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x1 ) } is set
[:(dom x1),(Seg (width x1)):] is Relation-like finite set
x1 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of S,Q,m
Indices x1 is set
dom x1 is finite Element of bool NAT
width x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width x1) is finite width x1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x1 ) } is set
[:(dom x1),(Seg (width x1)):] is Relation-like finite set
x2 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,M,m
Indices x2 is set
dom x2 is finite Element of bool NAT
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width x2) is finite width x2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x2 ) } is set
[:(dom x2),(Seg (width x2)):] is Relation-like finite set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[i1,i2] is set
{i1,i2} is non empty finite V37() set
{i1} is non empty trivial finite V37() 1 -element set
{{i1,i2},{i1}} is non empty finite V37() without_zero V103() set
x2 * (i1,i2) is Element of m
R . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n * ((R . i1),(P . i2)) is Element of m
Q is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,M,m
Indices Q is set
dom Q is finite Element of bool NAT
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width Q) is finite width Q -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width Q ) } is set
[:(dom Q),(Seg (width Q)):] is Relation-like finite set
S is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,M,m
Indices S is set
dom S is finite Element of bool NAT
width S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width S) is finite width S -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width S ) } is set
[:(dom S),(Seg (width S)):] is Relation-like finite set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[x1,x2] is set
{x1,x2} is non empty finite V37() set
{x1} is non empty trivial finite V37() 1 -element set
{{x1,x2},{x1}} is non empty finite V37() without_zero V103() set
Q * (x1,x2) is Element of m
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n * ((R . i1),(P . i2)) is Element of m
S * (x1,x2) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
Q is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng Q is finite V212() V213() V214() V217() set
Q . M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng S is finite V212() V213() V214() V217() set
[:(rng Q),(rng S):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,P,n,K,Q,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
Indices (m,P,n,K,Q,S) is set
dom (m,P,n,K,Q,S) is finite Element of bool NAT
width (m,P,n,K,Q,S) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,P,n,K,Q,S)) is finite width (m,P,n,K,Q,S) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,P,n,K,Q,S) ) } is set
[:(dom (m,P,n,K,Q,S)),(Seg (width (m,P,n,K,Q,S))):] is Relation-like finite set
S . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[(Q . M),(S . R)] is set
{(Q . M),(S . R)} is non empty finite V37() set
{(Q . M)} is non empty trivial finite V37() 1 -element set
{{(Q . M),(S . R)},{(Q . M)}} is non empty finite V37() without_zero V103() set
dom S is finite K -element Element of bool NAT
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg (width (m,P,n,K,Q,S))):] is Relation-like finite set
dom Q is finite n -element Element of bool NAT
dom Q is finite n -element Element of bool NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len P) is finite len P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len P ) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg (width (m,P,n,K,Q,S))):] is Relation-like finite set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices M is set
dom M is finite Element of bool NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
M @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
[:(rng R),(rng P):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,M,n,K,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,M,n,K,R,P) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
(m,(M @),K,n,P,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,(M @),K,n,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,(M @),K,n,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (m,(M @),K,n,P,R) is set
dom (m,(M @),K,n,P,R) is finite Element of bool NAT
Seg (width (m,(M @),K,n,P,R)) is finite width (m,(M @),K,n,P,R) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,(M @),K,n,P,R) ) } is set
[:(dom (m,(M @),K,n,P,R)),(Seg (width (m,(M @),K,n,P,R))):] is Relation-like finite set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg K),(Seg n):] is Relation-like finite set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[x2,i1] is set
{x2,i1} is non empty finite V37() set
{x2} is non empty trivial finite V37() 1 -element set
{{x2,i1},{x2}} is non empty finite V37() without_zero V103() set
Indices ((m,M,n,K,R,P) @) is set
dom ((m,M,n,K,R,P) @) is finite Element of bool NAT
width ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ((m,M,n,K,R,P) @)) is finite width ((m,M,n,K,R,P) @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ((m,M,n,K,R,P) @) ) } is set
[:(dom ((m,M,n,K,R,P) @)),(Seg (width ((m,M,n,K,R,P) @))):] is Relation-like finite set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[y1,i2] is Element of [:NAT,NAT:]
{y1,i2} is non empty finite V37() set
{y1} is non empty trivial finite V37() 1 -element set
{{y1,i2},{y1}} is non empty finite V37() without_zero V103() set
Indices (m,M,n,K,R,P) is set
dom (m,M,n,K,R,P) is finite Element of bool NAT
Seg (width (m,M,n,K,R,P)) is finite width (m,M,n,K,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,P) ) } is set
[:(dom (m,M,n,K,R,P)),(Seg (width (m,M,n,K,R,P))):] is Relation-like finite set
((m,M,n,K,R,P) @) * (i2,y1) is Element of m
(m,M,n,K,R,P) * (y1,i2) is Element of m
[:(Seg n),(Seg K):] is Relation-like finite set
[i2,y1] is Element of [:NAT,NAT:]
{i2,y1} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,y1},{i2}} is non empty finite V37() without_zero V103() set
R . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((R . i1),(P . x2)) is Element of m
[(R . i1),(P . x2)] is set
{(R . i1),(P . x2)} is non empty finite V37() set
{(R . i1)} is non empty trivial finite V37() 1 -element set
{{(R . i1),(P . x2)},{(R . i1)}} is non empty finite V37() without_zero V103() set
(M @) * ((P . x2),(R . i1)) is Element of m
((m,M,n,K,R,P) @) * (x2,i1) is Element of m
(m,(M @),K,n,P,R) * (x2,i1) is Element of m
len (m,(M @),K,n,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices M is set
dom M is finite Element of bool NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
M @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
P is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
[:(rng R),(rng P):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,M,K,n,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
(m,(M @),n,K,P,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,(M @),n,K,P,R) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len (m,(M @),n,K,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,(M @),n,K,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,(M @),n,K,P,R) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,K,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,K,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,K,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,M,K,n,R,P) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
((m,M,K,n,R,P) @) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1,1,m
n * (1,1) is Element of m
<*(n * (1,1))*> is Relation-like NAT -defined m -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on m
1 -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = 1 } is set
[1,(n * (1,1))] is set
{1,(n * (1,1))} is non empty finite set
{{1,(n * (1,1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (1,1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (1,1))*>*> is Relation-like NAT -defined m * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(n * (1,1))*>,m
len <*(n * (1,1))*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(n * (1,1))*>] is set
{1,<*(n * (1,1))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (1,1))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (1,1))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex 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 V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
[:(Seg 1),(Seg 1):] is Relation-like non empty finite set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
{1} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1,1,m
K * (M,R) is Element of m
n * (M,R) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
R . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
(m,M,n,K,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
P . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((R . 1),(P . 1)) is Element of m
<*(M * ((R . 1),(P . 1)))*> is Relation-like NAT -defined m -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on m
1 -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = 1 } is set
[1,(M * ((R . 1),(P . 1)))] is set
{1,(M * ((R . 1),(P . 1)))} is non empty finite set
{{1,(M * ((R . 1),(P . 1)))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * ((R . 1),(P . 1)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(M * ((R . 1),(P . 1)))*>*> is Relation-like NAT -defined m * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(M * ((R . 1),(P . 1)))*>,m
len <*(M * ((R . 1),(P . 1)))*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(M * ((R . 1),(P . 1)))*>] is set
{1,<*(M * ((R . 1),(P . 1)))*>} is non empty finite V37() without_zero V103() set
{{1,<*(M * ((R . 1),(P . 1)))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(M * ((R . 1),(P . 1)))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Indices (m,M,n,K,R,P) is set
dom (m,M,n,K,R,P) is finite Element of bool NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,M,n,K,R,P)) is finite width (m,M,n,K,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,P) ) } is set
[:(dom (m,M,n,K,R,P)),(Seg (width (m,M,n,K,R,P))):] is Relation-like finite set
[:(Seg 1),(Seg 1):] is Relation-like non empty finite set
[1,1] is Element of [:NAT,NAT:]
(m,M,n,K,R,P) * (1,1) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2,m
n * (1,1) is Element of m
n * (1,2) is Element of m
n * (2,1) is Element of m
n * (2,2) is Element of m
((n * (1,1)),(n * (1,2))) ][ ((n * (2,1)),(n * (2,2))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2,m
<*(n * (1,1)),(n * (1,2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (1,1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (1,1))] is set
{1,(n * (1,1))} is non empty finite set
{{1,(n * (1,1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (1,1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (1,2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (1,2))] is set
{1,(n * (1,2))} is non empty finite set
{{1,(n * (1,2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (1,2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (1,1))*>,<*(n * (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 V105() complex ext-real positive non negative Element of NAT
<*(n * (2,1)),(n * (2,2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (2,1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (2,1))] is set
{1,(n * (2,1))} is non empty finite set
{{1,(n * (2,1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (2,1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (2,2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (2,2))] is set
{1,(n * (2,2))} is non empty finite set
{{1,(n * (2,2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (2,2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (2,1))*>,<*(n * (2,2))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (1,1)),(n * (1,2))*>,<*(n * (2,1)),(n * (2,2))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (1,1)),(n * (1,2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (1,1)),(n * (1,2))*>] is set
{1,<*(n * (1,1)),(n * (1,2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (1,1)),(n * (1,2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (1,1)),(n * (1,2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (2,1)),(n * (2,2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (2,1)),(n * (2,2))*>] is set
{1,<*(n * (2,1)),(n * (2,2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (2,1)),(n * (2,2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (2,1)),(n * (2,2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (1,1)),(n * (1,2))*>*>,<*<*(n * (2,1)),(n * (2,2))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex 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 V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
[:(Seg 2),(Seg 2):] is Relation-like non empty finite set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
{1,2} is non empty finite V37() without_zero V103() Element of bool NAT
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2,m
K * (M,R) is Element of m
n * (M,R) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
R . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R . 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
(m,M,n,K,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
P . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((R . 1),(P . 1)) is Element of m
P . 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((R . 1),(P . 2)) is Element of m
M * ((R . 2),(P . 1)) is Element of m
M * ((R . 2),(P . 2)) is Element of m
((M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))) ][ ((M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2,m
<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(M * ((R . 1),(P . 1)))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(M * ((R . 1),(P . 1)))] is set
{1,(M * ((R . 1),(P . 1)))} is non empty finite set
{{1,(M * ((R . 1),(P . 1)))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * ((R . 1),(P . 1)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(M * ((R . 1),(P . 2)))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(M * ((R . 1),(P . 2)))] is set
{1,(M * ((R . 1),(P . 2)))} is non empty finite set
{{1,(M * ((R . 1),(P . 2)))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * ((R . 1),(P . 2)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(M * ((R . 1),(P . 1)))*>,<*(M * ((R . 1),(P . 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 V105() complex ext-real positive non negative Element of NAT
<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(M * ((R . 2),(P . 1)))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(M * ((R . 2),(P . 1)))] is set
{1,(M * ((R . 2),(P . 1)))} is non empty finite set
{{1,(M * ((R . 2),(P . 1)))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * ((R . 2),(P . 1)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(M * ((R . 2),(P . 2)))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(M * ((R . 2),(P . 2)))] is set
{1,(M * ((R . 2),(P . 2)))} is non empty finite set
{{1,(M * ((R . 2),(P . 2)))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * ((R . 2),(P . 2)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(M * ((R . 2),(P . 1)))*>,<*(M * ((R . 2),(P . 2)))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>,<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>] is set
{1,<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>} is non empty finite V37() without_zero V103() set
{{1,<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>] is set
{1,<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>} is non empty finite V37() without_zero V103() set
{{1,<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(M * ((R . 1),(P . 1))),(M * ((R . 1),(P . 2)))*>*>,<*<*(M * ((R . 2),(P . 1))),(M * ((R . 2),(P . 2)))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Indices (m,M,n,K,R,P) is set
dom (m,M,n,K,R,P) is finite Element of bool NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,M,n,K,R,P)) is finite width (m,M,n,K,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,P) ) } is set
[:(dom (m,M,n,K,R,P)),(Seg (width (m,M,n,K,R,P))):] is Relation-like finite set
[:(Seg 2),(Seg 2):] is Relation-like non empty finite set
[2,2] is Element of [:NAT,NAT:]
{2,2} is non empty finite V37() without_zero V103() set
{2} is non empty trivial finite V37() 1 -element without_zero V103() set
{{2,2},{2}} is non empty finite V37() without_zero V103() set
(m,M,n,K,R,P) * (2,2) is Element of m
[1,1] is Element of [:NAT,NAT:]
(m,M,n,K,R,P) * (1,1) is Element of m
[2,1] is Element of [:NAT,NAT:]
{2,1} is non empty finite V37() without_zero V103() set
{{2,1},{2}} is non empty finite V37() without_zero V103() set
(m,M,n,K,R,P) * (2,1) is Element of m
[1,2] is Element of [:NAT,NAT:]
{{1,2},{1}} is non empty finite V37() without_zero V103() set
(m,M,n,K,R,P) * (1,2) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
R is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width R) is finite width R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width R ) } is set
P is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
P . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,(P . K)) is Relation-like NAT -defined m -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on m
(width R) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width R } is set
Q is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng Q is finite V212() V213() V214() V217() set
(m,R,M,n,P,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,n,m
Line ((m,R,M,n,P,Q),K) is Relation-like NAT -defined m -valued Function-like finite width (m,R,M,n,P,Q) -element FinSequence-like FinSubsequence-like Element of (width (m,R,M,n,P,Q)) -tuples_on m
width (m,R,M,n,P,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,R,M,n,P,Q)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (m,R,M,n,P,Q) } is set
(Line (R,(P . K))) * Q is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
len (Line ((m,R,M,n,P,Q),K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line ((m,R,M,n,P,Q),K)) is finite width (m,R,M,n,P,Q) -element Element of bool NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom Q is finite n -element Element of bool NAT
len (Line (R,(P . K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (R,(P . K))) is finite width R -element Element of bool NAT
dom ((Line (R,(P . K))) * Q) is finite Element of bool NAT
i1 is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line ((m,R,M,n,P,Q),K)) . i2 is set
(m,R,M,n,P,Q) * (K,i2) is Element of m
[K,i2] is set
{K,i2} is non empty finite V37() set
{K} is non empty trivial finite V37() 1 -element set
{{K,i2},{K}} is non empty finite V37() without_zero V103() set
Seg (width (m,R,M,n,P,Q)) is finite width (m,R,M,n,P,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,R,M,n,P,Q) ) } is set
[:(Seg M),(Seg (width (m,R,M,n,P,Q))):] is Relation-like finite set
Indices (m,R,M,n,P,Q) is set
dom (m,R,M,n,P,Q) is finite Element of bool NAT
[:(dom (m,R,M,n,P,Q)),(Seg (width (m,R,M,n,P,Q))):] is Relation-like finite set
Q . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Line (R,(P . K))) . (Q . i2) is set
R * ((P . K),(Q . i2)) is Element of m
((Line (R,(P . K))) * Q) . i2 is set
(Line ((m,R,M,n,P,Q),K)) . i1 is set
((Line (R,(P . K))) * Q) . i1 is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Q is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
Q . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
(m,P,M,n,Q,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,n,m
Line ((m,P,M,n,Q,S),K) is Relation-like NAT -defined m -valued Function-like finite width (m,P,M,n,Q,S) -element FinSequence-like FinSubsequence-like Element of (width (m,P,M,n,Q,S)) -tuples_on m
width (m,P,M,n,Q,S) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,P,M,n,Q,S)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (m,P,M,n,Q,S) } is set
Line ((m,P,M,n,Q,S),R) is Relation-like NAT -defined m -valued Function-like finite width (m,P,M,n,Q,S) -element FinSequence-like FinSubsequence-like Element of (width (m,P,M,n,Q,S)) -tuples_on m
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg (width (m,P,M,n,Q,S)) is finite width (m,P,M,n,Q,S) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,P,M,n,Q,S) ) } is set
[K,i2] is set
{K,i2} is non empty finite V37() set
{K} is non empty trivial finite V37() 1 -element set
{{K,i2},{K}} is non empty finite V37() without_zero V103() set
[:(Seg M),(Seg (width (m,P,M,n,Q,S))):] is Relation-like finite set
Indices (m,P,M,n,Q,S) is set
dom (m,P,M,n,Q,S) is finite Element of bool NAT
[:(dom (m,P,M,n,Q,S)),(Seg (width (m,P,M,n,Q,S))):] is Relation-like finite set
(m,P,M,n,Q,S) * (K,i2) is Element of m
S . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((Q . K),(S . i2)) is Element of m
[R,i2] is set
{R,i2} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,i2},{R}} is non empty finite V37() without_zero V103() set
(m,P,M,n,Q,S) * (R,i2) is Element of m
P * ((Q . R),(S . i2)) is Element of m
(Line ((m,P,M,n,Q,S),K)) . i2 is set
(Line ((m,P,M,n,Q,S),R)) . i2 is set
len (Line ((m,P,M,n,Q,S),R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Line ((m,P,M,n,Q,S),K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
R . m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
Q is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
( the carrier of M,Q,n,n,R,P) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,n, the carrier of M
Det ( the carrier of M,Q,n,n,R,P) is Element of the carrier of M
Permutations n is non empty permutational set
the addF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
FinOmega (Permutations n) is Element of Fin (Permutations n)
Fin (Permutations n) is preBoolean set
Path_product ( the carrier of M,Q,n,n,R,P) is Relation-like Permutations n -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of M:]
[:(Permutations n), the carrier of M:] is Relation-like non empty set
bool [:(Permutations n), the carrier of M:] is non empty set
the addF of M $$ ((FinOmega (Permutations n)),(Path_product ( the carrier of M,Q,n,n,R,P))) is Element of the carrier of M
Line (( the carrier of M,Q,n,n,R,P),m) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,Q,n,n,R,P) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,Q,n,n,R,P)) -tuples_on the carrier of M
width ( the carrier of M,Q,n,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of M,Q,n,n,R,P)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width ( the carrier of M,Q,n,n,R,P) } is set
Line (( the carrier of M,Q,n,n,R,P),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,Q,n,n,R,P) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,Q,n,n,R,P)) -tuples_on the carrier of M
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
( the carrier of n,R,m,m,K,M) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,R,m,m,K,M) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product ( the carrier of n,R,m,m,K,M) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,R,m,m,K,M))) is Element of the carrier of n
dom K is finite m -element Element of bool NAT
P is set
Q is set
K . P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
R is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
P is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
Q is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
(m,R,n,M,P,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,M,m
Col ((m,R,n,M,P,Q),K) is Relation-like NAT -defined m -valued Function-like finite len (m,R,n,M,P,Q) -element FinSequence-like FinSubsequence-like Element of (len (m,R,n,M,P,Q)) -tuples_on m
len (m,R,n,M,P,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (m,R,n,M,P,Q)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len (m,R,n,M,P,Q) } is set
Q . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Col (R,(Q . K)) is Relation-like NAT -defined m -valued Function-like finite len R -element FinSequence-like FinSubsequence-like Element of (len R) -tuples_on m
(len R) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len R } is set
(Col (R,(Q . K))) * P is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
len (Col (R,(Q . K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Col (R,(Q . K))) is finite len R -element Element of bool NAT
dom ((Col (R,(Q . K))) * P) is finite Element of bool NAT
dom P is finite n -element Element of bool NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom (m,R,n,M,P,Q) is finite Element of bool NAT
len (Col ((m,R,n,M,P,Q),K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Col ((m,R,n,M,P,Q),K)) is finite len (m,R,n,M,P,Q) -element Element of bool NAT
dom R is finite Element of bool NAT
i1 is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Col ((m,R,n,M,P,Q),K)) . i2 is set
(m,R,n,M,P,Q) * (i2,K) is Element of m
P . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Col (R,(Q . K))) . (P . i2) is set
R * ((P . i2),(Q . K)) is Element of m
[i2,K] is set
{i2,K} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,K},{i2}} is non empty finite V37() without_zero V103() set
[:(Seg n),(Seg M):] is Relation-like finite set
Indices (m,R,n,M,P,Q) is set
width (m,R,n,M,P,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,R,n,M,P,Q)) is finite width (m,R,n,M,P,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,R,n,M,P,Q) ) } is set
[:(dom (m,R,n,M,P,Q)),(Seg (width (m,R,n,M,P,Q))):] is Relation-like finite set
((Col (R,(Q . K))) * P) . i2 is set
(Col ((m,R,n,M,P,Q),K)) . i1 is set
((Col (R,(Q . K))) * P) . i1 is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Q is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
S is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
S . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m,P,n,M,Q,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,M,m
Col ((m,P,n,M,Q,S),K) is Relation-like NAT -defined m -valued Function-like finite len (m,P,n,M,Q,S) -element FinSequence-like FinSubsequence-like Element of (len (m,P,n,M,Q,S)) -tuples_on m
len (m,P,n,M,Q,S) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (m,P,n,M,Q,S)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len (m,P,n,M,Q,S) } is set
Col ((m,P,n,M,Q,S),R) is Relation-like NAT -defined m -valued Function-like finite len (m,P,n,M,Q,S) -element FinSequence-like FinSubsequence-like Element of (len (m,P,n,M,Q,S)) -tuples_on m
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg (len (m,P,n,M,Q,S)) is finite len (m,P,n,M,Q,S) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (m,P,n,M,Q,S) ) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
width (m,P,n,M,Q,S) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[i2,R] is set
{i2,R} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,R},{i2}} is non empty finite V37() without_zero V103() set
[:(Seg n),(Seg M):] is Relation-like finite set
Indices (m,P,n,M,Q,S) is set
dom (m,P,n,M,Q,S) is finite Element of bool NAT
Seg (width (m,P,n,M,Q,S)) is finite width (m,P,n,M,Q,S) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,P,n,M,Q,S) ) } is set
[:(dom (m,P,n,M,Q,S)),(Seg (width (m,P,n,M,Q,S))):] is Relation-like finite set
(m,P,n,M,Q,S) * (i2,R) is Element of m
Q . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((Q . i2),(S . R)) is Element of m
[i2,K] is set
{i2,K} is non empty finite V37() set
{{i2,K},{i2}} is non empty finite V37() without_zero V103() set
(m,P,n,M,Q,S) * (i2,K) is Element of m
P * ((Q . i2),(S . K)) is Element of m
(Col ((m,P,n,M,Q,S),K)) . i2 is set
(Col ((m,P,n,M,Q,S),R)) . i2 is set
len (Col ((m,P,n,M,Q,S),R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Col ((m,P,n,M,Q,S),K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
R . m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R . K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
Q is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
( the carrier of M,Q,n,n,P,R) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,n, the carrier of M
Det ( the carrier of M,Q,n,n,P,R) is Element of the carrier of M
Permutations n is non empty permutational set
the addF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
FinOmega (Permutations n) is Element of Fin (Permutations n)
Fin (Permutations n) is preBoolean set
Path_product ( the carrier of M,Q,n,n,P,R) is Relation-like Permutations n -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of M:]
[:(Permutations n), the carrier of M:] is Relation-like non empty set
bool [:(Permutations n), the carrier of M:] is non empty set
the addF of M $$ ((FinOmega (Permutations n)),(Path_product ( the carrier of M,Q,n,n,P,R))) is Element of the carrier of M
width ( the carrier of M,Q,n,n,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col (( the carrier of M,Q,n,n,P,R),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite len ( the carrier of M,Q,n,n,P,R) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of M,Q,n,n,P,R)) -tuples_on the carrier of M
len ( the carrier of M,Q,n,n,P,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len ( the carrier of M,Q,n,n,P,R)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = len ( the carrier of M,Q,n,n,P,R) } is set
( the carrier of M,Q,n,n,P,R) @ is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,n, the carrier of M
Line ((( the carrier of M,Q,n,n,P,R) @),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (( the carrier of M,Q,n,n,P,R) @) -element FinSequence-like FinSubsequence-like Element of (width (( the carrier of M,Q,n,n,P,R) @)) -tuples_on the carrier of M
width (( the carrier of M,Q,n,n,P,R) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (( the carrier of M,Q,n,n,P,R) @)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width (( the carrier of M,Q,n,n,P,R) @) } is set
Col (( the carrier of M,Q,n,n,P,R),m) is Relation-like NAT -defined the carrier of M -valued Function-like finite len ( the carrier of M,Q,n,n,P,R) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of M,Q,n,n,P,R)) -tuples_on the carrier of M
Line ((( the carrier of M,Q,n,n,P,R) @),m) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (( the carrier of M,Q,n,n,P,R) @) -element FinSequence-like FinSubsequence-like Element of (width (( the carrier of M,Q,n,n,P,R) @)) -tuples_on the carrier of M
Det (( the carrier of M,Q,n,n,P,R) @) is Element of the carrier of M
Path_product (( the carrier of M,Q,n,n,P,R) @) is Relation-like Permutations n -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of M:]
the addF of M $$ ((FinOmega (Permutations n)),(Path_product (( the carrier of M,Q,n,n,P,R) @))) is Element of the carrier of M
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
( the carrier of n,R,m,m,M,K) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,R,m,m,M,K) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product ( the carrier of n,R,m,m,M,K) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,R,m,m,M,K))) is Element of the carrier of n
dom K is finite m -element Element of bool NAT
P is set
Q is set
K . P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
n is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng n is finite V212() V213() V214() V217() set
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng K is finite V212() V213() V214() V217() set
n " is Relation-like Function-like set
M is Relation-like Function-like set
dom M is set
dom n is finite m -element Element of bool NAT
rng M is set
K (#) M is Relation-like NAT -defined Function-like finite set
rng (K (#) M) is finite set
dom K is finite m -element Element of bool NAT
dom (K (#) M) is finite Element of bool NAT
R is Relation-like Seg m -defined Seg m -valued Function-like total quasi_total finite Element of bool [:(Seg m),(Seg m):]
P is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
n * P is Relation-like Seg m -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg m),NAT:]
[:(Seg m),NAT:] is Relation-like set
bool [:(Seg m),NAT:] is non empty set
M (#) n is Relation-like NAT -valued RAT -valued Function-like V202() V203() V204() V205() set
K (#) (M (#) n) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() set
id (rng K) is Relation-like rng K -defined rng K -valued Function-like one-to-one total quasi_total finite V202() V203() V204() V205() V206() V208() Element of bool [:(rng K),(rng K):]
[:(rng K),(rng K):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
bool [:(rng K),(rng K):] is non empty finite V37() set
(id (rng K)) * K is Relation-like NAT -defined RAT -valued rng K -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,(rng K):]
[:NAT,(rng K):] is Relation-like RAT -valued V202() V203() V204() V205() set
bool [:NAT,(rng K):] is non empty set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is non empty finite V37() set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
Q is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
(m,M,K,n,R,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
(m,M,K,n,P,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
S is Relation-like Seg K -defined Seg K -valued Function-like total quasi_total finite Element of bool [:(Seg K),(Seg K):]
P * S is Relation-like Seg K -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg K),NAT:]
[:(Seg K),NAT:] is Relation-like set
bool [:(Seg K),NAT:] is non empty set
(m,M,K,n,P,Q) * S is Relation-like NAT -defined Seg K -defined m * -valued m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[i2,y1] is set
{i2,y1} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,y1},{i2}} is non empty finite V37() without_zero V103() set
Indices (m,M,K,n,R,Q) is set
dom (m,M,K,n,R,Q) is finite Element of bool NAT
width (m,M,K,n,R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,M,K,n,R,Q)) is finite width (m,M,K,n,R,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,K,n,R,Q) ) } is set
[:(dom (m,M,K,n,R,Q)),(Seg (width (m,M,K,n,R,Q))):] is Relation-like finite set
[:(Seg K),(Seg (width (m,M,K,n,R,Q))):] is Relation-like finite set
Indices (m,M,K,n,P,Q) is set
dom (m,M,K,n,P,Q) is finite Element of bool NAT
width (m,M,K,n,P,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,M,K,n,P,Q)) is finite width (m,M,K,n,P,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,K,n,P,Q) ) } is set
[:(dom (m,M,K,n,P,Q)),(Seg (width (m,M,K,n,P,Q))):] is Relation-like finite set
S . i2 is set
((m,M,K,n,P,Q) * S) * (i2,y1) is Element of m
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[y2,y1] is set
{y2,y1} is non empty finite V37() set
{y2} is non empty trivial finite V37() 1 -element set
{{y2,y1},{y2}} is non empty finite V37() without_zero V103() set
(m,M,K,n,P,Q) * (y2,y1) is Element of m
dom R is finite K -element Element of bool NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P . (S . i2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m,M,K,n,R,Q) * (i2,y1) is Element of m
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((P . Q1),(Q . P1)) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is non empty finite V37() set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
(m,M,n,K,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,M,n,K,R,P) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Q is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
(m,M,n,K,R,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,M,n,K,R,Q) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
S is Relation-like Seg K -defined Seg K -valued Function-like total quasi_total finite Element of bool [:(Seg K),(Seg K):]
Q * S is Relation-like Seg K -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg K),NAT:]
[:(Seg K),NAT:] is Relation-like set
bool [:(Seg K),NAT:] is non empty set
((m,M,n,K,R,Q) @) * S is Relation-like Seg K -defined m * -valued Function-like finite Function-yielding V147() Element of bool [:(Seg K),(m *):]
[:(Seg K),(m *):] is Relation-like set
bool [:(Seg K),(m *):] is non empty set
len (m,M,n,K,R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,Q) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width ((m,M,n,K,R,P) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,n,K,R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width ((m,M,n,K,R,Q) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((m,M,n,K,R,Q) @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
i1 * S is Relation-like NAT -defined Seg K -defined m * -valued m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[y2,Q] is set
{y2,Q} is non empty finite V37() set
{y2} is non empty trivial finite V37() 1 -element set
{{y2,Q},{y2}} is non empty finite V37() without_zero V103() set
i2 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n,m
Indices i2 is set
dom i2 is finite Element of bool NAT
width i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width i2) is finite width i2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width i2 ) } is set
[:(dom i2),(Seg (width i2)):] is Relation-like finite set
[Q,y2] is set
{Q,y2} is non empty finite V37() set
{Q} is non empty trivial finite V37() 1 -element set
{{Q,y2},{Q}} is non empty finite V37() without_zero V103() set
Indices (m,M,n,K,R,P) is set
dom (m,M,n,K,R,P) is finite Element of bool NAT
Seg (width (m,M,n,K,R,P)) is finite width (m,M,n,K,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,P) ) } is set
[:(dom (m,M,n,K,R,P)),(Seg (width (m,M,n,K,R,P))):] is Relation-like finite set
i2 * (y2,Q) is Element of m
(m,M,n,K,R,P) * (Q,y2) is Element of m
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg K),(Seg n):] is Relation-like finite set
Indices i1 is set
dom i1 is finite Element of bool NAT
width i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width i1) is finite width i1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width i1 ) } is set
[:(dom i1),(Seg (width i1)):] is Relation-like finite set
S . y2 is set
(i1 * S) * (y2,Q) is Element of m
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[P1,Q] is set
{P1,Q} is non empty finite V37() set
{P1} is non empty trivial finite V37() 1 -element set
{{P1,Q},{P1}} is non empty finite V37() without_zero V103() set
i1 * (P1,Q) is Element of m
dom P is finite K -element Element of bool NAT
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q . (S . y2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,M,n,K,R,P) * (P2,Q1) is Element of m
R . P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * ((R . P2),(Q . Q2)) is Element of m
[Q,P1] is set
{Q,P1} is non empty finite V37() set
{{Q,P1},{Q}} is non empty finite V37() without_zero V103() set
Indices (m,M,n,K,R,Q) is set
dom (m,M,n,K,R,Q) is finite Element of bool NAT
Seg (width (m,M,n,K,R,Q)) is finite width (m,M,n,K,R,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,Q) ) } is set
[:(dom (m,M,n,K,R,Q)),(Seg (width (m,M,n,K,R,Q))):] is Relation-like finite set
(m,M,n,K,R,Q) * (Q,P1) is Element of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
Permutations m is non empty permutational set
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
R is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
( the carrier of n,P,m,m,K,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,P,m,m,K,R) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product ( the carrier of n,P,m,m,K,R) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,P,m,m,K,R))) is Element of the carrier of n
( the carrier of n,P,m,m,M,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,P,m,m,M,R) is Element of the carrier of n
Path_product ( the carrier of n,P,m,m,M,R) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,P,m,m,M,R))) is Element of the carrier of n
( the carrier of n,P,m,m,R,K) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,P,m,m,R,K) is Element of the carrier of n
Path_product ( the carrier of n,P,m,m,R,K) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,P,m,m,R,K))) is Element of the carrier of n
( the carrier of n,P,m,m,R,M) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,P,m,m,R,M) is Element of the carrier of n
Path_product ( the carrier of n,P,m,m,R,M) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,P,m,m,R,M))) is Element of the carrier of n
Q is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
M * Q is Relation-like Seg (len (Permutations m)) -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg (len (Permutations m))),NAT:]
[:(Seg (len (Permutations m))),NAT:] is Relation-like set
bool [:(Seg (len (Permutations m))),NAT:] is non empty set
- ((Det ( the carrier of n,P,m,m,M,R)),Q) is Element of the carrier of n
- ((Det ( the carrier of n,P,m,m,R,M)),Q) is Element of the carrier of n
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
S is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
( the carrier of n,P,m,m,M,R) * S is Relation-like NAT -defined Seg m -defined the carrier of n * -valued the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
( the carrier of n,P,m,m,R,K) @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det (( the carrier of n,P,m,m,R,K) @) is Element of the carrier of n
Path_product (( the carrier of n,P,m,m,R,K) @) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (( the carrier of n,P,m,m,R,K) @))) is Element of the carrier of n
( the carrier of n,P,m,m,R,M) @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(( the carrier of n,P,m,m,R,M) @) * S is Relation-like NAT -defined Seg m -defined the carrier of n * -valued the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ((( the carrier of n,P,m,m,R,M) @) * S) is Element of the carrier of n
Path_product ((( the carrier of n,P,m,m,R,M) @) * S) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ((( the carrier of n,P,m,m,R,M) @) * S))) is Element of the carrier of n
Det (( the carrier of n,P,m,m,R,M) @) is Element of the carrier of n
Path_product (( the carrier of n,P,m,m,R,M) @) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (( the carrier of n,P,m,m,R,M) @))) is Element of the carrier of n
- ((Det (( the carrier of n,P,m,m,R,M) @)),Q) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng n is finite V212() V213() V214() V217() set
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng K is finite V212() V213() V214() V217() set
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular FinSequence of the carrier of n *
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
P is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
Q is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng Q is finite V212() V213() V214() V217() set
( the carrier of n,K,m,m,M,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,K,m,m,M,R) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product ( the carrier of n,K,m,m,M,R) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,K,m,m,M,R))) is Element of the carrier of n
( the carrier of n,K,m,m,P,Q) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,K,m,m,P,Q) is Element of the carrier of n
Path_product ( the carrier of n,K,m,m,P,Q) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,K,m,m,P,Q))) is Element of the carrier of n
- (Det ( the carrier of n,K,m,m,P,Q)) is Element of the carrier of n
( the carrier of n,K,m,m,M,Q) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
i1 is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
Q * i1 is Relation-like Seg m -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg m),NAT:]
[:(Seg m),NAT:] is Relation-like set
bool [:(Seg m),NAT:] is non empty set
i2 is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
P * i2 is Relation-like Seg m -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg m),NAT:]
len (Permutations m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Permutations m)) is finite len (Permutations m) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations m) ) } is set
y2 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
Det ( the carrier of n,K,m,m,M,Q) is Element of the carrier of n
Path_product ( the carrier of n,K,m,m,M,Q) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,K,m,m,M,Q))) is Element of the carrier of n
- ((Det ( the carrier of n,K,m,m,M,Q)),y2) is Element of the carrier of n
y1 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
- ((Det ( the carrier of n,K,m,m,P,Q)),y1) is Element of the carrier of n
y2 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
Det ( the carrier of n,K,m,m,M,Q) is Element of the carrier of n
Path_product ( the carrier of n,K,m,m,M,Q) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,K,m,m,M,Q))) is Element of the carrier of n
y1 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
- ((Det ( the carrier of n,K,m,m,P,Q)),y1) is Element of the carrier of n
- ((Det ( the carrier of n,K,m,m,M,Q)),y2) is Element of the carrier of n
- (Det ( the carrier of n,K,m,m,M,Q)) is Element of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
- (- (Det ( the carrier of n,K,m,m,P,Q))) is Element of the carrier of n
(0. n) + (- (- (Det ( the carrier of n,K,m,m,P,Q)))) is Element of the carrier of n
the addF of n . ((0. n),(- (- (Det ( the carrier of n,K,m,m,P,Q))))) is Element of the carrier of n
(0. n) - (- (Det ( the carrier of n,K,m,m,P,Q))) is Element of the carrier of n
(0. n) + (- (- (Det ( the carrier of n,K,m,m,P,Q)))) is Element of the carrier of n
(Det ( the carrier of n,K,m,m,M,R)) + (- (Det ( the carrier of n,K,m,m,P,Q))) is Element of the carrier of n
the addF of n . ((Det ( the carrier of n,K,m,m,M,R)),(- (Det ( the carrier of n,K,m,m,P,Q)))) is Element of the carrier of n
y2 is Relation-like Seg (len (Permutations m)) -defined Seg (len (Permutations m)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = R } is set
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
Q is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
len Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
x1 is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is Relation-like NAT -defined NAT -valued Function-like finite R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of R -tuples_on NAT
Q * x2 is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
rng x2 is finite V212() V213() V214() V217() set
[:(rng x1),(rng x2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,P,M,R,x1,x2) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,R,m
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{i1} is non empty trivial finite V37() 1 -element set
ReplaceLine ((m,P,M,R,x1,x2),i1,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,R,m
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{i2} is non empty trivial finite V37() 1 -element set
x1 " {i2} is finite set
ReplaceLine (P,i2,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,R,m
dom x1 is finite M -element Element of bool NAT
x1 . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom x2 is finite R -element Element of bool NAT
Seg R is finite R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
Indices (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) is set
dom (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) is finite Element of bool NAT
width (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2)) is finite width (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) ) } is set
[:(dom (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2)),(Seg (width (m,(ReplaceLine (P,i2,Q)),M,R,x1,x2))):] is Relation-like finite set
Indices (m,P,M,R,x1,x2) is set
dom (m,P,M,R,x1,x2) is finite Element of bool NAT
width (m,P,M,R,x1,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,P,M,R,x1,x2)) is finite width (m,P,M,R,x1,x2) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,P,M,R,x1,x2) ) } is set
[:(dom (m,P,M,R,x1,x2)),(Seg (width (m,P,M,R,x1,x2))):] is Relation-like finite set
dom Q is finite Element of bool NAT
dom S is finite Element of bool NAT
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (ReplaceLine (P,i2,Q)) is set
dom (ReplaceLine (P,i2,Q)) is finite Element of bool NAT
width (ReplaceLine (P,i2,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (ReplaceLine (P,i2,Q))) is finite width (ReplaceLine (P,i2,Q)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (P,i2,Q)) ) } is set
[:(dom (ReplaceLine (P,i2,Q))),(Seg (width (ReplaceLine (P,i2,Q)))):] is Relation-like finite set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[Q1,P2] is set
{Q1,P2} is non empty finite V37() set
{Q1} is non empty trivial finite V37() 1 -element set
{{Q1,P2},{Q1}} is non empty finite V37() without_zero V103() set
[:(Seg M),(Seg R):] is Relation-like finite set
(m,(ReplaceLine (P,i2,Q)),M,R,x1,x2) * (Q1,P2) is Element of m
Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 . i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(ReplaceLine (P,i2,Q)) * ((x1 . Q2),(x2 . i)) is Element of m
[(x1 . Q2),(x2 . i)] is set
{(x1 . Q2),(x2 . i)} is non empty finite V37() set
{(x1 . Q2)} is non empty trivial finite V37() 1 -element set
{{(x1 . Q2),(x2 . i)},{(x1 . Q2)}} is non empty finite V37() without_zero V103() set
(ReplaceLine ((m,P,M,R,x1,x2),i1,S)) * (Q1,P2) is Element of m
S . P2 is set
x2 . P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q . (x2 . P2) is set
x1 . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((x1 . Q2),(x2 . i)) is Element of m
(ReplaceLine ((m,P,M,R,x1,x2),i1,S)) * (Q1,P2) is Element of m
(m,P,M,R,x1,x2) * (Q1,P2) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = R } is set
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
Q is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
rng Q is finite V212() V213() V214() V217() set
S is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
ReplaceLine (P,x1,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
x2 is Relation-like NAT -defined NAT -valued Function-like finite R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of R -tuples_on NAT
rng x2 is finite V212() V213() V214() V217() set
[:(rng x2),(rng Q):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,P,R,M,x2,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,M,m
(m,(ReplaceLine (P,x1,S)),R,M,x2,Q) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,M,m
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (m,(ReplaceLine (P,x1,S)),R,M,x2,Q) is set
dom (m,(ReplaceLine (P,x1,S)),R,M,x2,Q) is finite Element of bool NAT
width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q)) is finite width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q) ) } is set
[:(dom (m,(ReplaceLine (P,x1,S)),R,M,x2,Q)),(Seg (width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q))):] is Relation-like finite set
Indices (m,P,R,M,x2,Q) is set
dom (m,P,R,M,x2,Q) is finite Element of bool NAT
width (m,P,R,M,x2,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,P,R,M,x2,Q)) is finite width (m,P,R,M,x2,Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,P,R,M,x2,Q) ) } is set
[:(dom (m,P,R,M,x2,Q)),(Seg (width (m,P,R,M,x2,Q))):] is Relation-like finite set
dom x2 is finite R -element Element of bool NAT
Seg R is finite R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[y2,Q] is set
{y2,Q} is non empty finite V37() set
{y2} is non empty trivial finite V37() 1 -element set
{{y2,Q},{y2}} is non empty finite V37() without_zero V103() set
[:(Seg R),(Seg (width (m,(ReplaceLine (P,x1,S)),R,M,x2,Q))):] is Relation-like finite set
x2 . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[(x2 . P1),(Q . Q1)] is set
{(x2 . P1),(Q . Q1)} is non empty finite V37() set
{(x2 . P1)} is non empty trivial finite V37() 1 -element set
{{(x2 . P1),(Q . Q1)},{(x2 . P1)}} is non empty finite V37() without_zero V103() set
P * ((x2 . P1),(Q . Q1)) is Element of m
(ReplaceLine (P,x1,S)) * ((x2 . P1),(Q . Q1)) is Element of m
(m,P,R,M,x2,Q) * (P1,Q1) is Element of m
(m,(ReplaceLine (P,x1,S)),R,M,x2,Q) * (y2,Q) is Element of m
(m,P,R,M,x2,Q) * (y2,Q) is Element of m
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = M } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{R} is non empty trivial finite V37() 1 -element set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = P } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{Q} is non empty trivial finite V37() 1 -element set
S is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
Indices S is set
dom S is finite Element of bool NAT
width S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width S) is finite width S -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width S ) } is set
[:(dom S),(Seg (width S)):] is Relation-like finite set
Line (S,Q) is Relation-like NAT -defined m -valued Function-like finite width S -element FinSequence-like FinSubsequence-like Element of (width S) -tuples_on m
(width S) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width S } is set
ReplaceLine (S,R,(Line (S,Q))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
x1 is Relation-like NAT -defined NAT -valued Function-like finite P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of P -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
(rng x1) \ {R} is finite Element of bool (rng x1)
bool (rng x1) is non empty finite V37() set
((rng x1) \ {R}) \/ {Q} is non empty finite set
x2 is Relation-like NAT -defined NAT -valued Function-like finite M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of M -tuples_on NAT
rng x2 is finite V212() V213() V214() V217() set
[:(rng x1),(rng x2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of P,M,m
Seg P is finite P -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
dom i1 is finite Element of bool NAT
len i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is Relation-like NAT -defined NAT -valued Function-like finite P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of P -tuples_on NAT
rng i2 is finite V212() V213() V214() V217() set
y1 is set
dom x1 is finite P -element Element of bool NAT
y2 is set
i1 . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m,S,P,M,i2,x2) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of P,M,m
y1 is set
dom x1 is finite P -element Element of bool NAT
y2 is set
x1 . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom x1 is finite P -element Element of bool NAT
y2 is set
x1 . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Indices (ReplaceLine (S,R,(Line (S,Q)))) is set
dom (ReplaceLine (S,R,(Line (S,Q)))) is finite Element of bool NAT
width (ReplaceLine (S,R,(Line (S,Q)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (ReplaceLine (S,R,(Line (S,Q))))) is finite width (ReplaceLine (S,R,(Line (S,Q)))) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (S,R,(Line (S,Q)))) ) } is set
[:(dom (ReplaceLine (S,R,(Line (S,Q))))),(Seg (width (ReplaceLine (S,R,(Line (S,Q)))))):] is Relation-like finite set
len (Line (S,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (m,S,P,M,i2,x2) is set
dom (m,S,P,M,i2,x2) is finite Element of bool NAT
width (m,S,P,M,i2,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,S,P,M,i2,x2)) is finite width (m,S,P,M,i2,x2) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,S,P,M,i2,x2) ) } is set
[:(dom (m,S,P,M,i2,x2)),(Seg (width (m,S,P,M,i2,x2))):] is Relation-like finite set
Indices (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) is set
dom (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) is finite Element of bool NAT
width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2)) is finite width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) ) } is set
[:(dom (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2)),(Seg (width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2))):] is Relation-like finite set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[Q1,P2] is set
{Q1,P2} is non empty finite V37() set
{Q1} is non empty trivial finite V37() 1 -element set
{{Q1,P2},{Q1}} is non empty finite V37() without_zero V103() set
Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) * (Q2,i) is Element of m
x1 . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 . i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(ReplaceLine (S,R,(Line (S,Q)))) * ((x1 . Q2),(x2 . i)) is Element of m
[:(Seg P),(Seg (width (m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2))):] is Relation-like finite set
[(x1 . Q2),(x2 . i)] is set
{(x1 . Q2),(x2 . i)} is non empty finite V37() set
{(x1 . Q2)} is non empty trivial finite V37() 1 -element set
{{(x1 . Q2),(x2 . i)},{(x1 . Q2)}} is non empty finite V37() without_zero V103() set
(Line (S,Q)) . (x2 . i) is set
i2 . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S * (Q,(x2 . i)) is Element of m
S * ((x1 . Q2),(x2 . i)) is Element of m
(m,(ReplaceLine (S,R,(Line (S,Q)))),P,M,x1,x2) * (Q1,P2) is Element of m
(m,S,P,M,i2,x2) * (Q1,P2) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
P is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
ReplaceLine (R,M,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(ReplaceLine (R,M,P)) . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((ReplaceLine (R,M,P)),S) is Relation-like NAT -defined m -valued Function-like finite width (ReplaceLine (R,M,P)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,M,P))) -tuples_on m
width (ReplaceLine (R,M,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine (R,M,P))) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (ReplaceLine (R,M,P)) } is set
Line (R,S) is Relation-like NAT -defined m -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on m
(width R) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width R } is set
R . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (ReplaceLine (R,M,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
R is Element of the carrier of K
R * M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (R * M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (R * M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Element of the carrier of K
R is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
P is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
[:(rng R),(rng P):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
Q is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
Indices Q is set
dom Q is finite Element of bool NAT
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width Q) is finite width Q -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width Q ) } is set
[:(dom Q),(Seg (width Q)):] is Relation-like finite set
( the carrier of K,Q,m,n,R,P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(m,n,K,( the carrier of K,Q,m,n,R,P),M) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
M * Q is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
( the carrier of K,(M * Q),m,n,R,P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
Indices (M * Q) is set
dom (M * Q) is finite Element of bool NAT
width (M * Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (M * Q)) is finite width (M * Q) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (M * Q) ) } is set
[:(dom (M * Q)),(Seg (width (M * Q))):] is Relation-like finite set
Indices ( the carrier of K,(M * Q),m,n,R,P) is set
dom ( the carrier of K,(M * Q),m,n,R,P) is finite Element of bool NAT
width ( the carrier of K,(M * Q),m,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,(M * Q),m,n,R,P)) is finite width ( the carrier of K,(M * Q),m,n,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,(M * Q),m,n,R,P) ) } is set
[:(dom ( the carrier of K,(M * Q),m,n,R,P)),(Seg (width ( the carrier of K,(M * Q),m,n,R,P))):] is Relation-like finite set
Indices ( the carrier of K,Q,m,n,R,P) is set
dom ( the carrier of K,Q,m,n,R,P) is finite Element of bool NAT
width ( the carrier of K,Q,m,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,Q,m,n,R,P)) is finite width ( the carrier of K,Q,m,n,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,Q,m,n,R,P) ) } is set
[:(dom ( the carrier of K,Q,m,n,R,P)),(Seg (width ( the carrier of K,Q,m,n,R,P))):] is Relation-like finite set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[i1,i2] is set
{i1,i2} is non empty finite V37() set
{i1} is non empty trivial finite V37() 1 -element set
{{i1,i2},{i1}} is non empty finite V37() without_zero V103() set
(m,n,K,( the carrier of K,Q,m,n,R,P),M) * (i1,i2) is Element of the carrier of K
( the carrier of K,Q,m,n,R,P) * (i1,i2) is Element of the carrier of K
M * (( the carrier of K,Q,m,n,R,P) * (i1,i2)) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the multF of K . (M,(( the carrier of K,Q,m,n,R,P) * (i1,i2))) is Element of the carrier of K
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,(M * Q),m,n,R,P) * (y1,y2) is Element of the carrier of K
R . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(M * Q) * ((R . i1),(P . i2)) is Element of the carrier of K
( the carrier of K,Q,m,n,R,P) * (y1,y2) is Element of the carrier of K
Q * ((R . i1),(P . i2)) is Element of the carrier of K
R . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[(R . y1),(P . y2)] is set
{(R . y1),(P . y2)} is non empty finite V37() set
{(R . y1)} is non empty trivial finite V37() 1 -element set
{{(R . y1),(P . y2)},{(R . y1)}} is non empty finite V37() without_zero V103() set
( the carrier of K,(M * Q),m,n,R,P) * (i1,i2) is Element of the carrier of K
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
idseq (len M) is Relation-like NAT -defined Function-like finite len M -element FinSequence-like FinSubsequence-like set
Seg (len M) is finite len M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
id (Seg (len M)) is Relation-like Seg (len M) -defined Seg (len M) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len M)),(Seg (len M)):]
[:(Seg (len M)),(Seg (len M)):] is Relation-like finite set
bool [:(Seg (len M)),(Seg (len M)):] is non empty finite V37() set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
idseq (width M) is Relation-like NAT -defined Function-like finite width M -element FinSequence-like FinSubsequence-like set
Seg (width M) is finite width M -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
id (Seg (width M)) is Relation-like Seg (width M) -defined Seg (width M) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width M)),(Seg (width M)):]
[:(Seg (width M)),(Seg (width M)):] is Relation-like finite set
bool [:(Seg (width M)),(Seg (width M)):] is non empty finite V37() set
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
(m,M,n,K,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (idseq (width M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (idseq (len M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (m,M,n,K,R,P) is set
dom (m,M,n,K,R,P) is finite Element of bool NAT
Seg (width (m,M,n,K,R,P)) is finite width (m,M,n,K,R,P) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,M,n,K,R,P) ) } is set
[:(dom (m,M,n,K,R,P)),(Seg (width (m,M,n,K,R,P))):] is Relation-like finite set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
[:(Seg n),(Seg K):] is Relation-like finite set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[S,x1] is set
{S,x1} is non empty finite V37() set
{S} is non empty trivial finite V37() 1 -element set
{{S,x1},{S}} is non empty finite V37() without_zero V103() set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m,M,n,K,R,P) * (S,x1) is Element of m
M * (S,x1) is Element of m
len (m,M,n,K,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
{} NAT is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() Element of bool NAT
{1} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
m is without_zero set
n is set
m \ n is Element of bool m
bool m is non empty set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{m} is non empty trivial finite V37() 1 -element set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
{m,n} is non empty finite V37() set
m is finite without_zero Element of bool NAT
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is finite without_zero Element of bool NAT
Sgm m is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
card m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card m) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card m } is set
len (Sgm m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(n) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
K is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(K) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
(m,M,(n),(K)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (n), card (K),m
card (n) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
card (K) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
((n)) is Relation-like NAT -defined NAT -valued Function-like finite card (n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (n)) -tuples_on NAT
(card (n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (n) } is set
((K)) is Relation-like NAT -defined NAT -valued Function-like finite card (K) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (K)) -tuples_on NAT
(card (K)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (K) } is set
(m,M,(card (n)),(card (K)),((n)),((K))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (n), card (K),m
M * (n,K) is Element of m
<*(M * (n,K))*> is Relation-like NAT -defined m -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on m
1 -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = 1 } is set
[1,(M * (n,K))] is set
{1,(M * (n,K))} is non empty finite set
{{1,(M * (n,K))},{1}} is non empty finite V37() without_zero V103() set
{[1,(M * (n,K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(M * (n,K))*>*> is Relation-like NAT -defined m * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(M * (n,K))*>,m
len <*(M * (n,K))*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(M * (n,K))*>] is set
{1,<*(M * (n,K))*>} is non empty finite V37() without_zero V103() set
{{1,<*(M * (n,K))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(M * (n,K))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*n*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,n] is set
{1,n} is non empty finite V37() without_zero V103() set
{{1,n},{1}} is non empty finite V37() without_zero V103() set
{[1,n]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*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() without_zero V103() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*K*> . 1 is set
<*n*> . 1 is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
K is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(n,K) is non empty finite V37() without_zero V103() Element of bool NAT
M is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
R is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(M,R) is non empty finite V37() without_zero V103() Element of bool NAT
P is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
(m,P,(n,K),(M,R)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (n,K), card (M,R),m
card (n,K) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
card (M,R) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
((n,K)) is Relation-like NAT -defined NAT -valued Function-like finite card (n,K) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (n,K)) -tuples_on NAT
(card (n,K)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (n,K) } is set
((M,R)) is Relation-like NAT -defined NAT -valued Function-like finite card (M,R) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (M,R)) -tuples_on NAT
(card (M,R)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (M,R) } is set
(m,P,(card (n,K)),(card (M,R)),((n,K)),((M,R))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (n,K), card (M,R),m
P * (n,M) is Element of m
P * (n,R) is Element of m
P * (K,M) is Element of m
P * (K,R) is Element of m
((P * (n,M)),(P * (n,R))) ][ ((P * (K,M)),(P * (K,R))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2,m
<*(P * (n,M)),(P * (n,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(P * (n,M))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P * (n,M))] is set
{1,(P * (n,M))} is non empty finite set
{{1,(P * (n,M))},{1}} is non empty finite V37() without_zero V103() set
{[1,(P * (n,M))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(P * (n,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P * (n,R))] is set
{1,(P * (n,R))} is non empty finite set
{{1,(P * (n,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(P * (n,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(P * (n,M))*>,<*(P * (n,R))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(P * (K,M)),(P * (K,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(P * (K,M))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P * (K,M))] is set
{1,(P * (K,M))} is non empty finite set
{{1,(P * (K,M))},{1}} is non empty finite V37() without_zero V103() set
{[1,(P * (K,M))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(P * (K,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P * (K,R))] is set
{1,(P * (K,R))} is non empty finite set
{{1,(P * (K,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(P * (K,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(P * (K,M))*>,<*(P * (K,R))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(P * (n,M)),(P * (n,R))*>,<*(P * (K,M)),(P * (K,R))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(P * (n,M)),(P * (n,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(P * (n,M)),(P * (n,R))*>] is set
{1,<*(P * (n,M)),(P * (n,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(P * (n,M)),(P * (n,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(P * (n,M)),(P * (n,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(P * (K,M)),(P * (K,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(P * (K,M)),(P * (K,R))*>] is set
{1,<*(P * (K,M)),(P * (K,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(P * (K,M)),(P * (K,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(P * (K,M)),(P * (K,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(P * (n,M)),(P * (n,R))*>*>,<*<*(P * (K,M)),(P * (K,R))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*M,R*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*M*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,M] is set
{1,M} is non empty finite V37() without_zero V103() set
{{1,M},{1}} is non empty finite V37() without_zero V103() set
{[1,M]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*R*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,R] is set
{1,R} is non empty finite V37() without_zero V103() set
{{1,R},{1}} is non empty finite V37() without_zero V103() set
{[1,R]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*M*>,<*R*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
((M,R)) . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
<*n,K*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*n*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,n] is set
{1,n} is non empty finite V37() without_zero V103() set
{{1,n},{1}} is non empty finite V37() without_zero V103() set
{[1,n]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*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() without_zero V103() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*n*>,<*K*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
((n,K)) . 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
((n,K)) . 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
((M,R)) . 2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
(m,n,(Seg (len n)),(Seg (width n))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len n)), card (Seg (width n)),m
card (Seg (len n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg (width n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg (len n))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len n)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len n))) -tuples_on NAT
(card (Seg (len n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len n)) } is set
((Seg (width n))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width n)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (width n))) -tuples_on NAT
(card (Seg (width n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width n)) } is set
(m,n,(card (Seg (len n))),(card (Seg (width n))),((Seg (len n))),((Seg (width n)))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len n)), card (Seg (width n)),m
idseq (width n) is Relation-like NAT -defined Function-like finite width n -element FinSequence-like FinSubsequence-like set
id (Seg (width n)) is Relation-like Seg (width n) -defined Seg (width n) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width n)),(Seg (width n)):]
[:(Seg (width n)),(Seg (width n)):] is Relation-like finite set
bool [:(Seg (width n)),(Seg (width n)):] is non empty finite V37() set
idseq (len n) is Relation-like NAT -defined Function-like finite len n -element FinSequence-like FinSubsequence-like set
id (Seg (len n)) is Relation-like Seg (len n) -defined Seg (len n) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len n)),(Seg (len n)):]
[:(Seg (len n)),(Seg (len n)):] is Relation-like finite set
bool [:(Seg (len n)),(Seg (len n)):] is non empty finite V37() set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(M) . n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (K,((M) . n)) is Relation-like NAT -defined m -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on m
(width K) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width K } is set
R is finite without_zero Element of bool NAT
(m,K,M,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R,m
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(m,K,(card M),(card R),(M),(R)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R,m
Line ((m,K,M,R),n) is Relation-like NAT -defined m -valued Function-like finite width (m,K,M,R) -element FinSequence-like FinSubsequence-like Element of (width (m,K,M,R)) -tuples_on m
width (m,K,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,K,M,R)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (m,K,M,R) } is set
(Line (K,((M) . n))) * (R) is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
rng (R) is finite V212() V213() V214() V217() set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
(m,K,M,(Seg (width K))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg (width K)),m
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
((Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
(m,K,(card M),(card (Seg (width K))),(M),((Seg (width K)))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg (width K)),m
Line ((m,K,M,(Seg (width K))),n) is Relation-like NAT -defined m -valued Function-like finite width (m,K,M,(Seg (width K))) -element FinSequence-like FinSubsequence-like Element of (width (m,K,M,(Seg (width K)))) -tuples_on m
width (m,K,M,(Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,K,M,(Seg (width K)))) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (m,K,M,(Seg (width K))) } is set
(M) . n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (K,((M) . n)) is Relation-like NAT -defined m -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on m
(width K) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width K } is set
len (Line (K,((M) . n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (K,((M) . n))) is finite width K -element Element of bool NAT
idseq (width K) is Relation-like NAT -defined Function-like finite width K -element FinSequence-like FinSubsequence-like set
id (Seg (width K)) is Relation-like Seg (width K) -defined Seg (width K) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width K)),(Seg (width K)):]
[:(Seg (width K)),(Seg (width K)):] is Relation-like finite set
bool [:(Seg (width K)),(Seg (width K)):] is non empty finite V37() set
(Line (K,((M) . n))) * ((Seg (width K))) is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(M) . n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Col (K,((M) . n)) is Relation-like NAT -defined m -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on m
(len K) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len K } is set
R is finite without_zero Element of bool NAT
(m,K,R,M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card M,m
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(m,K,(card R),(card M),(R),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card M,m
Col ((m,K,R,M),n) is Relation-like NAT -defined m -valued Function-like finite len (m,K,R,M) -element FinSequence-like FinSubsequence-like Element of (len (m,K,R,M)) -tuples_on m
len (m,K,R,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (m,K,R,M)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len (m,K,R,M) } is set
(Col (K,((M) . n))) * (R) is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
rng (R) is finite V212() V213() V214() V217() set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
(m,K,(Seg (len K)),M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len K)), card M,m
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg (len K))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,K,(card (Seg (len K))),(card M),((Seg (len K))),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len K)), card M,m
Col ((m,K,(Seg (len K)),M),n) is Relation-like NAT -defined m -valued Function-like finite len (m,K,(Seg (len K)),M) -element FinSequence-like FinSubsequence-like Element of (len (m,K,(Seg (len K)),M)) -tuples_on m
len (m,K,(Seg (len K)),M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (m,K,(Seg (len K)),M)) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len (m,K,(Seg (len K)),M) } is set
(M) . n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Col (K,((M) . n)) is Relation-like NAT -defined m -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on m
(len K) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = len K } is set
len (Col (K,((M) . n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Col (K,((M) . n))) is finite len K -element Element of bool NAT
idseq (len K) is Relation-like NAT -defined Function-like finite len K -element FinSequence-like FinSubsequence-like set
id (Seg (len K)) is Relation-like Seg (len K) -defined Seg (len K) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len K)),(Seg (len K)):]
[:(Seg (len K)),(Seg (len K)):] is Relation-like finite set
bool [:(Seg (len K)),(Seg (len K)):] is non empty finite V37() set
(Col (K,((M) . n))) * ((Seg (len K))) is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(n) is non empty trivial finite V37() 1 -element Element of bool NAT
K is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) \ (n) is finite without_zero Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(m,K,((Seg (len K)) \ (n)),(Seg (width K))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (len K)) \ (n)), card (Seg (width K)),m
card ((Seg (len K)) \ (n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg (len K)) \ (n))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (len K)) \ (n)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (len K)) \ (n))) -tuples_on NAT
(card ((Seg (len K)) \ (n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (len K)) \ (n)) } is set
((Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
(m,K,(card ((Seg (len K)) \ (n))),(card (Seg (width K))),(((Seg (len K)) \ (n))),((Seg (width K)))) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (len K)) \ (n)), card (Seg (width K)),m
Del (K,n) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
dom K is finite Element of bool NAT
len (Del (K,n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg S is finite S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= S ) } is set
K * (((Seg (len K)) \ (n))) is Relation-like NAT -defined m * -valued Function-like finite Function-yielding V147() Element of bool [:NAT,(m *):]
[:NAT,(m *):] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,(m *):] is non empty non trivial non finite V103() set
dom (Del (K,n)) is finite Element of bool NAT
(((Seg (len K)) \ (n))) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Del (K,n)) . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x1 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of S + 1, width K,m
x1 . ((((Seg (len K)) \ (n))) . x2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (x1,((((Seg (len K)) \ (n))) . x2)) is Relation-like NAT -defined m -valued Function-like finite width x1 -element FinSequence-like FinSubsequence-like Element of (width x1) -tuples_on m
width x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width x1) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width x1 } is set
Line ((m,K,((Seg (len K)) \ (n)),(Seg (width K))),x2) is Relation-like NAT -defined m -valued Function-like finite width (m,K,((Seg (len K)) \ (n)),(Seg (width K))) -element FinSequence-like FinSubsequence-like Element of (width (m,K,((Seg (len K)) \ (n)),(Seg (width K)))) -tuples_on m
width (m,K,((Seg (len K)) \ (n)),(Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,K,((Seg (len K)) \ (n)),(Seg (width K)))) -tuples_on m is functional non empty FinSequence-membered FinSequenceSet of m
{ b₁ where b₁ is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Element of m * : len b₁ = width (m,K,((Seg (len K)) \ (n)),(Seg (width K))) } is set
(m,K,((Seg (len K)) \ (n)),(Seg (width K))) . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (m,K,((Seg (len K)) \ (n)),(Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m) is non empty trivial finite V37() 1 -element Element of bool NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() 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 Function-yielding V147() tabular FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) \ (m) is finite without_zero Element of bool NAT
( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len K)), card ((Seg (width K)) \ (m)), the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card ((Seg (width K)) \ (m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg (len K))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
(((Seg (width K)) \ (m))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width K)) \ (m)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (width K)) \ (m))) -tuples_on NAT
(card ((Seg (width K)) \ (m))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width K)) \ (m)) } is set
( the carrier of n,K,(card (Seg (len K))),(card ((Seg (width K)) \ (m))),((Seg (len K))),(((Seg (width K)) \ (m)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len K)), card ((Seg (width K)) \ (m)), the carrier of n
DelCol (K,m) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
len ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom K is finite Element of bool NAT
idseq (len K) is Relation-like NAT -defined Function-like finite len K -element FinSequence-like FinSubsequence-like set
id (Seg (len K)) is Relation-like Seg (len K) -defined Seg (len K) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len K)),(Seg (len K)):]
[:(Seg (len K)),(Seg (len K)):] is Relation-like finite set
bool [:(Seg (len K)),(Seg (len K)):] is non empty finite V37() set
((Seg (len K))) . x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (K,x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
len (Line (K,x1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (K,x1)) is finite width K -element Element of bool NAT
Line (( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))),x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m)))) -tuples_on the carrier of n
width ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m)))) -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 ( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) } is set
( the carrier of n,K,(Seg (len K)),((Seg (width K)) \ (m))) . x1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (K,(((Seg (len K))) . x1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(Line (K,(((Seg (len K))) . x1))) * (((Seg (width K)) \ (m))) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
DelLine (,(Line (K,x1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(DelCol (K,m)) . x1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (DelCol (K,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is set
n is finite without_zero Element of bool NAT
(n) is Relation-like NAT -defined NAT -valued Function-like finite card n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card n) -tuples_on NAT
card n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card n) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card n } is set
(n) " m is finite set
dom (n) is finite card n -element Element of bool NAT
Seg (card n) is finite card n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
m is set
Sgm m is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
n is finite without_zero Element of bool NAT
(n) is Relation-like NAT -defined NAT -valued Function-like finite card n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card n) -tuples_on NAT
card n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card n) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card n } is set
(n) " m is finite set
Sgm ((n) " m) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
(n) * (Sgm ((n) " m)) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite V103() set
dom (n) is finite card n -element Element of bool NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
rng (n) is finite V212() V213() V214() V217() set
(n) | ((n) " m) is Relation-like NAT -defined (n) " m -defined NAT -defined NAT -valued RAT -valued Function-like finite FinSubsequence-like V202() V203() V204() V205() Element of bool [:NAT,NAT:]
rng ((n) | ((n) " m)) is finite V212() V213() V214() V215() V217() set
(n) .: ((n) " m) is finite V212() V213() V214() V217() set
m is set
card m is V26() V27() V28() cardinal set
n is finite without_zero Element of bool NAT
(n) is Relation-like NAT -defined NAT -valued Function-like finite card n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card n) -tuples_on NAT
card n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card n) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card n } is set
(n) " m is finite set
card ((n) " m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (n) is finite V212() V213() V214() V217() set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(n) .: ((n) " m) is finite V212() V213() V214() V217() set
dom (n) is finite card n -element Element of bool NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
m is set
n is set
K is non empty set
K * is functional non empty FinSequence-membered FinSequenceSet of K
M is Relation-like NAT -defined K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of K *
R is finite without_zero Element of bool NAT
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(R) " m is finite set
P is finite without_zero Element of bool NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(P) " n is finite set
[:((R) " m),((P) " n):] is Relation-like finite set
(K,M,R,P) is Relation-like NAT -defined K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P,K
(K,M,(card R),(card P),(R),(P)) is Relation-like NAT -defined K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P,K
Indices (K,M,R,P) is set
dom (K,M,R,P) is finite Element of bool NAT
width (K,M,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (K,M,R,P)) is finite width (K,M,R,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (K,M,R,P) ) } is set
[:(dom (K,M,R,P)),(Seg (width (K,M,R,P))):] is Relation-like finite set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
[:(Seg (card R)),(Seg (card P)):] is Relation-like finite set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
[:(Seg (card R)),(Seg (card P)):] is Relation-like finite set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
[:(Seg (card R)),(Seg (card P)):] is Relation-like finite set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
[:(Seg (card R)),(Seg (card P)):] is Relation-like finite set
dom (P) is finite card P -element Element of bool NAT
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
dom (R) is finite card R -element Element of bool NAT
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(M) " K is finite set
R is finite without_zero Element of bool NAT
(m,n,K,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card R,m
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(m,n,(card K),(card R),(K),(R)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card R,m
P is finite without_zero Element of bool NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(P) " R is finite set
(m,n,M,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card P,m
(m,n,(card M),(card P),(M),(P)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card P,m
Indices (m,n,M,P) is set
dom (m,n,M,P) is finite Element of bool NAT
width (m,n,M,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,n,M,P)) is finite width (m,n,M,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,n,M,P) ) } is set
[:(dom (m,n,M,P)),(Seg (width (m,n,M,P))):] is Relation-like finite set
Q is finite without_zero Element of bool NAT
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
rng (Q) is finite V212() V213() V214() V217() set
S is finite without_zero Element of bool NAT
(S) is Relation-like NAT -defined NAT -valued Function-like finite card S -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card S) -tuples_on NAT
card S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card S) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card S } is set
rng (S) is finite V212() V213() V214() V217() set
[:(rng (Q)),(rng (S)):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
(m,(m,n,M,P),Q,S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card S,m
(m,(m,n,M,P),(card Q),(card S),(Q),(S)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card S,m
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P1 is finite P1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P1 ) } is set
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P1 is finite P1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P1 ) } is set
(P) * (S) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite V103() set
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[P1,Q1] is set
{P1,Q1} is non empty finite V37() set
{P1} is non empty trivial finite V37() 1 -element set
{{P1,Q1},{P1}} is non empty finite V37() without_zero V103() set
Indices (m,n,K,R) is set
dom (m,n,K,R) is finite Element of bool NAT
width (m,n,K,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,n,K,R)) is finite width (m,n,K,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,n,K,R) ) } is set
[:(dom (m,n,K,R)),(Seg (width (m,n,K,R))):] is Relation-like finite set
x2 is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card R,m
Indices x2 is set
dom x2 is finite Element of bool NAT
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width x2) is finite width x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x2 ) } is set
[:(dom x2),(Seg (width x2)):] is Relation-like finite set
(Q) . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(S) . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(M) * (Q) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,NAT:]
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
[:(Seg (card Q)),(Seg (width x2)):] is Relation-like finite set
dom (S) is finite card S -element Element of bool NAT
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i is finite i -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P) . Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(R) . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[i,m] is Element of [:NAT,NAT:]
{i,m} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,m},{i}} is non empty finite V37() without_zero V103() set
x2 * (P1,Q1) is Element of m
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,n,M,P) * (P2,Q2) is Element of m
dom (Q) is finite card Q -element Element of bool NAT
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q2i is finite Q2i -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q2i ) } is set
(M) . P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(K) . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[P2,Q2] is Element of [:NAT,NAT:]
{P2,Q2} is non empty finite V37() set
{P2} is non empty trivial finite V37() 1 -element set
{{P2,Q2},{P2}} is non empty finite V37() without_zero V103() set
n * (((M) . P2),((P) . Q2)) is Element of m
(m,n,K,R) * (P1,Q1) is Element of m
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(R) .: K is finite V212() V213() V214() V217() set
P is finite without_zero Element of bool NAT
(m,n,R,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P,m
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(m,n,(card R),(card P),(R),(P)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P,m
Indices (m,n,R,P) is set
dom (m,n,R,P) is finite Element of bool NAT
width (m,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (m,n,R,P)) is finite width (m,n,R,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (m,n,R,P) ) } is set
[:(dom (m,n,R,P)),(Seg (width (m,n,R,P))):] is Relation-like finite set
(P) .: M is finite V212() V213() V214() V217() set
(m,(m,n,R,P),K,M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,(m,n,R,P),(card K),(card M),(K),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
len (m,n,R,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (m,n,R,P)) is finite len (m,n,R,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (m,n,R,P) ) } is set
[:(Seg (len (m,n,R,P))),(Seg (width (m,n,R,P))):] is Relation-like finite set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
rng (P) is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
rng (R) is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
dom (P) is finite card P -element Element of bool NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
i1 is finite without_zero Element of bool NAT
(P) " i1 is finite set
dom (R) is finite card R -element Element of bool NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
x2 is finite without_zero Element of bool NAT
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(R) " x2 is finite set
(m,n,x2,i1) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1,m
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
(m,n,(card x2),(card i1),(x2),(i1)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1,m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(n) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg m) \ (n) is finite without_zero Element of bool NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(K) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg m) \ (K) is finite without_zero Element of bool NAT
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
R is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of M
( the carrier of M,R,((Seg m) \ (n)),((Seg m) \ (K))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg m) \ (n)), card ((Seg m) \ (K)), the carrier of M
card ((Seg m) \ (n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card ((Seg m) \ (K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg m) \ (n))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg m) \ (n)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg m) \ (n))) -tuples_on NAT
(card ((Seg m) \ (n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg m) \ (n)) } is set
(((Seg m) \ (K))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg m) \ (K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg m) \ (K))) -tuples_on NAT
(card ((Seg m) \ (K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg m) \ (K)) } is set
( the carrier of M,R,(card ((Seg m) \ (n))),(card ((Seg m) \ (K))),(((Seg m) \ (n))),(((Seg m) \ (K)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg m) \ (n)), card ((Seg m) \ (K)), the carrier of M
Deleting (R,n,K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
DelLine (R,n) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
DelCol ((DelLine (R,n)),K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom R is finite Element of bool NAT
Del (R,n) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
( the carrier of M,R,((Seg m) \ (n)),(Seg m)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg m) \ (n)), card (Seg m), the carrier of M
card (Seg m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg m)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg m) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg m)) -tuples_on NAT
(card (Seg m)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg m) } is set
( the carrier of M,R,(card ((Seg m) \ (n))),(card (Seg m)),(((Seg m) \ (n))),((Seg m))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg m) \ (n)), card (Seg m), the carrier of M
Del (R,n) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
len (Del (R,n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
len (Deleting (R,n,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(1) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
(Seg m) \ (1) is finite without_zero Element of bool NAT
len ( the carrier of M,R,((Seg m) \ (n)),((Seg m) \ (K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
1 + 0 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
width (DelLine (R,n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (((Seg m) \ (n))) is finite V212() V213() V214() V217() set
dom (((Seg m) \ (n))) is finite card ((Seg m) \ (n)) -element Element of bool NAT
Seg (card ((Seg m) \ (n))) is finite card ((Seg m) \ (n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card ((Seg m) \ (n)) ) } is set
(((Seg m) \ (n))) " ((Seg m) \ (n)) is finite set
len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))) is finite len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) ) } is set
width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m))) is finite width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)) ) } is set
(Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K) is finite without_zero Element of bool NAT
( the carrier of M,( the carrier of M,R,((Seg m) \ (n)),(Seg m)),(Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))),((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))), card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)), the carrier of M
card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))))) -tuples_on NAT
(card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) } is set
(((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K))) -tuples_on NAT
(card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)) } is set
( the carrier of M,( the carrier of M,R,((Seg m) \ (n)),(Seg m)),(card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))))),(card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K))),((Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m))))),(((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))), card ((Seg (width ( the carrier of M,R,((Seg m) \ (n)),(Seg m)))) \ (K)), the carrier of M
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
((Seg m)) " ((Seg m) \ (K)) is finite set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices R is set
dom R is finite Element of bool NAT
Seg (width R) is finite width R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width R ) } is set
[:(dom R),(Seg (width R)):] is Relation-like finite set
P is finite without_zero Element of bool NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
Q is finite without_zero Element of bool NAT
[:Q,P:] is Relation-like finite set
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,R,Q,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card P,m
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
(m,R,(card Q),(card P),(Q),(P)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card P,m
(Q) . M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
S * (P) is Relation-like NAT -defined m -valued Function-like finite Element of bool [:NAT,m:]
[:NAT,m:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT,m:] is non empty non trivial non finite V103() set
ReplaceLine ((m,R,Q,P),M,x1) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card P,m
ReplaceLine (R,((Q) . M),S) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
(m,(ReplaceLine (R,((Q) . M),S)),Q,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card P,m
(m,(ReplaceLine (R,((Q) . M),S)),(card Q),(card P),(Q),(P)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card P,m
len (m,R,Q,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (P) is finite V212() V213() V214() V217() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
rng (Q) is finite V212() V213() V214() V217() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
dom (Q) is finite card Q -element Element of bool NAT
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
(((Q) . M)) is non empty trivial finite V37() 1 -element Element of bool NAT
(Q) " (((Q) . M)) is finite set
(M) is non empty trivial finite V37() 1 -element Element of bool NAT
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
Indices M is set
dom M is finite Element of bool NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
R is finite without_zero Element of bool NAT
P is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of m
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
ReplaceLine (M,Q,P) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K,m
S is finite without_zero Element of bool NAT
[:S,R:] is Relation-like finite set
(m,M,S,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card S, card R,m
card S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(S) is Relation-like NAT -defined NAT -valued Function-like finite card S -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card S) -tuples_on NAT
(card S) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card S } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(m,M,(card S),(card R),(S),(R)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card S, card R,m
(m,(ReplaceLine (M,Q,P)),S,R) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card S, card R,m
(m,(ReplaceLine (M,Q,P)),(card S),(card R),(S),(R)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card S, card R,m
rng (R) is finite V212() V213() V214() V217() set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x1 is finite x1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x1 ) } is set
rng (S) is finite V212() V213() V214() V217() set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x1 is finite x1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x1 ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
n @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,n,K,M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(m,n,K,M) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
(m,(n @),M,K) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K,m
(m,(n @),(card M),(card K),(M),(K)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K,m
rng (M) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
rng (K) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
n @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(m,n,K,M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(m,(n @),M,K) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K,m
(m,(n @),(card M),(card K),(M),(K)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K,m
(m,(n @),M,K) @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
rng (M) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
rng (K) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Element of the carrier of m
K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
n * K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
( the carrier of m,K,(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
((card M),(card R),m,( the carrier of m,K,M,R),n) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
( the carrier of m,(n * K),M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
( the carrier of m,(n * K),(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
rng (M) is finite V212() V213() V214() V217() set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P is finite P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
rng (R) is finite V212() V213() V214() V217() set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P is finite P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
(m,n,K,M) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M,m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card K) is finite card K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card K ) } is set
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(K) . R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(((K) . R)) is non empty trivial finite V37() 1 -element Element of bool NAT
K \ (((K) . R)) is finite without_zero Element of bool NAT
card (K \ (((K) . R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Delete (( the carrier of m,n,K,M),R,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card K) -' 1,(card K) -' 1, the carrier of m
(card K) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) . P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(((M) . P)) is non empty trivial finite V37() 1 -element Element of bool NAT
M \ (((M) . P)) is finite without_zero Element of bool NAT
( the carrier of m,n,(K \ (((K) . R))),(M \ (((M) . P)))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K \ (((K) . R))), card (K \ (((K) . R))), the carrier of m
card (M \ (((M) . P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (M) is finite card M -element Element of bool NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg S is finite S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= S ) } is set
rng (M) is finite V212() V213() V214() V217() set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg S is finite S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= S ) } is set
(M) " M is finite set
(P) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (card M)) \ (P) is finite without_zero Element of bool NAT
dom (K) is finite card K -element Element of bool NAT
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i1 is finite i1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i1 ) } is set
rng (K) is finite V212() V213() V214() V217() set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i1 is finite i1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i1 ) } is set
(K) " K is finite set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i1 is finite i1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i1 ) } is set
(M) " (((M) . P)) is finite set
(M) " (M \ (((M) . P))) is finite set
(R) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (card K)) \ (R) is finite without_zero Element of bool NAT
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y1 is finite y1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y1 ) } is set
(K) " (((K) . R)) is finite set
(K) " (K \ (((K) . R))) is finite set
(card K) - 1 is V105() complex ext-real set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Deleting (( the carrier of m,n,K,M),R,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
DelLine (( the carrier of m,n,K,M),R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
DelCol ((DelLine (( the carrier of m,n,K,M),R)),P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
( the carrier of m,( the carrier of m,n,K,M),((Seg (card K)) \ (R)),((Seg (card M)) \ (P))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (card K)) \ (R)), card ((Seg (card M)) \ (P)), the carrier of m
card ((Seg (card K)) \ (R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card ((Seg (card M)) \ (P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg (card K)) \ (R))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (card K)) \ (R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (card K)) \ (R))) -tuples_on NAT
(card ((Seg (card K)) \ (R))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (card K)) \ (R)) } is set
(((Seg (card M)) \ (P))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (card M)) \ (P)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (card M)) \ (P))) -tuples_on NAT
(card ((Seg (card M)) \ (P))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (card M)) \ (P)) } is set
( the carrier of m,( the carrier of m,n,K,M),(card ((Seg (card K)) \ (R))),(card ((Seg (card M)) \ (P))),(((Seg (card K)) \ (R))),(((Seg (card M)) \ (P)))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (card K)) \ (R)), card ((Seg (card M)) \ (P)), the carrier of m
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
( the carrier of m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
( the carrier of m,( the carrier of m,n,K,M),((Seg (card K)) \ (R)),((Seg (card M)) \ (P))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (card K)) \ (R)), card ((Seg (card M)) \ (P)), the carrier of m
( the carrier of m,( the carrier of m,n,K,M),(card ((Seg (card K)) \ (R))),(card ((Seg (card M)) \ (P))),(((Seg (card K)) \ (R))),(((Seg (card M)) \ (P)))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (card K)) \ (R)), card ((Seg (card M)) \ (P)), the carrier of m
( the carrier of m,n,(K \ (((K) . R))),(M \ (((M) . P)))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K \ (((K) . R))), card (M \ (((M) . P))), the carrier of m
((K \ (((K) . R)))) is Relation-like NAT -defined NAT -valued Function-like finite card (K \ (((K) . R))) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (K \ (((K) . R)))) -tuples_on NAT
(card (K \ (((K) . R)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (K \ (((K) . R))) } is set
((M \ (((M) . P)))) is Relation-like NAT -defined NAT -valued Function-like finite card (M \ (((M) . P))) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (M \ (((M) . P)))) -tuples_on NAT
(card (M \ (((M) . P)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (M \ (((M) . P))) } is set
( the carrier of m,n,(card (K \ (((K) . R)))),(card (M \ (((M) . P)))),((K \ (((K) . R)))),((M \ (((M) . P))))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K \ (((K) . R))), card (M \ (((M) . P))), the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card K) is finite card K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card K ) } is set
(card K) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
LaplaceExpL (( the carrier of m,n,K,M),R) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
(card K) |-> (0. m) is Relation-like NAT -defined the carrier of m -valued Function-like finite card K -element FinSequence-like FinSubsequence-like Element of (card K) -tuples_on the carrier of m
(card K) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = card K } is set
(Seg (card K)) --> (0. m) is Relation-like Seg (card K) -defined Seg (card K) -defined the carrier of m -valued {(0. m)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card K)),{(0. m)}:]
{(0. m)} is non empty trivial finite 1 -element set
[:(Seg (card K)),{(0. m)}:] is Relation-like finite set
bool [:(Seg (card K)),{(0. m)}:] is non empty finite V37() set
Sum ((card K) |-> (0. m)) is Element of the carrier of m
the addF of m $$ ((card K) |-> (0. m)) is Element of the carrier of m
len (LaplaceExpL (( the carrier of m,n,K,M),R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (LaplaceExpL (( the carrier of m,n,K,M),R)) is finite Element of bool NAT
dom ((card K) |-> (0. m)) is finite card K -element Element of bool NAT
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(LaplaceExpL (( the carrier of m,n,K,M),R)) . x2 is set
((card K) |-> (0. m)) . x2 is set
( the carrier of m,n,K,M) * (R,x2) is Element of the carrier of m
Cofactor (( the carrier of m,n,K,M),R,x2) is Element of the carrier of m
power m is Relation-like [: the carrier of m,NAT:] -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of m,NAT:], the carrier of m:]
[: the carrier of m,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of m,NAT:], the carrier of m:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of m,NAT:], the carrier of m:] is non empty non trivial non finite V103() set
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
- (1_ m) is Element of the carrier of m
R + x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power m) . ((- (1_ m)),(R + x2)) is Element of the carrier of m
Minor (( the carrier of m,n,K,M),R,x2) is Element of the carrier of m
Delete (( the carrier of m,n,K,M),R,x2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card K) -' 1,(card K) -' 1, the carrier of m
Det (Delete (( the carrier of m,n,K,M),R,x2)) is Element of the carrier of m
Permutations ((card K) -' 1) is non empty permutational set
FinOmega (Permutations ((card K) -' 1)) is Element of Fin (Permutations ((card K) -' 1))
Fin (Permutations ((card K) -' 1)) is preBoolean set
Path_product (Delete (( the carrier of m,n,K,M),R,x2)) is Relation-like Permutations ((card K) -' 1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card K) -' 1)), the carrier of m:]
[:(Permutations ((card K) -' 1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations ((card K) -' 1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations ((card K) -' 1))),(Path_product (Delete (( the carrier of m,n,K,M),R,x2)))) is Element of the carrier of m
((power m) . ((- (1_ m)),(R + x2))) * (Minor (( the carrier of m,n,K,M),R,x2)) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the multF of m . (((power m) . ((- (1_ m)),(R + x2))),(Minor (( the carrier of m,n,K,M),R,x2))) is Element of the carrier of m
(( the carrier of m,n,K,M) * (R,x2)) * (Cofactor (( the carrier of m,n,K,M),R,x2)) is Element of the carrier of m
the multF of m . ((( the carrier of m,n,K,M) * (R,x2)),(Cofactor (( the carrier of m,n,K,M),R,x2))) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M \ Q is finite without_zero Element of bool NAT
S is set
x1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(x1) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
Q \/ (x1) is non empty finite without_zero V103() Element of bool NAT
x2 is finite without_zero Element of bool NAT
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 is finite without_zero Element of bool NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,x2,i1) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card x2, the carrier of m
Det ( the carrier of m,n,x2,i1) is Element of the carrier of m
Permutations (card x2) is non empty permutational set
FinOmega (Permutations (card x2)) is Element of Fin (Permutations (card x2))
Fin (Permutations (card x2)) is preBoolean set
Path_product ( the carrier of m,n,x2,i1) is Relation-like Permutations (card x2) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card x2)), the carrier of m:]
[:(Permutations (card x2)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card x2)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card x2))),(Path_product ( the carrier of m,n,x2,i1))) is Element of the carrier of m
x2 \ (x1) is finite without_zero Element of bool NAT
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
dom (x2) is finite card x2 -element Element of bool NAT
Seg (card x2) is finite card x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card x2 ) } is set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y1 is finite y1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y1 ) } is set
rng (x2) is finite V212() V213() V214() V217() set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y1 is finite y1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y1 ) } is set
y1 is set
(x2) . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(P + 1) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P + 1) - 1 is V105() complex ext-real set
(card x2) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Delete (( the carrier of m,n,x2,i1),y2,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card x2) -' 1,(card x2) -' 1, the carrier of m
Det (Delete (( the carrier of m,n,x2,i1),y2,Q)) is Element of the carrier of m
Permutations ((card x2) -' 1) is non empty permutational set
FinOmega (Permutations ((card x2) -' 1)) is Element of Fin (Permutations ((card x2) -' 1))
Fin (Permutations ((card x2) -' 1)) is preBoolean set
Path_product (Delete (( the carrier of m,n,x2,i1),y2,Q)) is Relation-like Permutations ((card x2) -' 1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card x2) -' 1)), the carrier of m:]
[:(Permutations ((card x2) -' 1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations ((card x2) -' 1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations ((card x2) -' 1))),(Path_product (Delete (( the carrier of m,n,x2,i1),y2,Q)))) is Element of the carrier of m
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
(i1) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(((i1) . Q)) is non empty trivial finite V37() 1 -element Element of bool NAT
i1 \ (((i1) . Q)) is finite without_zero Element of bool NAT
P1 is finite without_zero Element of bool NAT
card P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,Q,P1) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card Q, the carrier of m
Det ( the carrier of m,n,Q,P1) is Element of the carrier of m
Permutations (card Q) is non empty permutational set
FinOmega (Permutations (card Q)) is Element of Fin (Permutations (card Q))
Fin (Permutations (card Q)) is preBoolean set
Path_product ( the carrier of m,n,Q,P1) is Relation-like Permutations (card Q) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card Q)), the carrier of m:]
[:(Permutations (card Q)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card Q)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card Q))),(Path_product ( the carrier of m,n,Q,P1))) is Element of the carrier of m
P is finite without_zero Element of bool NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
(card M) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
( the carrier of n,K,M,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of n
Det ( the carrier of n,K,M,R) is Element of the carrier of n
Permutations (card M) is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of n,K,M,R) is Relation-like Permutations (card M) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of n:]
[:(Permutations (card M)), the carrier of n:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of n,K,M,R))) is Element of the carrier of n
LaplaceExpC (( the carrier of n,K,M,R),m) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(card M) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite card M -element FinSequence-like FinSubsequence-like Element of (card M) -tuples_on the carrier of n
(card M) -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 = card M } is set
(Seg (card M)) --> (0. n) is Relation-like Seg (card M) -defined Seg (card M) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card M)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (card M)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (card M)),{(0. n)}:] is non empty finite V37() set
Sum ((card M) |-> (0. n)) is Element of the carrier of n
the addF of n $$ ((card M) |-> (0. n)) is Element of the carrier of n
len (LaplaceExpC (( the carrier of n,K,M,R),m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (LaplaceExpC (( the carrier of n,K,M,R),m)) is finite Element of bool NAT
dom ((card M) |-> (0. n)) is finite card M -element Element of bool NAT
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(LaplaceExpC (( the carrier of n,K,M,R),m)) . i1 is set
((card M) |-> (0. n)) . i1 is set
( the carrier of n,K,M,R) * (i1,m) is Element of the carrier of n
Cofactor (( the carrier of n,K,M,R),i1,m) is Element of the carrier of n
power n is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial non finite V103() set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
the OneF of n is Element of the carrier of n
- (1_ n) is Element of the carrier of n
i1 + m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power n) . ((- (1_ n)),(i1 + m)) is Element of the carrier of n
Minor (( the carrier of n,K,M,R),i1,m) is Element of the carrier of n
Delete (( the carrier of n,K,M,R),i1,m) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card M) -' 1,(card M) -' 1, the carrier of n
Det (Delete (( the carrier of n,K,M,R),i1,m)) is Element of the carrier of n
Permutations ((card M) -' 1) is non empty permutational set
FinOmega (Permutations ((card M) -' 1)) is Element of Fin (Permutations ((card M) -' 1))
Fin (Permutations ((card M) -' 1)) is preBoolean set
Path_product (Delete (( the carrier of n,K,M,R),i1,m)) is Relation-like Permutations ((card M) -' 1) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card M) -' 1)), the carrier of n:]
[:(Permutations ((card M) -' 1)), the carrier of n:] is Relation-like non empty set
bool [:(Permutations ((card M) -' 1)), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations ((card M) -' 1))),(Path_product (Delete (( the carrier of n,K,M,R),i1,m)))) is Element of the carrier of n
((power n) . ((- (1_ n)),(i1 + m))) * (Minor (( the carrier of n,K,M,R),i1,m)) is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the multF of n . (((power n) . ((- (1_ n)),(i1 + m))),(Minor (( the carrier of n,K,M,R),i1,m))) is Element of the carrier of n
(( the carrier of n,K,M,R) * (i1,m)) * (Cofactor (( the carrier of n,K,M,R),i1,m)) is Element of the carrier of n
the multF of n . ((( the carrier of n,K,M,R) * (i1,m)),(Cofactor (( the carrier of n,K,M,R),i1,m))) is Element of the carrier of n
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,R) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,R) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,R))) is Element of the carrier of m
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R \ Q is finite without_zero Element of bool NAT
S is set
x1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(x1) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
Q \/ (x1) is non empty finite without_zero V103() Element of bool NAT
x2 is finite without_zero Element of bool NAT
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i1 is finite without_zero Element of bool NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,i1,x2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i1, card i1, the carrier of m
Det ( the carrier of m,n,i1,x2) is Element of the carrier of m
Permutations (card i1) is non empty permutational set
FinOmega (Permutations (card i1)) is Element of Fin (Permutations (card i1))
Fin (Permutations (card i1)) is preBoolean set
Path_product ( the carrier of m,n,i1,x2) is Relation-like Permutations (card i1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card i1)), the carrier of m:]
[:(Permutations (card i1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card i1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card i1))),(Path_product ( the carrier of m,n,i1,x2))) is Element of the carrier of m
(P + 1) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P + 1) - 1 is V105() complex ext-real set
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
dom (x2) is finite card x2 -element Element of bool NAT
Seg (card x2) is finite card x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card x2 ) } is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
rng (x2) is finite V212() V213() V214() V217() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
i2 is set
(x2) . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(card i1) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Delete (( the carrier of m,n,i1,x2),Q,y2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card i1) -' 1,(card i1) -' 1, the carrier of m
Det (Delete (( the carrier of m,n,i1,x2),Q,y2)) is Element of the carrier of m
Permutations ((card i1) -' 1) is non empty permutational set
FinOmega (Permutations ((card i1) -' 1)) is Element of Fin (Permutations ((card i1) -' 1))
Fin (Permutations ((card i1) -' 1)) is preBoolean set
Path_product (Delete (( the carrier of m,n,i1,x2),Q,y2)) is Relation-like Permutations ((card i1) -' 1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card i1) -' 1)), the carrier of m:]
[:(Permutations ((card i1) -' 1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations ((card i1) -' 1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations ((card i1) -' 1))),(Path_product (Delete (( the carrier of m,n,i1,x2),Q,y2)))) is Element of the carrier of m
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
(i1) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(((i1) . Q)) is non empty trivial finite V37() 1 -element Element of bool NAT
i1 \ (((i1) . Q)) is finite without_zero Element of bool NAT
P1 is finite without_zero Element of bool NAT
card P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,P1,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of m
Det ( the carrier of m,n,P1,Q) is Element of the carrier of m
Permutations (card P1) is non empty permutational set
FinOmega (Permutations (card P1)) is Element of Fin (Permutations (card P1))
Fin (Permutations (card P1)) is preBoolean set
Path_product ( the carrier of m,n,P1,Q) is Relation-like Permutations (card P1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of m:]
[:(Permutations (card P1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P1))),(Path_product ( the carrier of m,n,P1,Q))) is Element of the carrier of m
x2 \ (x1) is finite without_zero Element of bool NAT
P is finite without_zero Element of bool NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:K,M:] is Relation-like finite set
[:(Seg (len n)),(Seg (width n)):] is Relation-like finite set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
Indices M is set
dom M is finite Element of bool NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
R is finite without_zero Element of bool NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is finite without_zero Element of bool NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:R,P:] is Relation-like finite set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Q) is non empty trivial finite V37() 1 -element Element of bool NAT
R \ (Q) is finite without_zero Element of bool NAT
S is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(S) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
(R \ (Q)) \/ (S) is non empty finite without_zero V103() Element of bool NAT
card ((R \ (Q)) \/ (S)) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[:((R \ (Q)) \/ (S)),P:] is Relation-like finite set
Line (M,S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
ReplaceLine (M,Q,(Line (M,S))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),R,P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of K
Det ( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),R,P) is Element of the carrier of K
Permutations (card R) is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
FinOmega (Permutations (card R)) is Element of Fin (Permutations (card R))
Fin (Permutations (card R)) is preBoolean set
Path_product ( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),R,P) is Relation-like Permutations (card R) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card R)), the carrier of K:]
[:(Permutations (card R)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card R)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card R))),(Path_product ( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),R,P))) is Element of the carrier of K
( the carrier of K,M,((R \ (Q)) \/ (S)),P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((R \ (Q)) \/ (S)), card ((R \ (Q)) \/ (S)), the carrier of K
Det ( the carrier of K,M,((R \ (Q)) \/ (S)),P) is Element of the carrier of K
Permutations (card ((R \ (Q)) \/ (S))) is non empty permutational set
FinOmega (Permutations (card ((R \ (Q)) \/ (S)))) is Element of Fin (Permutations (card ((R \ (Q)) \/ (S))))
Fin (Permutations (card ((R \ (Q)) \/ (S)))) is preBoolean set
Path_product ( the carrier of K,M,((R \ (Q)) \/ (S)),P) is Relation-like Permutations (card ((R \ (Q)) \/ (S))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card ((R \ (Q)) \/ (S)))), the carrier of K:]
[:(Permutations (card ((R \ (Q)) \/ (S)))), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card ((R \ (Q)) \/ (S)))), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card ((R \ (Q)) \/ (S))))),(Path_product ( the carrier of K,M,((R \ (Q)) \/ (S)),P))) is Element of the carrier of K
- (Det ( the carrier of K,M,((R \ (Q)) \/ (S)),P)) is Element of the carrier of K
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(((R \ (Q)) \/ (S))) is Relation-like NAT -defined NAT -valued Function-like finite card ((R \ (Q)) \/ (S)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((R \ (Q)) \/ (S))) -tuples_on NAT
(card ((R \ (Q)) \/ (S))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((R \ (Q)) \/ (S)) } is set
rng (((R \ (Q)) \/ (S))) is finite V212() V213() V214() V217() set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y1 is finite y1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y1 ) } is set
(card R) - 1 is V105() complex ext-real set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
card (R \ (Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
y2 is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
Q is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
( the carrier of K,M,(card R),(card R),y2,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of K
( the carrier of K,M,((R \ (Q)) \/ (S)),P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((R \ (Q)) \/ (S)), card P, the carrier of K
( the carrier of K,M,(card ((R \ (Q)) \/ (S))),(card P),(((R \ (Q)) \/ (S))),(P)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((R \ (Q)) \/ (S)), card P, the carrier of K
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
rng (P) is finite V212() V213() V214() V217() set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q1 is finite Q1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q1 ) } is set
rng (R) is finite V212() V213() V214() V217() set
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q1 is finite Q1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q1 ) } is set
( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),(card R),(card P),(R),(P)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P, the carrier of K
Q1 is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
rng Q1 is finite V212() V213() V214() V217() set
( the carrier of K,M,(card R),(card P),Q1,(P)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P, the carrier of K
( the carrier of K,(ReplaceLine (M,Q,(Line (M,S)))),R,P) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P, the carrier of K
( the carrier of K,M,(card R),(card R),Q1,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of K
m is non empty set
m * is functional non empty FinSequence-membered FinSequenceSet of m
n is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
n @ is Relation-like NAT -defined m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of m *
Indices (n @) is set
dom (n @) is finite Element of bool NAT
width (n @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (n @)) is finite width (n @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n @) ) } is set
[:(dom (n @)),(Seg (width (n @))):] is Relation-like finite set
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:K,M:] is Relation-like finite set
[:M,K:] is Relation-like finite set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
len (n @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (n @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (n @)) is finite len (n @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (n @) ) } is set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
n @ is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
( the carrier of m,(n @),M,K) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,(n @),M,K) is Element of the carrier of m
Permutations (card M) is non empty permutational set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,(n @),M,K) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,(n @),M,K))) is Element of the carrier of m
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
( the carrier of m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
( the carrier of m,(n @),M,K) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K, the carrier of m
( the carrier of m,(n @),(card M),(card K),(M),(K)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card K, the carrier of m
( the carrier of m,(n @),M,K) @ is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
( the carrier of m,(n @),M,K) @ is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
power n is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial non finite V103() set
K is Element of the carrier of n
(power n) . (K,m) is Element of the carrier of n
M is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(m,m,n,M,K) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det (m,m,n,M,K) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product (m,m,n,M,K) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (m,m,n,M,K))) is Element of the carrier of n
Det M is Element of the carrier of n
Path_product M is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product M)) is Element of the carrier of n
((power n) . (K,m)) * (Det M) is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the multF of n . (((power n) . (K,m)),(Det M)) is Element of the carrier of n
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power n) . (K,P) is Element of the carrier of n
((power n) . (K,P)) * (Det M) is Element of the carrier of n
the multF of n . (((power n) . (K,P)),(Det M)) is Element of the carrier of n
(power n) . (K,R) is Element of the carrier of n
((power n) . (K,R)) * (Det M) is Element of the carrier of n
the multF of n . (((power n) . (K,R)),(Det M)) is Element of the carrier of n
Q is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det Q is Element of the carrier of n
Path_product Q is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product Q)) is Element of the carrier of n
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,P) is Relation-like NAT -defined the carrier of n -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of n
(width Q) -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 Q } is set
K * (Line (Q,P)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (Q,P)),(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (Q,P,(K * (Line (Q,P)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
S is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det S is Element of the carrier of n
Path_product S is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product S)) is Element of the carrier of n
0 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
K * (((power n) . (K,R)) * (Det M)) is Element of the carrier of n
the multF of n . (K,(((power n) . (K,R)) * (Det M))) is Element of the carrier of n
((power n) . (K,R)) * K is Element of the carrier of n
the multF of n . (((power n) . (K,R)),K) is Element of the carrier of n
(((power n) . (K,R)) * K) * (Det M) is Element of the carrier of n
the multF of n . ((((power n) . (K,R)) * K),(Det M)) is Element of the carrier of n
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (S,x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width S -element FinSequence-like FinSubsequence-like Element of (width S) -tuples_on the carrier of n
width S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width S) -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 S } is set
Line (M,x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
(width M) -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 M } is set
K * (Line (M,x1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
K391( the carrier of n, the carrier of n,(Line (M,x1)),(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Line (Q,x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of n
len (K * (Line (Q,P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Line (Q,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power n) . (K,R) is Element of the carrier of n
((power n) . (K,R)) * (Det M) is Element of the carrier of n
the multF of n . (((power n) . (K,R)),(Det M)) is Element of the carrier of n
P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det P is Element of the carrier of n
Path_product P is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product P)) is Element of the carrier of n
(power n) . (K,0) is Element of the carrier of n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
the OneF of n is Element of the carrier of n
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (P,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of n
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width P) -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 P } is set
Line (M,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
(width M) -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 M } is set
K * (Line (M,Q)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (M,Q)),(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det R is Element of the carrier of n
Path_product R is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product R)) is Element of the carrier of n
len (m,m,n,M,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (R,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of n
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width R) -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 R } is set
Line (M,Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
(width M) -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 M } is set
K * (Line (M,Q)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (M,Q)),(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Line ((m,m,n,M,K),Q) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (m,m,n,M,K) -element FinSequence-like FinSubsequence-like Element of (width (m,m,n,M,K)) -tuples_on the carrier of n
width (m,m,n,M,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (m,m,n,M,K)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (m,m,n,M,K) } is set
(m,m,n,M,K) . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
power m is Relation-like [: the carrier of m,NAT:] -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of m,NAT:], the carrier of m:]
[: the carrier of m,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of m,NAT:], the carrier of m:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of m,NAT:], the carrier of m:] is non empty non trivial non finite V103() set
n is Element of the carrier of m
K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
n * K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(power m) . (n,(card M)) is Element of the carrier of m
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,(n * K),M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,(n * K),M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,(n * K),M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,(n * K),M,R))) is Element of the carrier of m
( the carrier of m,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,K,M,R) is Element of the carrier of m
Path_product ( the carrier of m,K,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,K,M,R))) is Element of the carrier of m
((power m) . (n,(card M))) * (Det ( the carrier of m,K,M,R)) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the multF of m . (((power m) . (n,(card M))),(Det ( the carrier of m,K,M,R))) is Element of the carrier of m
( the carrier of m,(n * K),M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
( the carrier of m,(n * K),(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
( the carrier of m,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
( the carrier of m,K,(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
((card M),(card R),m,( the carrier of m,K,M,R),n) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
((card M),(card M),m,( the carrier of m,K,M,R),n) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() Element of bool NAT is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() Element of bool NAT
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:M,M:] is Relation-like finite set
( the carrier of m,n,M,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,M) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,M) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,M))) is Element of the carrier of m
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
card (Seg (len n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
P is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
[:P,Q:] is Relation-like finite set
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,n,P,Q) is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of m,n,P,Q) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,n,P,Q))) is Element of the carrier of m
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
P is finite without_zero Element of bool NAT
[:R,P:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,R,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of m
Det ( the carrier of m,n,R,P) is Element of the carrier of m
Permutations (card R) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card R)) is Element of Fin (Permutations (card R))
Fin (Permutations (card R)) is preBoolean set
Path_product ( the carrier of m,n,R,P) is Relation-like Permutations (card R) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card R)), the carrier of m:]
[:(Permutations (card R)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card R)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card R))),(Path_product ( the carrier of m,n,R,P))) is Element of the carrier of m
R is finite without_zero Element of bool NAT
P is finite without_zero Element of bool NAT
[:R,P:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,R,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of m
Det ( the carrier of m,n,R,P) is Element of the carrier of m
Permutations (card R) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card R)) is Element of Fin (Permutations (card R))
Fin (Permutations (card R)) is preBoolean set
Path_product ( the carrier of m,n,R,P) is Relation-like Permutations (card R) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card R)), the carrier of m:]
[:(Permutations (card R)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card R)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card R))),(Path_product ( the carrier of m,n,R,P))) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is finite without_zero Element of bool NAT
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg (width n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
card (Seg (len n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
[:(rng M),(rng R):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len P) is finite len P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len P ) } is set
[:(Seg (len P)),(Seg (width P)):] is Relation-like finite set
dom R is finite n -element Element of bool NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom M is finite m -element Element of bool NAT
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng K is finite V212() V213() V214() V217() set
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
[:(rng K),(rng M):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
R is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
Indices R is set
dom R is finite Element of bool NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width R) is finite width R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width R ) } is set
[:(dom R),(Seg (width R)):] is Relation-like finite set
( the carrier of n,R,m,m,K,M) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,R,m,m,K,M) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product ( the carrier of n,R,m,m,K,M) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,R,m,m,K,M))) is Element of the carrier of n
P is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
x1 is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
S is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng S is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
( the carrier of n,R,m,m,S,x1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Det ( the carrier of n,R,m,m,S,x1) is Element of the carrier of n
Path_product ( the carrier of n,R,m,m,S,x1) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
the addF of n $$ ((FinOmega (Permutations m)),(Path_product ( the carrier of n,R,m,m,S,x1))) is Element of the carrier of n
- (Det ( the carrier of n,R,m,m,K,M)) is Element of the carrier of n
( the carrier of n,R,(card P),(card Q),(P),(Q)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of n
( the carrier of n,R,P,Q) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of n
( the carrier of n,R,P,Q) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of n
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = K } is set
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
P is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng P is finite V212() V213() V214() V217() set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg S is finite S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= S ) } is set
Q is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng Q is finite V212() V213() V214() V217() set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg S is finite S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= S ) } is set
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
( the carrier of m,n,(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = S } is set
x1 is Relation-like NAT -defined NAT -valued Function-like finite S -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of S -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is Relation-like NAT -defined NAT -valued Function-like finite S -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of S -tuples_on NAT
rng x2 is finite V212() V213() V214() V217() set
[:(rng x1),(rng x2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
( the carrier of m,n,S,S,x1,x2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of S,S, the carrier of m
Det ( the carrier of m,n,S,S,x1,x2) is Element of the carrier of m
Permutations S is non empty permutational set
FinOmega (Permutations S) is Element of Fin (Permutations S)
Fin (Permutations S) is preBoolean set
Path_product ( the carrier of m,n,S,S,x1,x2) is Relation-like Permutations S -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations S), the carrier of m:]
[:(Permutations S), the carrier of m:] is Relation-like non empty set
bool [:(Permutations S), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations S)),(Path_product ( the carrier of m,n,S,S,x1,x2))) is Element of the carrier of m
i1 is finite without_zero Element of bool NAT
i2 is finite without_zero Element of bool NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,i1,i2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i1, card i1, the carrier of m
Det ( the carrier of m,n,i1,i2) is Element of the carrier of m
Permutations (card i1) is non empty permutational set
FinOmega (Permutations (card i1)) is Element of Fin (Permutations (card i1))
Fin (Permutations (card i1)) is preBoolean set
Path_product ( the carrier of m,n,i1,i2) is Relation-like Permutations (card i1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card i1)), the carrier of m:]
[:(Permutations (card i1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card i1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card i1))),(Path_product ( the carrier of m,n,i1,i2))) is Element of the carrier of m
M is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
[:(rng M),(rng R):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
( the carrier of m,n,K,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,K, the carrier of m
Det ( the carrier of m,n,K,K,M,R) is Element of the carrier of m
Permutations K is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations K) is Element of Fin (Permutations K)
Fin (Permutations K) is preBoolean set
Path_product ( the carrier of m,n,K,K,M,R) is Relation-like Permutations K -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of m:]
[:(Permutations K), the carrier of m:] is Relation-like non empty set
bool [:(Permutations K), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations K)),(Path_product ( the carrier of m,n,K,K,M,R))) is Element of the carrier of m
M is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of K -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
[:(rng M),(rng R):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
( the carrier of m,n,K,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,K, the carrier of m
Det ( the carrier of m,n,K,K,M,R) is Element of the carrier of m
Permutations K is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations K) is Element of Fin (Permutations K)
Fin (Permutations K) is preBoolean set
Path_product ( the carrier of m,n,K,K,M,R) is Relation-like Permutations K -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of m:]
[:(Permutations K), the carrier of m:] is Relation-like non empty set
bool [:(Permutations K), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations K)),(Path_product ( the carrier of m,n,K,K,M,R))) is Element of the carrier of m
P is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
[:P,Q:] is Relation-like finite set
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,n,P,Q) is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of m,n,P,Q) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,n,P,Q))) is Element of the carrier of m
S is finite without_zero Element of bool NAT
x1 is finite without_zero Element of bool NAT
card S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,S,x1) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card S, card S, the carrier of m
Det ( the carrier of m,n,S,x1) is Element of the carrier of m
Permutations (card S) is non empty permutational set
FinOmega (Permutations (card S)) is Element of Fin (Permutations (card S))
Fin (Permutations (card S)) is preBoolean set
Path_product ( the carrier of m,n,S,x1) is Relation-like Permutations (card S) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card S)), the carrier of m:]
[:(Permutations (card S)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card S)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card S))),(Path_product ( the carrier of m,n,S,x1))) is Element of the carrier of m
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
S is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
rng S is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
x1 is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg x2 is finite x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x2 ) } is set
( the carrier of m,n,P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of m
( the carrier of m,n,(card P),(card Q),(P),(Q)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
( the carrier of K,P,m,n,M,R) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,( the carrier of K,P,m,n,M,R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
[:(rng M),(rng R):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
( the carrier of K,P,m,n,M,R) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,( the carrier of K,P,m,n,M,R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom R is finite n -element Element of bool NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom M is finite m -element Element of bool NAT
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(K,( the carrier of K,P,m,n,M,R)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (K,( the carrier of K,P,m,n,M,R)) } is set
Indices ( the carrier of K,P,m,n,M,R) is set
dom ( the carrier of K,P,m,n,M,R) is finite Element of bool NAT
width ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,P,m,n,M,R)) is finite width ( the carrier of K,P,m,n,M,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,P,m,n,M,R) ) } is set
[:(dom ( the carrier of K,P,m,n,M,R)),(Seg (width ( the carrier of K,P,m,n,M,R))):] is Relation-like finite set
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
x1 is Relation-like NAT -defined NAT -valued Function-like finite (K,( the carrier of K,P,m,n,M,R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,( the carrier of K,P,m,n,M,R)) -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is Relation-like NAT -defined NAT -valued Function-like finite (K,( the carrier of K,P,m,n,M,R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,( the carrier of K,P,m,n,M,R)) -tuples_on NAT
rng x2 is finite V212() V213() V214() V217() set
[:(rng x1),(rng x2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)), the carrier of K
Det ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is Element of the carrier of K
Permutations (K,( the carrier of K,P,m,n,M,R)) is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
FinOmega (Permutations (K,( the carrier of K,P,m,n,M,R))) is Element of Fin (Permutations (K,( the carrier of K,P,m,n,M,R)))
Fin (Permutations (K,( the carrier of K,P,m,n,M,R))) is preBoolean set
Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is Relation-like Permutations (K,( the carrier of K,P,m,n,M,R)) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (K,( the carrier of K,P,m,n,M,R))), the carrier of K:]
[:(Permutations (K,( the carrier of K,P,m,n,M,R))), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (K,( the carrier of K,P,m,n,M,R))), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (K,( the carrier of K,P,m,n,M,R)))),(Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2))) is Element of the carrier of K
R * x2 is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite V103() set
M * x1 is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:NAT,NAT:]
rng (R * x2) is finite V212() V213() V214() V215() V217() set
len ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is finite without_zero Element of bool NAT
y2 is finite without_zero Element of bool NAT
card y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y1, card y1, the carrier of K
Det ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Element of the carrier of K
Permutations (card y1) is non empty permutational set
FinOmega (Permutations (card y1)) is Element of Fin (Permutations (card y1))
Fin (Permutations (card y1)) is preBoolean set
Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Relation-like Permutations (card y1) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card y1)), the carrier of K:]
[:(Permutations (card y1)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card y1)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card y1))),(Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2))) is Element of the carrier of K
dom (M * x1) is finite Element of bool NAT
dom x1 is finite (K,( the carrier of K,P,m,n,M,R)) -element Element of bool NAT
Seg (K,( the carrier of K,P,m,n,M,R)) is finite (K,( the carrier of K,P,m,n,M,R)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= (K,( the carrier of K,P,m,n,M,R)) ) } is set
y1 is finite without_zero Element of bool NAT
y2 is finite without_zero Element of bool NAT
card y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y1, card y1, the carrier of K
Det ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Element of the carrier of K
Permutations (card y1) is non empty permutational set
FinOmega (Permutations (card y1)) is Element of Fin (Permutations (card y1))
Fin (Permutations (card y1)) is preBoolean set
Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2) is Relation-like Permutations (card y1) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card y1)), the carrier of K:]
[:(Permutations (card y1)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card y1)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card y1))),(Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),y1,y2))) is Element of the carrier of K
dom (R * x2) is finite Element of bool NAT
dom x2 is finite (K,( the carrier of K,P,m,n,M,R)) -element Element of bool NAT
rng (M * x1) is finite V212() V213() V214() V215() V217() set
y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
len P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
len Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 is Relation-like NAT -defined NAT -valued Function-like finite (K,( the carrier of K,P,m,n,M,R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,( the carrier of K,P,m,n,M,R)) -tuples_on NAT
Q2 is Relation-like NAT -defined NAT -valued Function-like finite (K,( the carrier of K,P,m,n,M,R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,( the carrier of K,P,m,n,M,R)) -tuples_on NAT
( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)), the carrier of K
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[m,Q2i] is set
{m,Q2i} is non empty finite V37() set
{m} is non empty trivial finite V37() 1 -element set
{{m,Q2i},{m}} is non empty finite V37() without_zero V103() set
Indices ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is set
dom ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is finite Element of bool NAT
width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2)) is finite width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) ) } is set
[:(dom ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2)),(Seg (width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2))):] is Relation-like finite set
x1 . m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 . Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 . SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x2 . P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[(x1 . SQ2i),(x2 . P2m)] is set
{(x1 . SQ2i),(x2 . P2m)} is non empty finite V37() set
{(x1 . SQ2i)} is non empty trivial finite V37() 1 -element set
{{(x1 . SQ2i),(x2 . P2m)},{(x1 . SQ2i)}} is non empty finite V37() without_zero V103() set
dom P2 is finite (K,( the carrier of K,P,m,n,M,R)) -element Element of bool NAT
dom Q2 is finite (K,( the carrier of K,P,m,n,M,R)) -element Element of bool NAT
[:(dom P2),(dom Q2):] is Relation-like finite set
Indices ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) is set
dom ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) is finite Element of bool NAT
width ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2)) is finite width ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) ) } is set
[:(dom ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2)),(Seg (width ( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2))):] is Relation-like finite set
( the carrier of K,P,(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),P2,Q2) * (m,Q2i) is Element of the carrier of K
P2 . SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2 . P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((P2 . SQ2i),(Q2 . P2m)) is Element of the carrier of K
ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M . ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((M . ES),(Q2 . P2m)) is Element of the carrier of K
Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((M . ES),(R . Si)) is Element of the carrier of K
( the carrier of K,P,m,n,M,R) * ((x1 . SQ2i),(x2 . P2m)) is Element of the carrier of K
( the carrier of K,( the carrier of K,P,m,n,M,R),(K,( the carrier of K,P,m,n,M,R)),(K,( the carrier of K,P,m,n,M,R)),x1,x2) * (m,Q2i) is Element of the carrier of K
rng P2 is finite V212() V213() V214() V217() set
rng Q2 is finite V212() V213() V214() V217() set
[:(rng P2),(rng Q2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
( the carrier of m,n,(card K),(card M),(K),(M)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card M, the carrier of m
(m,( the carrier of m,n,K,M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (K) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
rng (M) is finite V212() V213() V214() V217() set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
( the carrier of m,n,K,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card R, the carrier of m
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
( the carrier of m,n,(card K),(card R),(K),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card R, the carrier of m
(m,( the carrier of m,n,K,R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is finite without_zero Element of bool NAT
( the carrier of m,n,M,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card P, the carrier of m
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
( the carrier of m,n,(card M),(card P),(M),(P)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card P, the carrier of m
(m,( the carrier of m,n,M,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices ( the carrier of m,n,K,R) is set
dom ( the carrier of m,n,K,R) is finite Element of bool NAT
width ( the carrier of m,n,K,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of m,n,K,R)) is finite width ( the carrier of m,n,K,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of m,n,K,R) ) } is set
[:(dom ( the carrier of m,n,K,R)),(Seg (width ( the carrier of m,n,K,R))):] is Relation-like finite set
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
x1 is finite without_zero Element of bool NAT
x2 is finite without_zero Element of bool NAT
[:x1,x2:] is Relation-like finite set
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,( the carrier of m,n,K,R),x1,x2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x1, the carrier of m
Det ( the carrier of m,( the carrier of m,n,K,R),x1,x2) is Element of the carrier of m
Permutations (card x1) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card x1)) is Element of Fin (Permutations (card x1))
Fin (Permutations (card x1)) is preBoolean set
Path_product ( the carrier of m,( the carrier of m,n,K,R),x1,x2) is Relation-like Permutations (card x1) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card x1)), the carrier of m:]
[:(Permutations (card x1)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card x1)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card x1))),(Path_product ( the carrier of m,( the carrier of m,n,K,R),x1,x2))) is Element of the carrier of m
(K) .: x1 is finite V212() V213() V214() V217() set
(R) .: x2 is finite V212() V213() V214() V217() set
( the carrier of m,( the carrier of m,n,K,R),x1,x2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x2, the carrier of m
(x1) is Relation-like NAT -defined NAT -valued Function-like finite card x1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x1) -tuples_on NAT
(card x1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x1 } is set
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
( the carrier of m,( the carrier of m,n,K,R),(card x1),(card x2),(x1),(x2)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x2, the carrier of m
i1 is finite without_zero Element of bool NAT
i2 is finite without_zero Element of bool NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,i1,i2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i1, card i2, the carrier of m
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
(i2) is Relation-like NAT -defined NAT -valued Function-like finite card i2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i2) -tuples_on NAT
(card i2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i2 } is set
( the carrier of m,n,(card i1),(card i2),(i1),(i2)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i1, card i2, the carrier of m
(M) " i1 is finite set
(P) " i2 is finite set
y2 is finite without_zero Element of bool NAT
card y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is finite without_zero Element of bool NAT
card y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(y2) is Relation-like NAT -defined NAT -valued Function-like finite card y2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card y2) -tuples_on NAT
(card y2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card y2 } is set
rng (y2) is finite V212() V213() V214() V217() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q is finite Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q ) } is set
(y1) is Relation-like NAT -defined NAT -valued Function-like finite card y1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card y1) -tuples_on NAT
(card y1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card y1 } is set
rng (y1) is finite V212() V213() V214() V217() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q is finite Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q ) } is set
[:y1,y2:] is Relation-like finite set
Indices ( the carrier of m,n,M,P) is set
dom ( the carrier of m,n,M,P) is finite Element of bool NAT
width ( the carrier of m,n,M,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of m,n,M,P)) is finite width ( the carrier of m,n,M,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of m,n,M,P) ) } is set
[:(dom ( the carrier of m,n,M,P)),(Seg (width ( the carrier of m,n,M,P))):] is Relation-like finite set
( the carrier of m,( the carrier of m,n,M,P),y1,y2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y1, card y2, the carrier of m
( the carrier of m,( the carrier of m,n,M,P),(card y1),(card y2),(y1),(y2)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y1, card y2, the carrier of m
( the carrier of m,( the carrier of m,n,M,P),y1,y2) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y1, card y1, the carrier of m
m is Relation-like Function-like set
rng m is set
n is Relation-like Function-like set
rng n is set
dom m is set
dom n is set
K is set
m . K is set
[:(dom m),(dom n):] is Relation-like set
bool [:(dom m),(dom n):] is non empty set
K is Relation-like dom m -defined dom n -valued Function-like quasi_total Element of bool [:(dom m),(dom n):]
{} (#) n is Relation-like non-empty empty-yielding NAT -defined RAT -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 V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() set
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() complex ext-real non positive non negative V202() V203() V204() V205() V206() V207() V208() V209() V212() V213() V214() V215() V217() set
rng K is set
dom K is Element of bool (dom m)
bool (dom m) is non empty set
K (#) n is Relation-like dom m -defined Function-like set
dom (K (#) n) is Element of bool (dom m)
M is set
m . M is set
K . M is set
n . (K . M) is set
(K (#) n) . M is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = m } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined NAT -valued Function-like finite m -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of m -tuples_on NAT
rng M is finite V212() V213() V214() V217() set
R is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of n -tuples_on NAT
rng R is finite V212() V213() V214() V217() set
[:(rng M),(rng R):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
Indices P is set
dom P is finite Element of bool NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width P) is finite width P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
(K,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,P,m,n,M,R) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,( the carrier of K,P,m,n,M,R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (K,P) } is set
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
x1 is Relation-like NAT -defined NAT -valued Function-like finite (K,P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,P) -tuples_on NAT
rng x1 is finite V212() V213() V214() V217() set
x2 is Relation-like NAT -defined NAT -valued Function-like finite (K,P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,P) -tuples_on NAT
rng x2 is finite V212() V213() V214() V217() set
[:(rng x1),(rng x2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
( the carrier of K,P,(K,P),(K,P),x1,x2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (K,P),(K,P), the carrier of K
Det ( the carrier of K,P,(K,P),(K,P),x1,x2) is Element of the carrier of K
Permutations (K,P) is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
FinOmega (Permutations (K,P)) is Element of Fin (Permutations (K,P))
Fin (Permutations (K,P)) is preBoolean set
Path_product ( the carrier of K,P,(K,P),(K,P),x1,x2) is Relation-like Permutations (K,P) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (K,P)), the carrier of K:]
[:(Permutations (K,P)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (K,P)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (K,P))),(Path_product ( the carrier of K,P,(K,P),(K,P),x1,x2))) is Element of the carrier of K
len x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom M is finite m -element Element of bool NAT
width ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom R is finite n -element Element of bool NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom x2 is finite (K,P) -element Element of bool NAT
Seg (K,P) is finite (K,P) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= (K,P) ) } is set
dom x1 is finite (K,P) -element Element of bool NAT
i2 is Relation-like Function-like set
dom i2 is set
rng i2 is set
i2 (#) M is Relation-like NAT -valued RAT -valued Function-like V202() V203() V204() V205() set
y1 is Relation-like Function-like set
dom y1 is set
rng y1 is set
y1 (#) R is Relation-like NAT -valued RAT -valued Function-like V202() V203() V204() V205() set
y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng y2 is finite set
Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng Q is finite set
P1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
len P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
len Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 is Relation-like NAT -defined NAT -valued Function-like finite (K,P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,P) -tuples_on NAT
Q2 is Relation-like NAT -defined NAT -valued Function-like finite (K,P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (K,P) -tuples_on NAT
( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (K,P),(K,P), the carrier of K
len ( the carrier of K,P,m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices ( the carrier of K,P,m,n,M,R) is set
dom ( the carrier of K,P,m,n,M,R) is finite Element of bool NAT
Seg (width ( the carrier of K,P,m,n,M,R)) is finite width ( the carrier of K,P,m,n,M,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,P,m,n,M,R) ) } is set
[:(dom ( the carrier of K,P,m,n,M,R)),(Seg (width ( the carrier of K,P,m,n,M,R))):] is Relation-like finite set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg n):] is Relation-like finite set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[m,Q2i] is set
{m,Q2i} is non empty finite V37() set
{m} is non empty trivial finite V37() 1 -element set
{{m,Q2i},{m}} is non empty finite V37() without_zero V103() set
Indices ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is set
dom ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is finite Element of bool NAT
width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2)) is finite width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) ) } is set
[:(dom ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2)),(Seg (width ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2))):] is Relation-like finite set
P2 . m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q2 . Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom P2 is finite (K,P) -element Element of bool NAT
dom Q2 is finite (K,P) -element Element of bool NAT
[:(dom P2),(dom Q2):] is Relation-like finite set
rng Q2 is finite V212() V213() V214() V217() set
rng P2 is finite V212() V213() V214() V217() set
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q2 . P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[(P2 . SQ2i),(Q2 . P2m)] is set
{(P2 . SQ2i),(Q2 . P2m)} is non empty finite V37() set
{(P2 . SQ2i)} is non empty trivial finite V37() 1 -element set
{{(P2 . SQ2i),(Q2 . P2m)},{(P2 . SQ2i)}} is non empty finite V37() without_zero V103() set
Indices ( the carrier of K,P,(K,P),(K,P),x1,x2) is set
dom ( the carrier of K,P,(K,P),(K,P),x1,x2) is finite Element of bool NAT
width ( the carrier of K,P,(K,P),(K,P),x1,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,P,(K,P),(K,P),x1,x2)) is finite width ( the carrier of K,P,(K,P),(K,P),x1,x2) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,P,(K,P),(K,P),x1,x2) ) } is set
[:(dom ( the carrier of K,P,(K,P),(K,P),x1,x2)),(Seg (width ( the carrier of K,P,(K,P),(K,P),x1,x2))):] is Relation-like finite set
( the carrier of K,P,(K,P),(K,P),x1,x2) * (m,Q2i) is Element of the carrier of K
x1 . SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 . P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((x1 . SQ2i),(x2 . P2m)) is Element of the carrier of K
ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M . ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((M . ES),(x2 . P2m)) is Element of the carrier of K
Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P * ((M . ES),(R . Si)) is Element of the carrier of K
( the carrier of K,P,m,n,M,R) * ((P2 . SQ2i),(Q2 . P2m)) is Element of the carrier of K
( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) * (m,Q2i) is Element of the carrier of K
Det ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is Element of the carrier of K
Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2) is Relation-like Permutations (K,P) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (K,P)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (K,P))),(Path_product ( the carrier of K,( the carrier of K,P,m,n,M,R),(K,P),(K,P),P2,Q2))) is Element of the carrier of K
[:(rng P2),(rng Q2):] is Relation-like RAT -valued finite V202() V203() V204() V205() set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
(n,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Det K is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product K)) is Element of the carrier of n
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,K,(Seg m),(Seg m)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg m), card (Seg m), the carrier of n
card (Seg m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg m)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg m) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg m)) -tuples_on NAT
(card (Seg m)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg m) } is set
( the carrier of n,K,(card (Seg m)),(card (Seg m)),((Seg m)),((Seg m))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg m), card (Seg m), the carrier of n
( the carrier of n,K,(Seg m),(Seg m)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg m), card (Seg m), the carrier of n
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,K,M,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of n
Det ( the carrier of n,K,M,R) is Element of the carrier of n
Permutations (card M) is non empty permutational set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of n,K,M,R) is Relation-like Permutations (card M) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of n:]
[:(Permutations (card M)), the carrier of n:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of n,K,M,R))) is Element of the carrier of n
( the carrier of n,K,M,R) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of n
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
( the carrier of n,K,(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of n
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n @ is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,(n @)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
[:M,K:] is Relation-like finite set
Indices (n @) is set
dom (n @) is finite Element of bool NAT
width (n @) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (n @)) is finite width (n @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n @) ) } is set
[:(dom (n @)),(Seg (width (n @))):] is Relation-like finite set
R is finite without_zero Element of bool NAT
P is finite without_zero Element of bool NAT
[:R,P:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,(n @),R,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of m
Det ( the carrier of m,(n @),R,P) is Element of the carrier of m
Permutations (card R) is non empty permutational set
FinOmega (Permutations (card R)) is Element of Fin (Permutations (card R))
Fin (Permutations (card R)) is preBoolean set
Path_product ( the carrier of m,(n @),R,P) is Relation-like Permutations (card R) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card R)), the carrier of m:]
[:(Permutations (card R)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card R)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card R))),(Path_product ( the carrier of m,(n @),R,P))) is Element of the carrier of m
[:P,R:] is Relation-like finite set
( the carrier of m,n,P,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,n,P,R) is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of m,n,P,R) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,n,P,R))) is Element of the carrier of m
( the carrier of m,(n @),M,K) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,(n @),M,K) is Element of the carrier of m
Permutations (card M) is non empty permutational set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,(n @),M,K) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,(n @),M,K))) is Element of the carrier of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
power n is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial non finite V103() set
K is Element of the carrier of n
(power n) . (K,m) is Element of the carrier of n
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(power n) . (K,M) is Element of the carrier of n
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power n) . (K,R) is Element of the carrier of n
((power n) . (K,M)) * K is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n . (((power n) . (K,M)),K) is Element of the carrier of n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
the OneF of n is Element of the carrier of n
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power n) . (K,M) is Element of the carrier of n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
[:(Seg m),(Seg m):] is Relation-like finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is Relation-like Seg m -defined Seg m -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg m),(Seg m):]
M * R is Relation-like NAT -defined Seg m -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,(M * R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
((Seg (len M))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len M)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len M))) -tuples_on NAT
card (Seg (len M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card (Seg (len M))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len M)) } is set
((Seg (width M))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width M)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (width M))) -tuples_on NAT
card (Seg (width M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card (Seg (width M))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width M)) } is set
Seg (card (Seg (len M))) is finite card (Seg (len M)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (len M)) ) } is set
[:(Seg (card (Seg (len M)))),(Seg (card (Seg (len M)))):] is Relation-like finite set
bool [:(Seg (card (Seg (len M)))),(Seg (card (Seg (len M)))):] is non empty finite V37() set
rng R is finite set
dom R is finite Element of bool (Seg m)
bool (Seg m) is non empty finite V37() set
dom ((Seg (len M))) is finite card (Seg (len M)) -element Element of bool NAT
((Seg (len M))) * R is Relation-like Seg m -defined NAT -valued RAT -valued Function-like finite V202() V203() V204() V205() Element of bool [:(Seg m),NAT:]
[:(Seg m),NAT:] is Relation-like set
bool [:(Seg m),NAT:] is non empty set
dom (((Seg (len M))) * R) is finite Element of bool (Seg m)
rng (((Seg (len M))) * R) is finite V212() V213() V214() V215() V217() set
rng ((Seg (len M))) is finite V212() V213() V214() V217() set
i1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V202() V203() V204() V205() FinSequence of NAT
len i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices M is set
dom M is finite Element of bool NAT
[:(dom M),(Seg (width M)):] is Relation-like finite set
[:(Seg (len M)),(Seg (width M)):] is Relation-like finite set
rng ((Seg (width M))) is finite V212() V213() V214() V217() set
y1 is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len M)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len M))) -tuples_on NAT
( the carrier of K,M,(card (Seg (len M))),(card (Seg (width M))),y1,((Seg (width M)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card (Seg (width M)), the carrier of K
x2 is Relation-like Seg (card (Seg (len M))) -defined Seg (card (Seg (len M))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (card (Seg (len M)))),(Seg (card (Seg (len M)))):]
( the carrier of K,M,(Seg (len M)),(Seg (width M))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card (Seg (width M)), the carrier of K
( the carrier of K,M,(card (Seg (len M))),(card (Seg (width M))),((Seg (len M))),((Seg (width M)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card (Seg (width M)), the carrier of K
( the carrier of K,M,(Seg (len M)),(Seg (width M))) * x2 is Relation-like NAT -defined Seg (card (Seg (len M))) -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card (Seg (width M)), the carrier of K
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Element of the carrier of m
K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n * K is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,(n * K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,K,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,K,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,K,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,K,M,R))) is Element of the carrier of m
Indices (n * K) is set
dom (n * K) is finite Element of bool NAT
width (n * K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (n * K)) is finite width (n * K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n * K) ) } is set
[:(dom (n * K)),(Seg (width (n * K))):] is Relation-like finite set
P is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
[:P,Q:] is Relation-like finite set
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,(n * K),P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,(n * K),P,Q) is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of m,(n * K),P,Q) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,(n * K),P,Q))) is Element of the carrier of m
power m is Relation-like [: the carrier of m,NAT:] -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of m,NAT:], the carrier of m:]
[: the carrier of m,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of m,NAT:], the carrier of m:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of m,NAT:], the carrier of m:] is non empty non trivial non finite V103() set
(power m) . (n,(card P)) is Element of the carrier of m
( the carrier of m,K,P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,K,P,Q) is Element of the carrier of m
Path_product ( the carrier of m,K,P,Q) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,K,P,Q))) is Element of the carrier of m
((power m) . (n,(card P))) * (Det ( the carrier of m,K,P,Q)) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the multF of m . (((power m) . (n,(card P))),(Det ( the carrier of m,K,P,Q))) is Element of the carrier of m
(power m) . (n,(card M)) is Element of the carrier of m
( the carrier of m,(n * K),M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,(n * K),M,R) is Element of the carrier of m
Path_product ( the carrier of m,(n * K),M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,(n * K),M,R))) is Element of the carrier of m
((power m) . (n,(card M))) * (Det ( the carrier of m,K,M,R)) is Element of the carrier of m
the multF of m . (((power m) . (n,(card M))),(Det ( the carrier of m,K,M,R))) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
n is Element of the carrier of m
M is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
dom K is finite Element of bool NAT
n * K is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
n multfield is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
bool [: the carrier of m, the carrier of m:] is non empty set
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
id the carrier of m is Relation-like the carrier of m -defined the carrier of m -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
the multF of m [;] (n,(id the carrier of m)) is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
K391( the carrier of m, the carrier of m,K,(n multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
n * M is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K391( the carrier of m, the carrier of m,M,(n multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
R is Relation-like Function-like set
R (#) K is Relation-like the carrier of m -valued Function-like set
rng R is set
R (#) (n * K) is Relation-like the carrier of m -valued Function-like set
len (n * K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (n * K) is finite Element of bool NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
dom (R (#) (n * K)) is set
dom R is set
len (n * M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (n * M) is finite Element of bool NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
dom M is finite Element of bool NAT
rng M is finite set
Q is set
R . Q is set
M . Q is set
K . (R . Q) is set
rng K is finite set
(n * M) . Q is set
S is Element of the carrier of m
n * S is Element of the carrier of m
the multF of m . (n,S) is Element of the carrier of m
(n * K) . (R . Q) is set
(R (#) (n * K)) . Q is set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
K is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
n is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
dom n is finite Element of bool NAT
R is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
M is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
dom M is finite Element of bool NAT
n + M is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,n,M) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K + R is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,K,R) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
P is Relation-like Function-like set
P (#) n is Relation-like the carrier of m -valued Function-like set
rng P is set
P (#) M is Relation-like the carrier of m -valued Function-like set
P (#) (n + M) is Relation-like the carrier of m -valued Function-like set
dom K is finite Element of bool NAT
dom P is set
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
dom R is finite Element of bool NAT
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len K) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = len K } is set
S is Relation-like NAT -defined the carrier of m -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of m
x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of m
S + x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,S,x1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
dom (S + x1) is finite len K -element Element of bool NAT
rng M is finite set
rng n is finite set
[:(rng n),(rng M):] is Relation-like finite set
dom the addF of m is Relation-like the carrier of m -defined the carrier of m -valued non empty Element of bool [: the carrier of m, the carrier of m:]
bool [: the carrier of m, the carrier of m:] is non empty set
dom (n + M) is finite Element of bool NAT
(dom n) /\ (dom M) is finite Element of bool NAT
rng R is finite set
rng K is finite set
i1 is set
P . i1 is set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
R . i1 is set
K . i1 is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K . i2 is set
R . i2 is set
y2 is Element of the carrier of m
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n . y1 is set
Q is Element of the carrier of m
M . y1 is set
(P (#) (n + M)) . i1 is set
(n + M) . y1 is set
y2 + Q is Element of the carrier of m
the addF of m . (y2,Q) is Element of the carrier of m
(S + x1) . i1 is set
dom (P (#) (n + M)) is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
R is Element of the carrier of M
P is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of M
(M,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (P,K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of M
(width P) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width P } is set
R * (Line (P,K)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of M
R multfield is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
bool [: the carrier of M, the carrier of M:] is non empty set
the multF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
id the carrier of M is Relation-like the carrier of M -defined the carrier of M -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
the multF of M [;] (R,(id the carrier of M)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
K391( the carrier of M, the carrier of M,(Line (P,K)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
ReplaceLine (P,K,(R * (Line (P,K)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of M
(M,(ReplaceLine (P,K,(R * (Line (P,K)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
Indices (ReplaceLine (P,K,(R * (Line (P,K))))) is set
dom (ReplaceLine (P,K,(R * (Line (P,K))))) is finite Element of bool NAT
width (ReplaceLine (P,K,(R * (Line (P,K))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (ReplaceLine (P,K,(R * (Line (P,K)))))) is finite width (ReplaceLine (P,K,(R * (Line (P,K))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (P,K,(R * (Line (P,K))))) ) } is set
[:(dom (ReplaceLine (P,K,(R * (Line (P,K)))))),(Seg (width (ReplaceLine (P,K,(R * (Line (P,K))))))):] is Relation-like finite set
x2 is finite without_zero Element of bool NAT
i1 is finite without_zero Element of bool NAT
[:x2,i1:] is Relation-like finite set
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of M,P,x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card x2, the carrier of M
Det ( the carrier of M,P,x2,i1) is Element of the carrier of M
Permutations (card x2) is non empty permutational set
the addF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
FinOmega (Permutations (card x2)) is Element of Fin (Permutations (card x2))
Fin (Permutations (card x2)) is preBoolean set
Path_product ( the carrier of M,P,x2,i1) is Relation-like Permutations (card x2) -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card x2)), the carrier of M:]
[:(Permutations (card x2)), the carrier of M:] is Relation-like non empty set
bool [:(Permutations (card x2)), the carrier of M:] is non empty set
the addF of M $$ ((FinOmega (Permutations (card x2))),(Path_product ( the carrier of M,P,x2,i1))) is Element of the carrier of M
len (Line (P,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (P,K)) is finite width P -element Element of bool NAT
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
rng (i1) is finite V212() V213() V214() V217() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
dom (x2) is finite card x2 -element Element of bool NAT
Seg (card x2) is finite card x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card x2 ) } is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
rng (x2) is finite V212() V213() V214() V217() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i2 is finite i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i2 ) } is set
i2 is set
(x2) . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
( the carrier of M,P,x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,P,(card x2),(card i1),(x2),(i1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of M,P,x2,i1),y1) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,x2,i1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M
width ( the carrier of M,P,x2,i1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width ( the carrier of M,P,x2,i1) } is set
(Line (P,K)) * (i1) is Relation-like NAT -defined the carrier of M -valued Function-like finite Element of bool [:NAT, the carrier of M:]
[:NAT, the carrier of M:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of M:] is non empty non trivial non finite V103() set
R * (Line (( the carrier of M,P,x2,i1),y1)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,x2,i1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M
K391( the carrier of M, the carrier of M,(Line (( the carrier of M,P,x2,i1),y1)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(R * (Line (P,K))) * (i1) is Relation-like NAT -defined the carrier of M -valued Function-like finite Element of bool [:NAT, the carrier of M:]
len (R * (Line (P,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width ( the carrier of M,P,x2,i1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of M,P,x2,i1),y1) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,x2,i1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M
(width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width ( the carrier of M,P,x2,i1) } is set
R * (Line (( the carrier of M,P,x2,i1),y1)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,x2,i1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,x2,i1)) -tuples_on the carrier of M
K391( the carrier of M, the carrier of M,(Line (( the carrier of M,P,x2,i1),y1)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
ReplaceLine (( the carrier of M,P,x2,i1),y1,(R * (Line (( the carrier of M,P,x2,i1),y1)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card x2, the carrier of M
ReplaceLine (( the carrier of M,P,x2,i1),y1,(R * (Line (( the carrier of M,P,x2,i1),y1)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
ReplaceLine (( the carrier of M,P,x2,i1),y1,(R * (Line (( the carrier of M,P,x2,i1),y1)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),(card x2),(card i1),(x2),(i1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card x2, the carrier of M
Det ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Element of the carrier of M
Path_product ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Relation-like Permutations (card x2) -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card x2)), the carrier of M:]
the addF of M $$ ((FinOmega (Permutations (card x2))),(Path_product ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1))) is Element of the carrier of M
R * (Det ( the carrier of M,P,x2,i1)) is Element of the carrier of M
the multF of M . (R,(Det ( the carrier of M,P,x2,i1))) is Element of the carrier of M
( the carrier of M,P,x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
(i1) is Relation-like NAT -defined NAT -valued Function-like finite card i1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i1) -tuples_on NAT
(card i1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i1 } is set
( the carrier of M,P,(card x2),(card i1),(x2),(i1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),(card x2),(card i1),(x2),(i1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card i1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),x2,i1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x2, card x2, the carrier of M
i2 is finite without_zero Element of bool NAT
y1 is finite without_zero Element of bool NAT
[:i2,y1:] is Relation-like finite set
card i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card i2, the carrier of M
Det ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1) is Element of the carrier of M
Permutations (card i2) is non empty permutational set
FinOmega (Permutations (card i2)) is Element of Fin (Permutations (card i2))
Fin (Permutations (card i2)) is preBoolean set
Path_product ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1) is Relation-like Permutations (card i2) -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card i2)), the carrier of M:]
[:(Permutations (card i2)), the carrier of M:] is Relation-like non empty set
bool [:(Permutations (card i2)), the carrier of M:] is non empty set
the addF of M $$ ((FinOmega (Permutations (card i2))),(Path_product ( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1))) is Element of the carrier of M
len (Line (P,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (P,K)) is finite width P -element Element of bool NAT
(y1) is Relation-like NAT -defined NAT -valued Function-like finite card y1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card y1) -tuples_on NAT
(card y1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card y1 } is set
rng (y1) is finite V212() V213() V214() V217() set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y2 is finite y2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y2 ) } is set
(i2) is Relation-like NAT -defined NAT -valued Function-like finite card i2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i2) -tuples_on NAT
(card i2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i2 } is set
dom (i2) is finite card i2 -element Element of bool NAT
Seg (card i2) is finite card i2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card i2 ) } is set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y2 is finite y2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y2 ) } is set
rng (i2) is finite V212() V213() V214() V217() set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg y2 is finite y2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= y2 ) } is set
y2 is set
(i2) . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
( the carrier of M,P,i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
( the carrier of M,P,(card i2),(card y1),(i2),(y1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of M,P,i2,y1),Q) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,i2,y1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M
width ( the carrier of M,P,i2,y1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width ( the carrier of M,P,i2,y1) } is set
(Line (P,K)) * (y1) is Relation-like NAT -defined the carrier of M -valued Function-like finite Element of bool [:NAT, the carrier of M:]
[:NAT, the carrier of M:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of M:] is non empty non trivial non finite V103() set
R * (Line (( the carrier of M,P,i2,y1),Q)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,i2,y1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M
K391( the carrier of M, the carrier of M,(Line (( the carrier of M,P,i2,y1),Q)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(R * (Line (P,K))) * (y1) is Relation-like NAT -defined the carrier of M -valued Function-like finite Element of bool [:NAT, the carrier of M:]
len (R * (Line (P,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of M,P,i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card i2, the carrier of M
width ( the carrier of M,P,i2,y1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of M,P,i2,y1),Q) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,i2,y1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M
(width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width ( the carrier of M,P,i2,y1) } is set
R * (Line (( the carrier of M,P,i2,y1),Q)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width ( the carrier of M,P,i2,y1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of M,P,i2,y1)) -tuples_on the carrier of M
K391( the carrier of M, the carrier of M,(Line (( the carrier of M,P,i2,y1),Q)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
ReplaceLine (( the carrier of M,P,i2,y1),Q,(R * (Line (( the carrier of M,P,i2,y1),Q)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card i2, the carrier of M
ReplaceLine (( the carrier of M,P,i2,y1),Q,(R * (Line (( the carrier of M,P,i2,y1),Q)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
ReplaceLine (( the carrier of M,P,i2,y1),Q,(R * (Line (( the carrier of M,P,i2,y1),Q)))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),(card i2),(card y1),(i2),(y1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
Det ( the carrier of M,P,i2,y1) is Element of the carrier of M
Path_product ( the carrier of M,P,i2,y1) is Relation-like Permutations (card i2) -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card i2)), the carrier of M:]
the addF of M $$ ((FinOmega (Permutations (card i2))),(Path_product ( the carrier of M,P,i2,y1))) is Element of the carrier of M
R * (Det ( the carrier of M,P,i2,y1)) is Element of the carrier of M
the multF of M . (R,(Det ( the carrier of M,P,i2,y1))) is Element of the carrier of M
( the carrier of M,P,i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card i2, the carrier of M
( the carrier of M,P,i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
(i2) is Relation-like NAT -defined NAT -valued Function-like finite card i2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card i2) -tuples_on NAT
(card i2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card i2 } is set
(y1) is Relation-like NAT -defined NAT -valued Function-like finite card y1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card y1) -tuples_on NAT
(card y1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card y1 } is set
( the carrier of M,P,(card i2),(card y1),(i2),(y1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),i2,y1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
( the carrier of M,(ReplaceLine (P,K,(R * (Line (P,K))))),(card i2),(card y1),(i2),(y1)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card i2, card y1, the carrier of M
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
Line (K,m) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is non empty finite V37() set
DelLine (K,m) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
(n,(DelLine (K,m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(n,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices K is set
dom K is finite Element of bool NAT
[:(dom K),(Seg (width K)):] is Relation-like finite set
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
[:(Seg (len K)),(Seg (width K)):] is Relation-like finite set
(m) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (len K)) \ (m) is finite without_zero Element of bool NAT
( the carrier of n,K,((Seg (len K)) \ (m)),(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (len K)) \ (m)), card (Seg (width K)), the carrier of n
card ((Seg (len K)) \ (m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg (len K)) \ (m))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (len K)) \ (m)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (len K)) \ (m))) -tuples_on NAT
(card ((Seg (len K)) \ (m))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (len K)) \ (m)) } is set
((Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
( the carrier of n,K,(card ((Seg (len K)) \ (m))),(card (Seg (width K))),(((Seg (len K)) \ (m))),((Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((Seg (len K)) \ (m)), card (Seg (width K)), the carrier of n
Indices (DelLine (K,m)) is set
dom (DelLine (K,m)) is finite Element of bool NAT
width (DelLine (K,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (DelLine (K,m))) is finite width (DelLine (K,m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (DelLine (K,m)) ) } is set
[:(dom (DelLine (K,m))),(Seg (width (DelLine (K,m)))):] is Relation-like finite set
R is finite without_zero Element of bool NAT
P is finite without_zero Element of bool NAT
[:R,P:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,(DelLine (K,m)),R,P) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card R, the carrier of n
Det ( the carrier of n,(DelLine (K,m)),R,P) is Element of the carrier of n
Permutations (card R) is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations (card R)) is Element of Fin (Permutations (card R))
Fin (Permutations (card R)) is preBoolean set
Path_product ( the carrier of n,(DelLine (K,m)),R,P) is Relation-like Permutations (card R) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card R)), the carrier of n:]
[:(Permutations (card R)), the carrier of n:] is Relation-like non empty set
bool [:(Permutations (card R)), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations (card R))),(Path_product ( the carrier of n,(DelLine (K,m)),R,P))) is Element of the carrier of n
( the carrier of n,(DelLine (K,m)),R,P) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P, the carrier of n
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
( the carrier of n,(DelLine (K,m)),(card R),(card P),(R),(P)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card R, card P, the carrier of n
(n,( the carrier of n,(DelLine (K,m)),R,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg (len K)) \ (m))) .: R is finite V212() V213() V214() V217() set
((Seg (width K))) .: P is finite V212() V213() V214() V217() set
Q is finite without_zero Element of bool NAT
S is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,K,Q,S) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card S, the carrier of n
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
(S) is Relation-like NAT -defined NAT -valued Function-like finite card S -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card S) -tuples_on NAT
(card S) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card S } is set
( the carrier of n,K,(card Q),(card S),(Q),(S)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card S, the carrier of n
[:Q,S:] is Relation-like finite set
x1 is finite without_zero Element of bool NAT
x2 is finite without_zero Element of bool NAT
[:x1,x2:] is Relation-like finite set
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,K,x1,x2) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x1, the carrier of n
Det ( the carrier of n,K,x1,x2) is Element of the carrier of n
Permutations (card x1) is non empty permutational set
FinOmega (Permutations (card x1)) is Element of Fin (Permutations (card x1))
Fin (Permutations (card x1)) is preBoolean set
Path_product ( the carrier of n,K,x1,x2) is Relation-like Permutations (card x1) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card x1)), the carrier of n:]
[:(Permutations (card x1)), the carrier of n:] is Relation-like non empty set
bool [:(Permutations (card x1)), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations (card x1))),(Path_product ( the carrier of n,K,x1,x2))) is Element of the carrier of n
( the carrier of n,K,x1,x2) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x2, the carrier of n
(x1) is Relation-like NAT -defined NAT -valued Function-like finite card x1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x1) -tuples_on NAT
(card x1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x1 } is set
(x2) is Relation-like NAT -defined NAT -valued Function-like finite card x2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card x2) -tuples_on NAT
(card x2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x2 } is set
( the carrier of n,K,(card x1),(card x2),(x1),(x2)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card x1, card x2, the carrier of n
(n,( the carrier of n,K,x1,x2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (m) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
i1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(i1) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
card (i1) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
i2 is finite without_zero Element of bool NAT
card i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of n,K,(i1),i2) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (i1), card (i1), the carrier of n
Det ( the carrier of n,K,(i1),i2) is Element of the carrier of n
Permutations (card (i1)) is non empty permutational set
FinOmega (Permutations (card (i1))) is Element of Fin (Permutations (card (i1)))
Fin (Permutations (card (i1))) is preBoolean set
Path_product ( the carrier of n,K,(i1),i2) is Relation-like Permutations (card (i1)) -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (i1))), the carrier of n:]
[:(Permutations (card (i1))), the carrier of n:] is Relation-like non empty set
bool [:(Permutations (card (i1))), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations (card (i1)))),(Path_product ( the carrier of n,K,(i1),i2))) is Element of the carrier of n
y1 is set
{y1} is non empty trivial finite 1 -element set
y2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
K * (i1,y2) is Element of the carrier of n
(Line (K,m)) . y2 is set
(y2) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
( the carrier of n,K,(i1),(y2)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (i1), card (y2), the carrier of n
card (y2) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
((i1)) is Relation-like NAT -defined NAT -valued Function-like finite card (i1) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (i1)) -tuples_on NAT
(card (i1)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (i1) } is set
((y2)) is Relation-like NAT -defined NAT -valued Function-like finite card (y2) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (y2)) -tuples_on NAT
(card (y2)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (y2) } is set
( the carrier of n,K,(card (i1)),(card (y2)),((i1)),((y2))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (i1), card (y2), the carrier of n
<*(0. n)*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
1 -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 = 1 } is set
[1,(0. n)] is set
{1,(0. n)} is non empty finite set
{{1,(0. n)},{1}} is non empty finite V37() without_zero V103() set
{[1,(0. n)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(0. n)*>*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(0. n)*>, the carrier of n
len <*(0. n)*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(0. n)*>] is set
{1,<*(0. n)*>} is non empty finite V37() without_zero V103() set
{{1,<*(0. n)*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(0. n)*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
x1 \ (m) is finite without_zero Element of bool NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
R is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of M
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
DelLine (R,K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
(M,(DelLine (R,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(0. M) * P is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(0. M) multfield is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
bool [: the carrier of M, the carrier of M:] is non empty set
the multF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
id the carrier of M is Relation-like the carrier of M -defined the carrier of M -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
the multF of M [;] ((0. M),(id the carrier of M)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
K391( the carrier of M, the carrier of M,P,((0. M) multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
ReplaceLine (R,K,((0. M) * P)) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of M
(M,(ReplaceLine (R,K,((0. M) * P)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
dom R is finite Element of bool NAT
(len P) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = len P } is set
len ((0. M) * P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line ((ReplaceLine (R,K,((0. M) * P))),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (ReplaceLine (R,K,((0. M) * P))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,K,((0. M) * P)))) -tuples_on the carrier of M
width (ReplaceLine (R,K,((0. M) * P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine (R,K,((0. M) * P)))) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width (ReplaceLine (R,K,((0. M) * P))) } is set
(len P) |-> (0. M) is Relation-like NAT -defined the carrier of M -valued Function-like finite len P -element FinSequence-like FinSubsequence-like Element of (len P) -tuples_on the carrier of M
Seg (len P) is finite len P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len P ) } is set
(Seg (len P)) --> (0. M) is Relation-like Seg (len P) -defined Seg (len P) -defined the carrier of M -valued {(0. M)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len P)),{(0. M)}:]
{(0. M)} is non empty trivial finite 1 -element set
[:(Seg (len P)),{(0. M)}:] is Relation-like finite set
bool [:(Seg (len P)),{(0. M)}:] is non empty finite V37() set
x1 is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M *
Replace (R,K,x1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of M *
Replace ((ReplaceLine (R,K,((0. M) * P))),K,x1) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of M *
DelLine ((ReplaceLine (R,K,((0. M) * P))),K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of M *
(M,(DelLine ((ReplaceLine (R,K,((0. M) * P))),K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of R is non empty non trivial V103() set
the carrier of R * is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
1_ R is Element of the carrier of R
1. R is non zero Element of the carrier of R
the OneF of R is Element of the carrier of R
- (1_ R) is Element of the carrier of R
P is Element of the carrier of R
Q is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
len Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len Q) is finite len Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len Q ) } is set
(R,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,M) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
(width Q) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width Q } is set
Line (Q,K) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
P * (Line (Q,K)) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
P multfield is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
[: the carrier of R, the carrier of R:] is Relation-like non empty set
bool [: the carrier of R, the carrier of R:] is non empty set
the multF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[:[: the carrier of R, the carrier of R:], the carrier of R:] is Relation-like non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
id the carrier of R is Relation-like the carrier of R -defined the carrier of R -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
the multF of R [;] (P,(id the carrier of R)) is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
K391( the carrier of R, the carrier of R,(Line (Q,K)),(P multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
(Line (Q,M)) + (P * (Line (Q,K))) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
the addF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
K388( the carrier of R, the carrier of R, the carrier of R, the addF of R,(Line (Q,M)),(P * (Line (Q,K)))) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
(R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Line (Q,M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (Q,M)) is finite width Q -element Element of bool NAT
Seg (width Q) is finite width Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width Q ) } is set
len ((Line (Q,M)) + (P * (Line (Q,K)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
ReplaceLine ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M,(Line (Q,M))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
y1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R *
Replace ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Q is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R *
Replace (Q,M,Q) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace ((Replace (Q,M,Q)),M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace (Q,M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
ReplaceLine (Q,M,(Line (Q,M))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
Line ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M) is Relation-like NAT -defined the carrier of R -valued Function-like finite width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))))) -tuples_on the carrier of R
(width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))))) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) } is set
len (Line (Q,K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Line (Q,K)) is finite width Q -element Element of bool NAT
len (P * (Line (Q,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (P * (Line (Q,K))) is finite width Q -element Element of bool NAT
Indices (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) is set
dom (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) is finite Element of bool NAT
Seg (width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))))) is finite width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))) ) } is set
[:(dom (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K))))))),(Seg (width (ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))))):] is Relation-like finite set
Indices Q is set
dom Q is finite Element of bool NAT
[:(dom Q),(Seg (width Q)):] is Relation-like finite set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
P1 is finite without_zero Element of bool NAT
Q1 is finite without_zero Element of bool NAT
[:P1,Q1:] is Relation-like finite set
card P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of R,Q,P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
Det ( the carrier of R,Q,P1,Q1) is Element of the carrier of R
Permutations (card P1) is non empty permutational set
FinOmega (Permutations (card P1)) is Element of Fin (Permutations (card P1))
Fin (Permutations (card P1)) is preBoolean set
Path_product ( the carrier of R,Q,P1,Q1) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
[:(Permutations (card P1)), the carrier of R:] is Relation-like non empty set
bool [:(Permutations (card P1)), the carrier of R:] is non empty set
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
( the carrier of R,Q,P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
(P1) is Relation-like NAT -defined NAT -valued Function-like finite card P1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P1) -tuples_on NAT
(card P1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P1 } is set
(Q1) is Relation-like NAT -defined NAT -valued Function-like finite card Q1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q1) -tuples_on NAT
(card Q1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q1 } is set
( the carrier of R,Q,(card P1),(card Q1),(P1),(Q1)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),(card P1),(card Q1),(P1),(Q1)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
ReplaceLine ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M,(Line (Q,K))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
y2 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R *
Replace ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M,y2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace ((Replace (Q,M,Q)),M,y2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace (Q,M,y2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
ReplaceLine (Q,M,(Line (Q,K))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
rng (Line (Q,K)) is finite set
dom (P1) is finite card P1 -element Element of bool NAT
Seg (card P1) is finite card P1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P1 ) } is set
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg i is finite i -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
dom (Q1) is finite card Q1 -element Element of bool NAT
Seg (card Q1) is finite card Q1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q1 ) } is set
rng (Q1) is finite V212() V213() V214() V217() set
(Line (Q,K)) * (Q1) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of R:] is non empty non trivial non finite V103() set
dom ((Line (Q,K)) * (Q1)) is finite Element of bool NAT
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng i is finite set
m is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (P1) is finite V212() V213() V214() V217() set
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q2i is finite Q2i -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q2i ) } is set
Q2i is set
(P1) . Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of R,Q,P1,Q1),SQ2i) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,Q,P1,Q1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,Q,P1,Q1)) -tuples_on the carrier of R
width ( the carrier of R,Q,P1,Q1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of R,Q,P1,Q1)) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width ( the carrier of R,Q,P1,Q1) } is set
len (Line (( the carrier of R,Q,P1,Q1),SQ2i)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (Q,M)) * (Q1) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
Line (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1)) -tuples_on the carrier of R
width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1)) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) } is set
(Line ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M)) * (Q1) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),(card P1),(card Q1),(P1),(Q1)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of R
P * m is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
K391( the carrier of R, the carrier of R,m,(P multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
(P * (Line (Q,K))) * (Q1) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
len (P * m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
(Line (( the carrier of R,Q,P1,Q1),SQ2i)) + (P * m) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
K388( the carrier of R, the carrier of R, the carrier of R, the addF of R,(Line (( the carrier of R,Q,P1,Q1),SQ2i)),(P * m)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,((Line (( the carrier of R,Q,P1,Q1),SQ2i)) + (P * m))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
Det ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) is Element of the carrier of R
Path_product ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1))) is Element of the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i)))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i)))) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i)))))) is Element of the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m))) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m))))) is Element of the carrier of R
(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i))))) + (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m)))) is Element of the carrier of R
the addF of R . ((Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i))))),(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(P * m))))) is Element of the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m)) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m)) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m)))) is Element of the carrier of R
P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m))) is Element of the carrier of R
the multF of R . (P,(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m)))) is Element of the carrier of R
(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i))))) + (P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m)))) is Element of the carrier of R
the addF of R . ((Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),SQ2i))))),(P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P1,Q1),SQ2i,m))))) is Element of the carrier of R
Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1) is Element of the carrier of R
Path_product ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1))) is Element of the carrier of R
P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1)) is Element of the carrier of R
the multF of R . (P,(Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1))) is Element of the carrier of R
(Det ( the carrier of R,Q,P1,Q1)) + (P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1))) is Element of the carrier of R
the addF of R . ((Det ( the carrier of R,Q,P1,Q1)),(P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P1,Q1)))) is Element of the carrier of R
P * (Det ( the carrier of R,Q,P1,Q1)) is Element of the carrier of R
the multF of R . (P,(Det ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
(Det ( the carrier of R,Q,P1,Q1)) + (P * (Det ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
the addF of R . ((Det ( the carrier of R,Q,P1,Q1)),(P * (Det ( the carrier of R,Q,P1,Q1)))) is Element of the carrier of R
(1_ R) * (Det ( the carrier of R,Q,P1,Q1)) is Element of the carrier of R
the multF of R . ((1_ R),(Det ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
((1_ R) * (Det ( the carrier of R,Q,P1,Q1))) + (P * (Det ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
the addF of R . (((1_ R) * (Det ( the carrier of R,Q,P1,Q1))),(P * (Det ( the carrier of R,Q,P1,Q1)))) is Element of the carrier of R
(1_ R) + P is Element of the carrier of R
the addF of R . ((1_ R),P) is Element of the carrier of R
((1_ R) + P) * (Det ( the carrier of R,Q,P1,Q1)) is Element of the carrier of R
the multF of R . (((1_ R) + P),(Det ( the carrier of R,Q,P1,Q1))) is Element of the carrier of R
(0. R) - (1_ R) is Element of the carrier of R
(0. R) + (- (1_ R)) is Element of the carrier of R
the addF of R . ((0. R),(- (1_ R))) is Element of the carrier of R
(0. R) + (- (1_ R)) is Element of the carrier of R
ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg ES is finite ES -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= ES ) } is set
P2m is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
ES is set
(P1) . ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of R,Q,P1,Q1),Si) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,Q,P1,Q1) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,Q,P1,Q1)) -tuples_on the carrier of R
ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of R
Det (ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si)))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si)))) is Relation-like Permutations (card P1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P1)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P1))),(Path_product (ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si)))))) is Element of the carrier of R
P * (Det (ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si))))) is Element of the carrier of R
the multF of R . (P,(Det (ReplaceLine (( the carrier of R,Q,P1,Q1),SQ2i,(Line (( the carrier of R,Q,P1,Q1),Si)))))) is Element of the carrier of R
(M) is non empty trivial finite V37() 1 -element Element of bool NAT
P1 \ (M) is finite without_zero Element of bool NAT
P2m is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(P2m) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
(P1 \ (M)) \/ (P2m) is non empty finite without_zero V103() Element of bool NAT
[:((P1 \ (M)) \/ (P2m)),Q1:] is Relation-like finite set
card ((P1 \ (M)) \/ (P2m)) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card ((P1 \ (M)) \/ (P2m)), the carrier of R
Det ( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1) is Element of the carrier of R
Permutations (card ((P1 \ (M)) \/ (P2m))) is non empty permutational set
FinOmega (Permutations (card ((P1 \ (M)) \/ (P2m)))) is Element of Fin (Permutations (card ((P1 \ (M)) \/ (P2m))))
Fin (Permutations (card ((P1 \ (M)) \/ (P2m)))) is preBoolean set
Path_product ( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1) is Relation-like Permutations (card ((P1 \ (M)) \/ (P2m))) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card ((P1 \ (M)) \/ (P2m)))), the carrier of R:]
[:(Permutations (card ((P1 \ (M)) \/ (P2m)))), the carrier of R:] is Relation-like non empty set
bool [:(Permutations (card ((P1 \ (M)) \/ (P2m)))), the carrier of R:] is non empty set
the addF of R $$ ((FinOmega (Permutations (card ((P1 \ (M)) \/ (P2m))))),(Path_product ( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1))) is Element of the carrier of R
- (Det ( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1)) is Element of the carrier of R
(K) is non empty trivial finite V37() 1 -element Element of bool NAT
( the carrier of R,Q,((P1 \ (M)) \/ (P2m)),Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card Q1, the carrier of R
(((P1 \ (M)) \/ (P2m))) is Relation-like NAT -defined NAT -valued Function-like finite card ((P1 \ (M)) \/ (P2m)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((P1 \ (M)) \/ (P2m))) -tuples_on NAT
(card ((P1 \ (M)) \/ (P2m))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((P1 \ (M)) \/ (P2m)) } is set
( the carrier of R,Q,(card ((P1 \ (M)) \/ (P2m))),(card Q1),(((P1 \ (M)) \/ (P2m))),(Q1)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),((P1 \ (M)) \/ (P2m)),Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),(card ((P1 \ (M)) \/ (P2m))),(card Q1),(((P1 \ (M)) \/ (P2m))),(Q1)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card Q1, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),((P1 \ (M)) \/ (P2m)),Q1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P1 \ (M)) \/ (P2m)), card ((P1 \ (M)) \/ (P2m)), the carrier of R
P2 is finite without_zero Element of bool NAT
Q2 is finite without_zero Element of bool NAT
[:P2,Q2:] is Relation-like finite set
card P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
Det ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) is Element of the carrier of R
Permutations (card P2) is non empty permutational set
FinOmega (Permutations (card P2)) is Element of Fin (Permutations (card P2))
Fin (Permutations (card P2)) is preBoolean set
Path_product ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
[:(Permutations (card P2)), the carrier of R:] is Relation-like non empty set
bool [:(Permutations (card P2)), the carrier of R:] is non empty set
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2))) is Element of the carrier of R
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
(P2) is Relation-like NAT -defined NAT -valued Function-like finite card P2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P2) -tuples_on NAT
(card P2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P2 } is set
(Q2) is Relation-like NAT -defined NAT -valued Function-like finite card Q2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q2) -tuples_on NAT
(card Q2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q2 } is set
( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),(card P2),(card Q2),(P2),(Q2)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
( the carrier of R,Q,P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
( the carrier of R,Q,P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
( the carrier of R,Q,(card P2),(card Q2),(P2),(Q2)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
rng (Line (Q,K)) is finite set
dom (P2) is finite card P2 -element Element of bool NAT
Seg (card P2) is finite card P2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P2 ) } is set
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Q2i is finite Q2i -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Q2i ) } is set
dom (Q2) is finite card Q2 -element Element of bool NAT
Seg (card Q2) is finite card Q2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q2 ) } is set
rng (Q2) is finite V212() V213() V214() V217() set
(Line (Q,K)) * (Q2) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of R:] is non empty non trivial non finite V103() set
dom ((Line (Q,K)) * (Q2)) is finite Element of bool NAT
Q2i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng Q2i is finite set
SQ2i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (P2) is finite V212() V213() V214() V217() set
P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg P2m is finite P2m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P2m ) } is set
P2m is set
(P2) . P2m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of R,Q,P2,Q2),ES) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,Q,P2,Q2) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,Q,P2,Q2)) -tuples_on the carrier of R
width ( the carrier of R,Q,P2,Q2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of R,Q,P2,Q2)) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width ( the carrier of R,Q,P2,Q2) } is set
len (Line (( the carrier of R,Q,P2,Q2),ES)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (Q,M)) * (Q2) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
Line (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2)) -tuples_on the carrier of R
width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2)) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width ( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2) } is set
(Line ((ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),M)) * (Q2) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),(card P2),(card Q2),(P2),(Q2)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of R
P * SQ2i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
K391( the carrier of R, the carrier of R,SQ2i,(P multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
(P * (Line (Q,K))) * (Q2) is Relation-like NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
len (P * SQ2i) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
(Line (( the carrier of R,Q,P2,Q2),ES)) + (P * SQ2i) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
K388( the carrier of R, the carrier of R, the carrier of R, the addF of R,(Line (( the carrier of R,Q,P2,Q2),ES)),(P * SQ2i)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,((Line (( the carrier of R,Q,P2,Q2),ES)) + (P * SQ2i))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES)))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES)))) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES)))))) is Element of the carrier of R
ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i)) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i))) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i))))) is Element of the carrier of R
(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES))))) + (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i)))) is Element of the carrier of R
the addF of R . ((Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES))))),(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(P * SQ2i))))) is Element of the carrier of R
Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i)) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i)) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i)))) is Element of the carrier of R
P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i))) is Element of the carrier of R
the multF of R . (P,(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i)))) is Element of the carrier of R
(Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES))))) + (P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i)))) is Element of the carrier of R
the addF of R . ((Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),ES))))),(P * (Det (ReplaceLine (( the carrier of R,(ReplaceLine (Q,M,((Line (Q,M)) + (P * (Line (Q,K)))))),P2,Q2),ES,SQ2i))))) is Element of the carrier of R
Det ( the carrier of R,Q,P2,Q2) is Element of the carrier of R
Path_product ( the carrier of R,Q,P2,Q2) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2) is Element of the carrier of R
Path_product ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2))) is Element of the carrier of R
P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2)) is Element of the carrier of R
the multF of R . (P,(Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2))) is Element of the carrier of R
(Det ( the carrier of R,Q,P2,Q2)) + (P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2))) is Element of the carrier of R
the addF of R . ((Det ( the carrier of R,Q,P2,Q2)),(P * (Det ( the carrier of R,(ReplaceLine (Q,M,(Line (Q,K)))),P2,Q2)))) is Element of the carrier of R
P * (Det ( the carrier of R,Q,P2,Q2)) is Element of the carrier of R
the multF of R . (P,(Det ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
(Det ( the carrier of R,Q,P2,Q2)) + (P * (Det ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
the addF of R . ((Det ( the carrier of R,Q,P2,Q2)),(P * (Det ( the carrier of R,Q,P2,Q2)))) is Element of the carrier of R
(1_ R) * (Det ( the carrier of R,Q,P2,Q2)) is Element of the carrier of R
the multF of R . ((1_ R),(Det ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
((1_ R) * (Det ( the carrier of R,Q,P2,Q2))) + (P * (Det ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
the addF of R . (((1_ R) * (Det ( the carrier of R,Q,P2,Q2))),(P * (Det ( the carrier of R,Q,P2,Q2)))) is Element of the carrier of R
(1_ R) + P is Element of the carrier of R
the addF of R . ((1_ R),P) is Element of the carrier of R
((1_ R) + P) * (Det ( the carrier of R,Q,P2,Q2)) is Element of the carrier of R
the multF of R . (((1_ R) + P),(Det ( the carrier of R,Q,P2,Q2))) is Element of the carrier of R
Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg Si is finite Si -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= Si ) } is set
Si is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Si is set
(P2) . Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
LC is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of R,Q,P2,Q2),LC) is Relation-like NAT -defined the carrier of R -valued Function-like finite width ( the carrier of R,Q,P2,Q2) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of R,Q,P2,Q2)) -tuples_on the carrier of R
ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of R
Det (ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC)))) is Element of the carrier of R
Path_product (ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC)))) is Relation-like Permutations (card P2) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P2)), the carrier of R:]
the addF of R $$ ((FinOmega (Permutations (card P2))),(Path_product (ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC)))))) is Element of the carrier of R
P * (Det (ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC))))) is Element of the carrier of R
the multF of R . (P,(Det (ReplaceLine (( the carrier of R,Q,P2,Q2),ES,(Line (( the carrier of R,Q,P2,Q2),LC)))))) is Element of the carrier of R
(M) is non empty trivial finite V37() 1 -element Element of bool NAT
P2 \ (M) is finite without_zero Element of bool NAT
Si is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(Si) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
(P2 \ (M)) \/ (Si) is non empty finite without_zero V103() Element of bool NAT
card ((P2 \ (M)) \/ (Si)) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
( the carrier of R,Q,((P2 \ (M)) \/ (Si)),Q2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P2 \ (M)) \/ (Si)), card ((P2 \ (M)) \/ (Si)), the carrier of R
Det ( the carrier of R,Q,((P2 \ (M)) \/ (Si)),Q2) is Element of the carrier of R
Permutations (card ((P2 \ (M)) \/ (Si))) is non empty permutational set
FinOmega (Permutations (card ((P2 \ (M)) \/ (Si)))) is Element of Fin (Permutations (card ((P2 \ (M)) \/ (Si))))
Fin (Permutations (card ((P2 \ (M)) \/ (Si)))) is preBoolean set
Path_product ( the carrier of R,Q,((P2 \ (M)) \/ (Si)),Q2) is Relation-like Permutations (card ((P2 \ (M)) \/ (Si))) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card ((P2 \ (M)) \/ (Si)))), the carrier of R:]
[:(Permutations (card ((P2 \ (M)) \/ (Si)))), the carrier of R:] is Relation-like non empty set
bool [:(Permutations (card ((P2 \ (M)) \/ (Si)))), the carrier of R:] is non empty set
the addF of R $$ ((FinOmega (Permutations (card ((P2 \ (M)) \/ (Si))))),(Path_product ( the carrier of R,Q,((P2 \ (M)) \/ (Si)),Q2))) is Element of the carrier of R
- (Det ( the carrier of R,Q,((P2 \ (M)) \/ (Si)),Q2)) is Element of the carrier of R
[:((P2 \ (M)) \/ (Si)),Q2:] is Relation-like finite set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of R is non empty non trivial V103() set
the carrier of R * is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
P is Element of the carrier of R
Q is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
len Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len Q) is finite len Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len Q ) } is set
DelLine (Q,M) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of R *
(R,(DelLine (Q,M))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,K) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
(width Q) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width Q } is set
P * (Line (Q,K)) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
P multfield is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
[: the carrier of R, the carrier of R:] is Relation-like non empty set
bool [: the carrier of R, the carrier of R:] is non empty set
the multF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[:[: the carrier of R, the carrier of R:], the carrier of R:] is Relation-like non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
id the carrier of R is Relation-like the carrier of R -defined the carrier of R -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
the multF of R [;] (P,(id the carrier of R)) is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
K391( the carrier of R, the carrier of R,(Line (Q,K)),(P multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
ReplaceLine (Q,M,(P * (Line (Q,K)))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
(R,(ReplaceLine (Q,M,(P * (Line (Q,K)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,M) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
(0. R) * (Line (Q,M)) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
(0. R) multfield is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
the multF of R [;] ((0. R),(id the carrier of R)) is Relation-like the carrier of R -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of R:]
K391( the carrier of R, the carrier of R,(Line (Q,M)),((0. R) multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
ReplaceLine (Q,M,((0. R) * (Line (Q,M)))) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
len ((0. R) * (Line (Q,M))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (P * (Line (Q,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R *
ReplaceLine ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
Replace ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
i2 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R *
Replace (Q,M,i2) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace ((Replace (Q,M,i2)),M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
Replace (Q,M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of R *
ReplaceLine (Q,M,y1) is Relation-like NAT -defined the carrier of R * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of R
Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),K) is Relation-like NAT -defined the carrier of R -valued Function-like finite width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) -tuples_on the carrier of R
(width (ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) -tuples_on the carrier of R is functional non empty FinSequence-membered FinSequenceSet of the carrier of R
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of R * : len b₁ = width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) } is set
Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),M) is Relation-like NAT -defined the carrier of R -valued Function-like finite width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) -tuples_on the carrier of R
P * (Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),K)) is Relation-like NAT -defined the carrier of R -valued Function-like finite width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) -tuples_on the carrier of R
K391( the carrier of R, the carrier of R,(Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),K)),(P multfield)) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
(Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),M)) + (P * (Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),K))) is Relation-like NAT -defined the carrier of R -valued Function-like finite width (ReplaceLine (Q,M,((0. R) * (Line (Q,M))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) -tuples_on the carrier of R
the addF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
K388( the carrier of R, the carrier of R, the carrier of R, the addF of R,(Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),M)),(P * (Line ((ReplaceLine (Q,M,((0. R) * (Line (Q,M))))),K)))) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
(width Q) |-> (0. R) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
Seg (width Q) is finite width Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width Q ) } is set
(Seg (width Q)) --> (0. R) is Relation-like Seg (width Q) -defined Seg (width Q) -defined the carrier of R -valued {(0. R)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width Q)),{(0. R)}:]
{(0. R)} is non empty trivial finite 1 -element set
[:(Seg (width Q)),{(0. R)}:] is Relation-like finite set
bool [:(Seg (width Q)),{(0. R)}:] is non empty finite V37() set
((width Q) |-> (0. R)) + (P * (Line (Q,K))) is Relation-like NAT -defined the carrier of R -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of R
K388( the carrier of R, the carrier of R, the carrier of R, the addF of R,((width Q) |-> (0. R)),(P * (Line (Q,K)))) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len (Line (Q,M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(R,(ReplaceLine (Q,M,((0. R) * (Line (Q,M)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom Q is finite Element of bool NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
M is finite without_zero Element of bool NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
P is set
Q is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(Q) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
card (Q) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
S is finite without_zero Element of bool NAT
card S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,(Q),S) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q), card (Q), the carrier of m
Det ( the carrier of m,n,(Q),S) is Element of the carrier of m
Permutations (card (Q)) is non empty permutational set
FinOmega (Permutations (card (Q))) is Element of Fin (Permutations (card (Q)))
Fin (Permutations (card (Q))) is preBoolean set
Path_product ( the carrier of m,n,(Q),S) is Relation-like Permutations (card (Q)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (Q))), the carrier of m:]
[:(Permutations (card (Q))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (Q))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (Q)))),(Path_product ( the carrier of m,n,(Q),S))) is Element of the carrier of m
x1 is set
{x1} is non empty trivial finite 1 -element set
x2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[Q,x2] is set
{Q,x2} is non empty finite V37() without_zero V103() set
{Q} is non empty trivial finite V37() 1 -element without_zero V103() set
{{Q,x2},{Q}} is non empty finite V37() without_zero V103() set
n * (Q,x2) is Element of the carrier of m
[Q,x2] is Element of [:NAT,NAT:]
(x2) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
( the carrier of m,n,(Q),(x2)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q), card (x2), the carrier of m
card (x2) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
((Q)) is Relation-like NAT -defined NAT -valued Function-like finite card (Q) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Q)) -tuples_on NAT
(card (Q)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Q) } is set
((x2)) is Relation-like NAT -defined NAT -valued Function-like finite card (x2) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (x2)) -tuples_on NAT
(card (x2)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (x2) } is set
( the carrier of m,n,(card (Q)),(card (x2)),((Q)),((x2))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q), card (x2), the carrier of m
<*(n * (Q,x2))*> is Relation-like NAT -defined the carrier of m -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of m
1 -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = 1 } is set
[1,(n * (Q,x2))] is set
{1,(n * (Q,x2))} is non empty finite set
{{1,(n * (Q,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (Q,x2))*>*> is Relation-like NAT -defined the carrier of m * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(n * (Q,x2))*>, the carrier of m
len <*(n * (Q,x2))*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(n * (Q,x2))*>] is set
{1,<*(n * (Q,x2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (Q,x2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (Q,x2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
n * (M,R) is Element of the carrier of m
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
[:(Seg (len n)),(Seg (width n)):] is Relation-like finite set
P is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(P) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
card (P) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(Q) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
card (Q) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
( the carrier of m,n,(P),(Q)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P), card (P), the carrier of m
( the carrier of m,n,(P),(Q)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P), card (Q), the carrier of m
((P)) is Relation-like NAT -defined NAT -valued Function-like finite card (P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (P)) -tuples_on NAT
(card (P)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (P) } is set
((Q)) is Relation-like NAT -defined NAT -valued Function-like finite card (Q) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Q)) -tuples_on NAT
(card (Q)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Q) } is set
( the carrier of m,n,(card (P)),(card (Q)),((P)),((Q))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P), card (Q), the carrier of m
n * (P,Q) is Element of the carrier of m
<*(n * (P,Q))*> is Relation-like NAT -defined the carrier of m -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of m
1 -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = 1 } is set
[1,(n * (P,Q))] is set
{1,(n * (P,Q))} is non empty finite set
{{1,(n * (P,Q))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (P,Q))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (P,Q))*>*> is Relation-like NAT -defined the carrier of m * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 1, len <*(n * (P,Q))*>, the carrier of m
len <*(n * (P,Q))*> is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
[1,<*(n * (P,Q))*>] is set
{1,<*(n * (P,Q))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (P,Q))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (P,Q))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Det ( the carrier of m,n,(P),(Q)) is Element of the carrier of m
Permutations (card (P)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card (P))) is Element of Fin (Permutations (card (P)))
Fin (Permutations (card (P))) is preBoolean set
Path_product ( the carrier of m,n,(P),(Q)) is Relation-like Permutations (card (P)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (P))), the carrier of m:]
[:(Permutations (card (P))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (P))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (P)))),(Path_product ( the carrier of m,n,(P),(Q)))) is Element of the carrier of m
[:(P),(Q):] is Relation-like non empty finite set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
0. (m,(len n),(width n)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len n, width n, the carrier of m
(width n) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = width n } is set
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
(width n) |-> (0. m) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
(Seg (width n)) --> (0. m) is Relation-like Seg (width n) -defined Seg (width n) -defined the carrier of m -valued {(0. m)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width n)),{(0. m)}:]
{(0. m)} is non empty trivial finite 1 -element set
[:(Seg (width n)),{(0. m)}:] is Relation-like finite set
bool [:(Seg (width n)),{(0. m)}:] is non empty finite V37() set
(len n) |-> ((width n) |-> (0. m)) is Relation-like NAT -defined (width n) -tuples_on the carrier of m -valued Function-like finite len n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len n) -tuples_on ((width n) -tuples_on the carrier of m)
(len n) -tuples_on ((width n) -tuples_on the carrier of m) is functional non empty FinSequence-membered FinSequenceSet of (width n) -tuples_on the carrier of m
((width n) -tuples_on the carrier of m) * is functional non empty FinSequence-membered FinSequenceSet of (width n) -tuples_on the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined (width n) -tuples_on the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width n) -tuples_on the carrier of m) * : len b₁ = len n } is set
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
(Seg (len n)) --> ((width n) |-> (0. m)) is Relation-like Seg (len n) -defined Seg (len n) -defined (width n) -tuples_on the carrier of m -valued {((width n) |-> (0. m))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len n)),{((width n) |-> (0. m))}:]
{((width n) |-> (0. m))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len n)),{((width n) |-> (0. m))}:] is Relation-like finite set
bool [:(Seg (len n)),{((width n) |-> (0. m))}:] is non empty finite V37() set
M is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len n, width n, the carrier of m
Indices M is set
dom M is finite Element of bool NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
Indices (0. (m,(len n),(width n))) is set
dom (0. (m,(len n),(width n))) is finite Element of bool NAT
width (0. (m,(len n),(width n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (0. (m,(len n),(width n)))) is finite width (0. (m,(len n),(width n))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (m,(len n),(width n))) ) } is set
[:(dom (0. (m,(len n),(width n)))),(Seg (width (0. (m,(len n),(width n))))):] is Relation-like finite set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[R,P] is set
{R,P} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,P},{R}} is non empty finite V37() without_zero V103() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n * (Q,S) is Element of the carrier of m
M * (R,P) is Element of the carrier of m
(0. (m,(len n),(width n))) * (R,P) is Element of the carrier of m
Indices n is set
dom n is finite Element of bool NAT
[:(dom n),(Seg (width n)):] is Relation-like finite set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[R,P] is set
{R,P} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,P},{R}} is non empty finite V37() without_zero V103() set
n * (R,P) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
R is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
P is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(R,P) is non empty finite V37() without_zero V103() Element of bool NAT
card (R,P) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
S is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(Q,S) is non empty finite V37() without_zero V103() Element of bool NAT
[:(R,P),(Q,S):] is Relation-like non empty finite set
( the carrier of m,n,(R,P),(Q,S)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (R,P), card (R,P), the carrier of m
Det ( the carrier of m,n,(R,P),(Q,S)) is Element of the carrier of m
Permutations (card (R,P)) is non empty permutational set
FinOmega (Permutations (card (R,P))) is Element of Fin (Permutations (card (R,P)))
Fin (Permutations (card (R,P))) is preBoolean set
Path_product ( the carrier of m,n,(R,P),(Q,S)) is Relation-like Permutations (card (R,P)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (R,P))), the carrier of m:]
[:(Permutations (card (R,P))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (R,P))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (R,P)))),(Path_product ( the carrier of m,n,(R,P),(Q,S)))) is Element of the carrier of m
card (Q,S) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[R,P] is set
{R,P} is non empty finite V37() set
{R} is non empty trivial finite V37() 1 -element set
{{R,P},{R}} is non empty finite V37() without_zero V103() set
n * (R,P) is Element of the carrier of m
1 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
bool K is non empty finite V37() set
R is finite Element of bool K
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is finite without_zero Element of bool NAT
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,P,Q) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,n,P,Q) is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of m,n,P,Q) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,n,P,Q))) is Element of the carrier of m
S is set
x1 is set
{S,x1} is non empty finite set
i2 is set
y1 is set
{i2,y1} is non empty finite set
[:P,Q:] is Relation-like finite set
y2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Q is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(y2,Q) is non empty finite V37() without_zero V103() Element of bool NAT
x2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
i1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(x2,i1) is non empty finite V37() without_zero V103() Element of bool NAT
[:(y2,Q),(x2,i1):] is Relation-like non empty finite set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(K,M) is non empty finite V37() Element of bool NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(R,P) is non empty finite V37() Element of bool NAT
[:(K,M),(R,P):] is Relation-like non empty finite set
n * (K,R) is Element of the carrier of m
n * (M,P) is Element of the carrier of m
(n * (K,R)) * (n * (M,P)) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
the multF of m . ((n * (K,R)),(n * (M,P))) is Element of the carrier of m
n * (K,P) is Element of the carrier of m
n * (M,R) is Element of the carrier of m
(n * (K,P)) * (n * (M,R)) is Element of the carrier of m
the multF of m . ((n * (K,P)),(n * (M,R))) is Element of the carrier of m
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
[:(Seg (len n)),(Seg (width n)):] is Relation-like finite set
Q is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
S is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(Q,S) is non empty finite V37() without_zero V103() Element of bool NAT
card (Q,S) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
x1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
n * (S,x1) is Element of the carrier of m
x2 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
n * (S,x2) is Element of the carrier of m
n * (Q,x1) is Element of the carrier of m
n * (Q,x2) is Element of the carrier of m
(x2,x1) is non empty finite V37() without_zero V103() Element of bool NAT
( the carrier of m,n,(Q,S),(x2,x1)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q,S), card (Q,S), the carrier of m
Det ( the carrier of m,n,(Q,S),(x2,x1)) is Element of the carrier of m
Permutations (card (Q,S)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
FinOmega (Permutations (card (Q,S))) is Element of Fin (Permutations (card (Q,S)))
Fin (Permutations (card (Q,S))) is preBoolean set
Path_product ( the carrier of m,n,(Q,S),(x2,x1)) is Relation-like Permutations (card (Q,S)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (Q,S))), the carrier of m:]
[:(Permutations (card (Q,S))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (Q,S))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (Q,S)))),(Path_product ( the carrier of m,n,(Q,S),(x2,x1)))) is Element of the carrier of m
card (x2,x1) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
( the carrier of m,n,(Q,S),(x2,x1)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q,S), card (x2,x1), the carrier of m
((Q,S)) is Relation-like NAT -defined NAT -valued Function-like finite card (Q,S) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Q,S)) -tuples_on NAT
(card (Q,S)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Q,S) } is set
((x2,x1)) is Relation-like NAT -defined NAT -valued Function-like finite card (x2,x1) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (x2,x1)) -tuples_on NAT
(card (x2,x1)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (x2,x1) } is set
( the carrier of m,n,(card (Q,S)),(card (x2,x1)),((Q,S)),((x2,x1))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Q,S), card (x2,x1), the carrier of m
((n * (Q,x2)),(n * (Q,x1))) ][ ((n * (S,x2)),(n * (S,x1))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (Q,x2)),(n * (Q,x1))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (Q,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x2))] is set
{1,(n * (Q,x2))} is non empty finite set
{{1,(n * (Q,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (Q,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x1))] is set
{1,(n * (Q,x1))} is non empty finite set
{{1,(n * (Q,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (Q,x2))*>,<*(n * (Q,x1))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (S,x2)),(n * (S,x1))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (S,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x2))] is set
{1,(n * (S,x2))} is non empty finite set
{{1,(n * (S,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (S,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x1))] is set
{1,(n * (S,x1))} is non empty finite set
{{1,(n * (S,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (S,x2))*>,<*(n * (S,x1))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (Q,x2)),(n * (Q,x1))*>,<*(n * (S,x2)),(n * (S,x1))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (Q,x2)),(n * (Q,x1))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (Q,x2)),(n * (Q,x1))*>] is set
{1,<*(n * (Q,x2)),(n * (Q,x1))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (Q,x2)),(n * (Q,x1))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (Q,x2)),(n * (Q,x1))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (S,x2)),(n * (S,x1))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (S,x2)),(n * (S,x1))*>] is set
{1,<*(n * (S,x2)),(n * (S,x1))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (S,x2)),(n * (S,x1))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (S,x2)),(n * (S,x1))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (Q,x2)),(n * (Q,x1))*>*>,<*<*(n * (S,x2)),(n * (S,x1))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det (((n * (Q,x2)),(n * (Q,x1))) ][ ((n * (S,x2)),(n * (S,x1)))) is Element of the carrier of m
Permutations 2 is non empty permutational set
FinOmega (Permutations 2) is Element of Fin (Permutations 2)
Fin (Permutations 2) is preBoolean set
Path_product (((n * (Q,x2)),(n * (Q,x1))) ][ ((n * (S,x2)),(n * (S,x1)))) is Relation-like Permutations 2 -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations 2), the carrier of m:]
[:(Permutations 2), the carrier of m:] is Relation-like non empty set
bool [:(Permutations 2), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations 2)),(Path_product (((n * (Q,x2)),(n * (Q,x1))) ][ ((n * (S,x2)),(n * (S,x1)))))) is Element of the carrier of m
(n * (Q,x2)) * (n * (S,x1)) is Element of the carrier of m
the multF of m . ((n * (Q,x2)),(n * (S,x1))) is Element of the carrier of m
(n * (Q,x1)) * (n * (S,x2)) is Element of the carrier of m
the multF of m . ((n * (Q,x1)),(n * (S,x2))) is Element of the carrier of m
((n * (Q,x2)) * (n * (S,x1))) - ((n * (Q,x1)) * (n * (S,x2))) is Element of the carrier of m
- ((n * (Q,x1)) * (n * (S,x2))) is Element of the carrier of m
((n * (Q,x2)) * (n * (S,x1))) + (- ((n * (Q,x1)) * (n * (S,x2)))) is Element of the carrier of m
the addF of m . (((n * (Q,x2)) * (n * (S,x1))),(- ((n * (Q,x1)) * (n * (S,x2))))) is Element of the carrier of m
((n * (Q,x1)),(n * (Q,x2))) ][ ((n * (S,x1)),(n * (S,x2))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (Q,x1)),(n * (Q,x2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (Q,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x1))] is set
{1,(n * (Q,x1))} is non empty finite set
{{1,(n * (Q,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (Q,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x2))] is set
{1,(n * (Q,x2))} is non empty finite set
{{1,(n * (Q,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (Q,x1))*>,<*(n * (Q,x2))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (S,x1)),(n * (S,x2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (S,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x1))] is set
{1,(n * (S,x1))} is non empty finite set
{{1,(n * (S,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (S,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x2))] is set
{1,(n * (S,x2))} is non empty finite set
{{1,(n * (S,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (S,x1))*>,<*(n * (S,x2))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (Q,x1)),(n * (Q,x2))*>,<*(n * (S,x1)),(n * (S,x2))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (Q,x1)),(n * (Q,x2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (Q,x1)),(n * (Q,x2))*>] is set
{1,<*(n * (Q,x1)),(n * (Q,x2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (Q,x1)),(n * (Q,x2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (Q,x1)),(n * (Q,x2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (S,x1)),(n * (S,x2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (S,x1)),(n * (S,x2))*>] is set
{1,<*(n * (S,x1)),(n * (S,x2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (S,x1)),(n * (S,x2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (S,x1)),(n * (S,x2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (Q,x1)),(n * (Q,x2))*>*>,<*<*(n * (S,x1)),(n * (S,x2))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det (((n * (Q,x1)),(n * (Q,x2))) ][ ((n * (S,x1)),(n * (S,x2)))) is Element of the carrier of m
Permutations 2 is non empty permutational set
FinOmega (Permutations 2) is Element of Fin (Permutations 2)
Fin (Permutations 2) is preBoolean set
Path_product (((n * (Q,x1)),(n * (Q,x2))) ][ ((n * (S,x1)),(n * (S,x2)))) is Relation-like Permutations 2 -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations 2), the carrier of m:]
[:(Permutations 2), the carrier of m:] is Relation-like non empty set
bool [:(Permutations 2), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations 2)),(Path_product (((n * (Q,x1)),(n * (Q,x2))) ][ ((n * (S,x1)),(n * (S,x2)))))) is Element of the carrier of m
(n * (Q,x1)) * (n * (S,x2)) is Element of the carrier of m
the multF of m . ((n * (Q,x1)),(n * (S,x2))) is Element of the carrier of m
(n * (Q,x2)) * (n * (S,x1)) is Element of the carrier of m
the multF of m . ((n * (Q,x2)),(n * (S,x1))) is Element of the carrier of m
((n * (Q,x1)) * (n * (S,x2))) - ((n * (Q,x2)) * (n * (S,x1))) is Element of the carrier of m
- ((n * (Q,x2)) * (n * (S,x1))) is Element of the carrier of m
((n * (Q,x1)) * (n * (S,x2))) + (- ((n * (Q,x2)) * (n * (S,x1)))) is Element of the carrier of m
the addF of m . (((n * (Q,x1)) * (n * (S,x2))),(- ((n * (Q,x2)) * (n * (S,x1))))) is Element of the carrier of m
((n * (S,x2)),(n * (S,x1))) ][ ((n * (Q,x2)),(n * (Q,x1))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (S,x2)),(n * (S,x1))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (S,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x2))] is set
{1,(n * (S,x2))} is non empty finite set
{{1,(n * (S,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (S,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x1))] is set
{1,(n * (S,x1))} is non empty finite set
{{1,(n * (S,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (S,x2))*>,<*(n * (S,x1))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (Q,x2)),(n * (Q,x1))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (Q,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x2))] is set
{1,(n * (Q,x2))} is non empty finite set
{{1,(n * (Q,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (Q,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x1))] is set
{1,(n * (Q,x1))} is non empty finite set
{{1,(n * (Q,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (Q,x2))*>,<*(n * (Q,x1))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (S,x2)),(n * (S,x1))*>,<*(n * (Q,x2)),(n * (Q,x1))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (S,x2)),(n * (S,x1))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (S,x2)),(n * (S,x1))*>] is set
{1,<*(n * (S,x2)),(n * (S,x1))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (S,x2)),(n * (S,x1))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (S,x2)),(n * (S,x1))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (Q,x2)),(n * (Q,x1))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (Q,x2)),(n * (Q,x1))*>] is set
{1,<*(n * (Q,x2)),(n * (Q,x1))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (Q,x2)),(n * (Q,x1))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (Q,x2)),(n * (Q,x1))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (S,x2)),(n * (S,x1))*>*>,<*<*(n * (Q,x2)),(n * (Q,x1))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det (((n * (S,x2)),(n * (S,x1))) ][ ((n * (Q,x2)),(n * (Q,x1)))) is Element of the carrier of m
Permutations 2 is non empty permutational set
FinOmega (Permutations 2) is Element of Fin (Permutations 2)
Fin (Permutations 2) is preBoolean set
Path_product (((n * (S,x2)),(n * (S,x1))) ][ ((n * (Q,x2)),(n * (Q,x1)))) is Relation-like Permutations 2 -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations 2), the carrier of m:]
[:(Permutations 2), the carrier of m:] is Relation-like non empty set
bool [:(Permutations 2), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations 2)),(Path_product (((n * (S,x2)),(n * (S,x1))) ][ ((n * (Q,x2)),(n * (Q,x1)))))) is Element of the carrier of m
(n * (S,x2)) * (n * (Q,x1)) is Element of the carrier of m
the multF of m . ((n * (S,x2)),(n * (Q,x1))) is Element of the carrier of m
(n * (S,x1)) * (n * (Q,x2)) is Element of the carrier of m
the multF of m . ((n * (S,x1)),(n * (Q,x2))) is Element of the carrier of m
((n * (S,x2)) * (n * (Q,x1))) - ((n * (S,x1)) * (n * (Q,x2))) is Element of the carrier of m
- ((n * (S,x1)) * (n * (Q,x2))) is Element of the carrier of m
((n * (S,x2)) * (n * (Q,x1))) + (- ((n * (S,x1)) * (n * (Q,x2)))) is Element of the carrier of m
the addF of m . (((n * (S,x2)) * (n * (Q,x1))),(- ((n * (S,x1)) * (n * (Q,x2))))) is Element of the carrier of m
((n * (S,x1)),(n * (S,x2))) ][ ((n * (Q,x1)),(n * (Q,x2))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (S,x1)),(n * (S,x2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (S,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x1))] is set
{1,(n * (S,x1))} is non empty finite set
{{1,(n * (S,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (S,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (S,x2))] is set
{1,(n * (S,x2))} is non empty finite set
{{1,(n * (S,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (S,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (S,x1))*>,<*(n * (S,x2))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (Q,x1)),(n * (Q,x2))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (Q,x1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x1))] is set
{1,(n * (Q,x1))} is non empty finite set
{{1,(n * (Q,x1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (Q,x2))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (Q,x2))] is set
{1,(n * (Q,x2))} is non empty finite set
{{1,(n * (Q,x2))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (Q,x2))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (Q,x1))*>,<*(n * (Q,x2))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (S,x1)),(n * (S,x2))*>,<*(n * (Q,x1)),(n * (Q,x2))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (S,x1)),(n * (S,x2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (S,x1)),(n * (S,x2))*>] is set
{1,<*(n * (S,x1)),(n * (S,x2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (S,x1)),(n * (S,x2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (S,x1)),(n * (S,x2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (Q,x1)),(n * (Q,x2))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (Q,x1)),(n * (Q,x2))*>] is set
{1,<*(n * (Q,x1)),(n * (Q,x2))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (Q,x1)),(n * (Q,x2))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (Q,x1)),(n * (Q,x2))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (S,x1)),(n * (S,x2))*>*>,<*<*(n * (Q,x1)),(n * (Q,x2))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det (((n * (S,x1)),(n * (S,x2))) ][ ((n * (Q,x1)),(n * (Q,x2)))) is Element of the carrier of m
Permutations 2 is non empty permutational set
FinOmega (Permutations 2) is Element of Fin (Permutations 2)
Fin (Permutations 2) is preBoolean set
Path_product (((n * (S,x1)),(n * (S,x2))) ][ ((n * (Q,x1)),(n * (Q,x2)))) is Relation-like Permutations 2 -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations 2), the carrier of m:]
[:(Permutations 2), the carrier of m:] is Relation-like non empty set
bool [:(Permutations 2), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations 2)),(Path_product (((n * (S,x1)),(n * (S,x2))) ][ ((n * (Q,x1)),(n * (Q,x2)))))) is Element of the carrier of m
(n * (S,x1)) * (n * (Q,x2)) is Element of the carrier of m
the multF of m . ((n * (S,x1)),(n * (Q,x2))) is Element of the carrier of m
(n * (S,x2)) * (n * (Q,x1)) is Element of the carrier of m
the multF of m . ((n * (S,x2)),(n * (Q,x1))) is Element of the carrier of m
((n * (S,x1)) * (n * (Q,x2))) - ((n * (S,x2)) * (n * (Q,x1))) is Element of the carrier of m
- ((n * (S,x2)) * (n * (Q,x1))) is Element of the carrier of m
((n * (S,x1)) * (n * (Q,x2))) + (- ((n * (S,x2)) * (n * (Q,x1)))) is Element of the carrier of m
the addF of m . (((n * (S,x1)) * (n * (Q,x2))),(- ((n * (S,x2)) * (n * (Q,x1))))) is Element of the carrier of m
K is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
M is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
R is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
P is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(K,M) is non empty finite V37() without_zero V103() Element of bool NAT
(R,P) is non empty finite V37() without_zero V103() Element of bool NAT
[:(K,M),(R,P):] is Relation-like non empty finite set
card (K,M) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
n * (M,P) is Element of the carrier of m
n * (M,R) is Element of the carrier of m
n * (K,P) is Element of the carrier of m
n * (K,R) is Element of the carrier of m
(n * (K,R)) * (n * (M,P)) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
the multF of m . ((n * (K,R)),(n * (M,P))) is Element of the carrier of m
(n * (K,P)) * (n * (M,R)) is Element of the carrier of m
the multF of m . ((n * (K,P)),(n * (M,R))) is Element of the carrier of m
card (R,P) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
( the carrier of m,n,(K,M),(R,P)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K,M), card (K,M), the carrier of m
( the carrier of m,n,(K,M),(R,P)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K,M), card (R,P), the carrier of m
((K,M)) is Relation-like NAT -defined NAT -valued Function-like finite card (K,M) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (K,M)) -tuples_on NAT
(card (K,M)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (K,M) } is set
((R,P)) is Relation-like NAT -defined NAT -valued Function-like finite card (R,P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (R,P)) -tuples_on NAT
(card (R,P)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (R,P) } is set
( the carrier of m,n,(card (K,M)),(card (R,P)),((K,M)),((R,P))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (K,M), card (R,P), the carrier of m
((n * (K,R)),(n * (K,P))) ][ ((n * (M,R)),(n * (M,P))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (K,R)),(n * (K,P))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (K,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,R))] is set
{1,(n * (K,R))} is non empty finite set
{{1,(n * (K,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (K,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,P))] is set
{1,(n * (K,P))} is non empty finite set
{{1,(n * (K,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (K,R))*>,<*(n * (K,P))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (M,R)),(n * (M,P))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (M,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,R))] is set
{1,(n * (M,R))} is non empty finite set
{{1,(n * (M,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (M,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,P))] is set
{1,(n * (M,P))} is non empty finite set
{{1,(n * (M,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (M,R))*>,<*(n * (M,P))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (K,R)),(n * (K,P))*>,<*(n * (M,R)),(n * (M,P))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (K,R)),(n * (K,P))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (K,R)),(n * (K,P))*>] is set
{1,<*(n * (K,R)),(n * (K,P))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (K,R)),(n * (K,P))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (K,R)),(n * (K,P))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (M,R)),(n * (M,P))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (M,R)),(n * (M,P))*>] is set
{1,<*(n * (M,R)),(n * (M,P))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (M,R)),(n * (M,P))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (M,R)),(n * (M,P))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (K,R)),(n * (K,P))*>*>,<*<*(n * (M,R)),(n * (M,P))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det ( the carrier of m,n,(K,M),(R,P)) is Element of the carrier of m
Permutations (card (K,M)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
FinOmega (Permutations (card (K,M))) is Element of Fin (Permutations (card (K,M)))
Fin (Permutations (card (K,M))) is preBoolean set
Path_product ( the carrier of m,n,(K,M),(R,P)) is Relation-like Permutations (card (K,M)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (K,M))), the carrier of m:]
[:(Permutations (card (K,M))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (K,M))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (K,M)))),(Path_product ( the carrier of m,n,(K,M),(R,P)))) is Element of the carrier of m
((n * (K,R)) * (n * (M,P))) - ((n * (K,P)) * (n * (M,R))) is Element of the carrier of m
- ((n * (K,P)) * (n * (M,R))) is Element of the carrier of m
((n * (K,R)) * (n * (M,P))) + (- ((n * (K,P)) * (n * (M,R)))) is Element of the carrier of m
the addF of m . (((n * (K,R)) * (n * (M,P))),(- ((n * (K,P)) * (n * (M,R))))) is Element of the carrier of m
((n * (K,P)),(n * (K,R))) ][ ((n * (M,P)),(n * (M,R))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (K,P)),(n * (K,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (K,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,P))] is set
{1,(n * (K,P))} is non empty finite set
{{1,(n * (K,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (K,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,R))] is set
{1,(n * (K,R))} is non empty finite set
{{1,(n * (K,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (K,P))*>,<*(n * (K,R))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (M,P)),(n * (M,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (M,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,P))] is set
{1,(n * (M,P))} is non empty finite set
{{1,(n * (M,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (M,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,R))] is set
{1,(n * (M,R))} is non empty finite set
{{1,(n * (M,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (M,P))*>,<*(n * (M,R))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (K,P)),(n * (K,R))*>,<*(n * (M,P)),(n * (M,R))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (K,P)),(n * (K,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (K,P)),(n * (K,R))*>] is set
{1,<*(n * (K,P)),(n * (K,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (K,P)),(n * (K,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (K,P)),(n * (K,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (M,P)),(n * (M,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (M,P)),(n * (M,R))*>] is set
{1,<*(n * (M,P)),(n * (M,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (M,P)),(n * (M,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (M,P)),(n * (M,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (K,P)),(n * (K,R))*>*>,<*<*(n * (M,P)),(n * (M,R))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det ( the carrier of m,n,(K,M),(R,P)) is Element of the carrier of m
Permutations (card (K,M)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
FinOmega (Permutations (card (K,M))) is Element of Fin (Permutations (card (K,M)))
Fin (Permutations (card (K,M))) is preBoolean set
Path_product ( the carrier of m,n,(K,M),(R,P)) is Relation-like Permutations (card (K,M)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (K,M))), the carrier of m:]
[:(Permutations (card (K,M))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (K,M))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (K,M)))),(Path_product ( the carrier of m,n,(K,M),(R,P)))) is Element of the carrier of m
((n * (K,P)) * (n * (M,R))) - ((n * (K,R)) * (n * (M,P))) is Element of the carrier of m
- ((n * (K,R)) * (n * (M,P))) is Element of the carrier of m
((n * (K,P)) * (n * (M,R))) + (- ((n * (K,R)) * (n * (M,P)))) is Element of the carrier of m
the addF of m . (((n * (K,P)) * (n * (M,R))),(- ((n * (K,R)) * (n * (M,P))))) is Element of the carrier of m
((n * (M,R)),(n * (M,P))) ][ ((n * (K,R)),(n * (K,P))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (M,R)),(n * (M,P))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (M,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,R))] is set
{1,(n * (M,R))} is non empty finite set
{{1,(n * (M,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (M,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,P))] is set
{1,(n * (M,P))} is non empty finite set
{{1,(n * (M,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (M,R))*>,<*(n * (M,P))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (K,R)),(n * (K,P))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (K,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,R))] is set
{1,(n * (K,R))} is non empty finite set
{{1,(n * (K,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (K,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,P))] is set
{1,(n * (K,P))} is non empty finite set
{{1,(n * (K,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (K,R))*>,<*(n * (K,P))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (M,R)),(n * (M,P))*>,<*(n * (K,R)),(n * (K,P))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (M,R)),(n * (M,P))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (M,R)),(n * (M,P))*>] is set
{1,<*(n * (M,R)),(n * (M,P))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (M,R)),(n * (M,P))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (M,R)),(n * (M,P))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (K,R)),(n * (K,P))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (K,R)),(n * (K,P))*>] is set
{1,<*(n * (K,R)),(n * (K,P))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (K,R)),(n * (K,P))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (K,R)),(n * (K,P))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (M,R)),(n * (M,P))*>*>,<*<*(n * (K,R)),(n * (K,P))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det ( the carrier of m,n,(K,M),(R,P)) is Element of the carrier of m
Permutations (card (K,M)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
FinOmega (Permutations (card (K,M))) is Element of Fin (Permutations (card (K,M)))
Fin (Permutations (card (K,M))) is preBoolean set
Path_product ( the carrier of m,n,(K,M),(R,P)) is Relation-like Permutations (card (K,M)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (K,M))), the carrier of m:]
[:(Permutations (card (K,M))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (K,M))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (K,M)))),(Path_product ( the carrier of m,n,(K,M),(R,P)))) is Element of the carrier of m
(n * (M,R)) * (n * (K,P)) is Element of the carrier of m
the multF of m . ((n * (M,R)),(n * (K,P))) is Element of the carrier of m
(n * (M,P)) * (n * (K,R)) is Element of the carrier of m
the multF of m . ((n * (M,P)),(n * (K,R))) is Element of the carrier of m
((n * (M,R)) * (n * (K,P))) - ((n * (M,P)) * (n * (K,R))) is Element of the carrier of m
- ((n * (M,P)) * (n * (K,R))) is Element of the carrier of m
((n * (M,R)) * (n * (K,P))) + (- ((n * (M,P)) * (n * (K,R)))) is Element of the carrier of m
the addF of m . (((n * (M,R)) * (n * (K,P))),(- ((n * (M,P)) * (n * (K,R))))) is Element of the carrier of m
((n * (M,P)),(n * (M,R))) ][ ((n * (K,P)),(n * (K,R))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of 2,2, the carrier of m
<*(n * (M,P)),(n * (M,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (M,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,P))] is set
{1,(n * (M,P))} is non empty finite set
{{1,(n * (M,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (M,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (M,R))] is set
{1,(n * (M,R))} is non empty finite set
{{1,(n * (M,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (M,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (M,P))*>,<*(n * (M,R))*>) 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 V105() complex ext-real positive non negative Element of NAT
<*(n * (K,P)),(n * (K,R))*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(n * (K,P))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,P))] is set
{1,(n * (K,P))} is non empty finite set
{{1,(n * (K,P))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,P))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n * (K,R))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(n * (K,R))] is set
{1,(n * (K,R))} is non empty finite set
{{1,(n * (K,R))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n * (K,R))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*(n * (K,P))*>,<*(n * (K,R))*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*<*(n * (M,P)),(n * (M,R))*>,<*(n * (K,P)),(n * (K,R))*>*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*<*(n * (M,P)),(n * (M,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (M,P)),(n * (M,R))*>] is set
{1,<*(n * (M,P)),(n * (M,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (M,P)),(n * (M,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (M,P)),(n * (M,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*<*(n * (K,P)),(n * (K,R))*>*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,<*(n * (K,P)),(n * (K,R))*>] is set
{1,<*(n * (K,P)),(n * (K,R))*>} is non empty finite V37() without_zero V103() set
{{1,<*(n * (K,P)),(n * (K,R))*>},{1}} is non empty finite V37() without_zero V103() set
{[1,<*(n * (K,P)),(n * (K,R))*>]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*<*(n * (M,P)),(n * (M,R))*>*>,<*<*(n * (K,P)),(n * (K,R))*>*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
Det ( the carrier of m,n,(K,M),(R,P)) is Element of the carrier of m
Permutations (card (K,M)) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
FinOmega (Permutations (card (K,M))) is Element of Fin (Permutations (card (K,M)))
Fin (Permutations (card (K,M))) is preBoolean set
Path_product ( the carrier of m,n,(K,M),(R,P)) is Relation-like Permutations (card (K,M)) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (K,M))), the carrier of m:]
[:(Permutations (card (K,M))), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card (K,M))), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card (K,M)))),(Path_product ( the carrier of m,n,(K,M),(R,P)))) is Element of the carrier of m
(n * (M,P)) * (n * (K,R)) is Element of the carrier of m
the multF of m . ((n * (M,P)),(n * (K,R))) is Element of the carrier of m
(n * (M,R)) * (n * (K,P)) is Element of the carrier of m
the multF of m . ((n * (M,R)),(n * (K,P))) is Element of the carrier of m
((n * (M,P)) * (n * (K,R))) - ((n * (M,R)) * (n * (K,P))) is Element of the carrier of m
- ((n * (M,R)) * (n * (K,P))) is Element of the carrier of m
((n * (M,P)) * (n * (K,R))) + (- ((n * (M,R)) * (n * (K,P)))) is Element of the carrier of m
the addF of m . (((n * (M,P)) * (n * (K,R))),(- ((n * (M,R)) * (n * (K,P))))) is Element of the carrier of m
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[K,M] is set
{K,M} is non empty finite V37() set
{K} is non empty trivial finite V37() 1 -element set
{{K,M},{K}} is non empty finite V37() without_zero V103() set
n * (K,M) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
Indices n is set
dom n is finite Element of bool NAT
[:(dom n),(Seg (width n)):] is Relation-like finite set
[:(Seg (len n)),(Seg (width n)):] is Relation-like finite set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[K,M] is set
{K,M} is non empty finite V37() set
{K} is non empty trivial finite V37() 1 -element set
{{K,M},{K}} is non empty finite V37() without_zero V103() set
n * (K,M) is Element of the carrier of m
Line (n,K) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
(width n) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = width n } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (n,Q) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
n * (Q,M) is Element of the carrier of m
(n * (K,M)) " is Element of the carrier of m
(n * (Q,M)) * ((n * (K,M)) ") is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
the multF of m . ((n * (Q,M)),((n * (K,M)) ")) is Element of the carrier of m
x2 is Element of the carrier of m
x2 * (Line (n,K)) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
x2 multfield is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
bool [: the carrier of m, the carrier of m:] is non empty set
id the carrier of m is Relation-like the carrier of m -defined the carrier of m -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
the multF of m [;] (x2,(id the carrier of m)) is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
K391( the carrier of m, the carrier of m,(Line (n,K)),(x2 multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(M,i1) is non empty finite V37() Element of bool NAT
(Line (n,K)) . i1 is set
n * (K,i1) is Element of the carrier of m
(x2 * (Line (n,K))) . i1 is set
x2 * (n * (K,i1)) is Element of the carrier of m
the multF of m . (x2,(n * (K,i1))) is Element of the carrier of m
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
(n * (Q,M)) * (1_ m) is Element of the carrier of m
the multF of m . ((n * (Q,M)),(1_ m)) is Element of the carrier of m
(n * (K,M)) * ((n * (K,M)) ") is Element of the carrier of m
the multF of m . ((n * (K,M)),((n * (K,M)) ")) is Element of the carrier of m
(n * (Q,M)) * ((n * (K,M)) * ((n * (K,M)) ")) is Element of the carrier of m
the multF of m . ((n * (Q,M)),((n * (K,M)) * ((n * (K,M)) "))) is Element of the carrier of m
x2 * (n * (K,M)) is Element of the carrier of m
the multF of m . (x2,(n * (K,M))) is Element of the carrier of m
(K,Q) is non empty finite V37() Element of bool NAT
[:(K,Q),(M,i1):] is Relation-like non empty finite set
n * (Q,i1) is Element of the carrier of m
(n * (K,M)) * (n * (Q,i1)) is Element of the carrier of m
the multF of m . ((n * (K,M)),(n * (Q,i1))) is Element of the carrier of m
(x2 * (n * (K,M))) * (n * (K,i1)) is Element of the carrier of m
the multF of m . ((x2 * (n * (K,M))),(n * (K,i1))) is Element of the carrier of m
(n * (K,M)) * (x2 * (n * (K,i1))) is Element of the carrier of m
the multF of m . ((n * (K,M)),(x2 * (n * (K,i1)))) is Element of the carrier of m
(Line (n,Q)) . i1 is set
len (x2 * (Line (n,K))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Line (n,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (n,K) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
(width n) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = width n } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(R,P) is non empty finite V37() Element of bool NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Q,S) is non empty finite V37() Element of bool NAT
[:(R,P),(Q,S):] is Relation-like non empty finite set
(Line (n,K)) . Q is set
n * (K,Q) is Element of the carrier of m
Line (n,P) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
Line (n,R) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
i1 is Element of the carrier of m
i1 * (Line (n,K)) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
i1 multfield is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
bool [: the carrier of m, the carrier of m:] is non empty set
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
id the carrier of m is Relation-like the carrier of m -defined the carrier of m -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
the multF of m [;] (i1,(id the carrier of m)) is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
K391( the carrier of m, the carrier of m,(Line (n,K)),(i1 multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
i2 is Element of the carrier of m
i2 * (Line (n,K)) is Relation-like NAT -defined the carrier of m -valued Function-like finite width n -element FinSequence-like FinSubsequence-like Element of (width n) -tuples_on the carrier of m
i2 multfield is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
the multF of m [;] (i2,(id the carrier of m)) is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
K391( the carrier of m, the carrier of m,(Line (n,K)),(i2 multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
(Line (n,P)) . Q is set
n * (P,Q) is Element of the carrier of m
i2 * (n * (K,Q)) is Element of the carrier of m
the multF of m . (i2,(n * (K,Q))) is Element of the carrier of m
(Line (n,K)) . S is set
n * (K,S) is Element of the carrier of m
(Line (n,P)) . S is set
n * (P,S) is Element of the carrier of m
i2 * (n * (K,S)) is Element of the carrier of m
the multF of m . (i2,(n * (K,S))) is Element of the carrier of m
(Line (n,R)) . S is set
n * (R,S) is Element of the carrier of m
i1 * (n * (K,S)) is Element of the carrier of m
the multF of m . (i1,(n * (K,S))) is Element of the carrier of m
(Line (n,R)) . Q is set
n * (R,Q) is Element of the carrier of m
i1 * (n * (K,Q)) is Element of the carrier of m
the multF of m . (i1,(n * (K,Q))) is Element of the carrier of m
(n * (R,Q)) * (n * (P,S)) is Element of the carrier of m
the multF of m . ((n * (R,Q)),(n * (P,S))) is Element of the carrier of m
(i1 * (n * (K,Q))) * i2 is Element of the carrier of m
the multF of m . ((i1 * (n * (K,Q))),i2) is Element of the carrier of m
((i1 * (n * (K,Q))) * i2) * (n * (K,S)) is Element of the carrier of m
the multF of m . (((i1 * (n * (K,Q))) * i2),(n * (K,S))) is Element of the carrier of m
(i2 * (n * (K,Q))) * i1 is Element of the carrier of m
the multF of m . ((i2 * (n * (K,Q))),i1) is Element of the carrier of m
((i2 * (n * (K,Q))) * i1) * (n * (K,S)) is Element of the carrier of m
the multF of m . (((i2 * (n * (K,Q))) * i1),(n * (K,S))) is Element of the carrier of m
(n * (R,S)) * (n * (P,Q)) is Element of the carrier of m
the multF of m . ((n * (R,S)),(n * (P,Q))) is Element of the carrier of m
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n * (K,M) is Element of the carrier of m
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n * (K,M) is Element of the carrier of m
[K,M] is set
{K,M} is non empty finite V37() set
{K} is non empty trivial finite V37() 1 -element set
{{K,M},{K}} is non empty finite V37() without_zero V103() set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
the Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal upper_triangular lower_triangular Matrix of 1,1, the carrier of m is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal upper_triangular lower_triangular Matrix of 1,1, the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( [b₁,b₁] in Indices n & not n * (b₁,b₁) = 0. m ) } is set
K is set
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is finite without_zero Element of bool NAT
[:M,R:] is Relation-like finite set
card R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det ( the carrier of m,n,M,R) is Element of the carrier of m
Permutations (card M) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card M)) is Element of Fin (Permutations (card M))
Fin (Permutations (card M)) is preBoolean set
Path_product ( the carrier of m,n,M,R) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
[:(Permutations (card M)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card M)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product ( the carrier of m,n,M,R))) is Element of the carrier of m
( the carrier of m,n,M,R) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
(R) is Relation-like NAT -defined NAT -valued Function-like finite card R -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card R) -tuples_on NAT
(card R) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card R } is set
( the carrier of m,n,(card M),(card R),(M),(R)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card R, the carrier of m
Indices ( the carrier of m,n,M,R) is set
dom ( the carrier of m,n,M,R) is finite Element of bool NAT
width ( the carrier of m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of m,n,M,R)) is finite width ( the carrier of m,n,M,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of m,n,M,R) ) } is set
[:(dom ( the carrier of m,n,M,R)),(Seg (width ( the carrier of m,n,M,R))):] is Relation-like finite set
len ( the carrier of m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of m,n,M,R)) is finite len ( the carrier of m,n,M,R) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of m,n,M,R) ) } is set
[:(Seg (len ( the carrier of m,n,M,R))),(Seg (width ( the carrier of m,n,M,R))):] is Relation-like finite set
x2 is set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
rng (M) is finite V212() V213() V214() V217() set
dom (M) is finite card M -element Element of bool NAT
i1 is set
(M) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of m,n,M,R),y1) is Relation-like NAT -defined the carrier of m -valued Function-like finite width ( the carrier of m,n,M,R) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of m,n,M,R)) -tuples_on the carrier of m
(width ( the carrier of m,n,M,R)) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = width ( the carrier of m,n,M,R) } is set
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
rng (R) is finite V212() V213() V214() V217() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Line (( the carrier of m,n,M,R),y1)) . Q is set
( the carrier of m,n,M,R) * (y1,Q) is Element of the carrier of m
[y1,Q] is set
{y1,Q} is non empty finite V37() set
{y1} is non empty trivial finite V37() 1 -element set
{{y1,Q},{y1}} is non empty finite V37() without_zero V103() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(R) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n * (i2,((R) . Q)) is Element of the carrier of m
(0. m) * (0. m) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the multF of m . ((0. m),(0. m)) is Element of the carrier of m
[i2,((R) . Q)] is set
{i2,((R) . Q)} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,((R) . Q)},{i2}} is non empty finite V37() without_zero V103() set
(0. m) * (Line (( the carrier of m,n,M,R),y1)) is Relation-like NAT -defined the carrier of m -valued Function-like finite width ( the carrier of m,n,M,R) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of m,n,M,R)) -tuples_on the carrier of m
(0. m) multfield is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
bool [: the carrier of m, the carrier of m:] is non empty set
id the carrier of m is Relation-like the carrier of m -defined the carrier of m -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
the multF of m [;] ((0. m),(id the carrier of m)) is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
K391( the carrier of m, the carrier of m,(Line (( the carrier of m,n,M,R),y1)),((0. m) multfield)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
((0. m) * (Line (( the carrier of m,n,M,R),y1))) . Q is set
len (Line (( the carrier of m,n,M,R),y1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ((0. m) * (Line (( the carrier of m,n,M,R),y1))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of m,n,M,R),y1) is Relation-like NAT -defined the carrier of m -valued Function-like finite width ( the carrier of m,n,M,R) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of m,n,M,R)) -tuples_on the carrier of m
width ( the carrier of m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of m,n,M,R)) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = width ( the carrier of m,n,M,R) } is set
ReplaceLine (( the carrier of m,n,M,R),y1,(Line (( the carrier of m,n,M,R),y1))) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card M, the carrier of m
Det (ReplaceLine (( the carrier of m,n,M,R),y1,(Line (( the carrier of m,n,M,R),y1)))) is Element of the carrier of m
Path_product (ReplaceLine (( the carrier of m,n,M,R),y1,(Line (( the carrier of m,n,M,R),y1)))) is Relation-like Permutations (card M) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card M)), the carrier of m:]
the addF of m $$ ((FinOmega (Permutations (card M))),(Path_product (ReplaceLine (( the carrier of m,n,M,R),y1,(Line (( the carrier of m,n,M,R),y1)))))) is Element of the carrier of m
(0. m) * (Det ( the carrier of m,n,M,R)) is Element of the carrier of m
the multF of m . ((0. m),(Det ( the carrier of m,n,M,R))) is Element of the carrier of m
dom (R) is finite card R -element Element of bool NAT
Seg (card R) is finite card R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card R ) } is set
x2 is set
rng (R) is finite V212() V213() V214() V217() set
i1 is set
(R) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col (( the carrier of m,n,M,R),y1) is Relation-like NAT -defined the carrier of m -valued Function-like finite len ( the carrier of m,n,M,R) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of m,n,M,R)) -tuples_on the carrier of m
len ( the carrier of m,n,M,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len ( the carrier of m,n,M,R)) -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = len ( the carrier of m,n,M,R) } is set
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
rng (M) is finite V212() V213() V214() V217() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
( the carrier of m,n,M,R) * (Q,y1) is Element of the carrier of m
(Col (( the carrier of m,n,M,R),y1)) . Q is set
[Q,y1] is Element of [:NAT,NAT:]
{Q,y1} is non empty finite V37() set
{Q} is non empty trivial finite V37() 1 -element set
{{Q,y1},{Q}} is non empty finite V37() without_zero V103() set
(M) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n * (((M) . Q),i2) is Element of the carrier of m
[((M) . Q),i2] is set
{((M) . Q),i2} is non empty finite V37() set
{((M) . Q)} is non empty trivial finite V37() 1 -element set
{{((M) . Q),i2},{((M) . Q)}} is non empty finite V37() without_zero V103() set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
K is finite without_zero Element of bool NAT
[:K,K:] is Relation-like finite set
( the carrier of m,n,K,K) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K) is Relation-like NAT -defined NAT -valued Function-like finite card K -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card K) -tuples_on NAT
(card K) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card K } is set
( the carrier of m,n,(card K),(card K),(K),(K)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card K) is finite card K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card K ) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,K) * (P,Q) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
len n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len n) is finite len n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
[P,Q] is Element of [:NAT,NAT:]
{P,Q} is non empty finite V37() set
{P} is non empty trivial finite V37() 1 -element set
{{P,Q},{P}} is non empty finite V37() without_zero V103() set
[:(Seg (card K)),(Seg (card K)):] is Relation-like finite set
Indices ( the carrier of m,n,K,K) is set
dom ( the carrier of m,n,K,K) is finite Element of bool NAT
width ( the carrier of m,n,K,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of m,n,K,K)) is finite width ( the carrier of m,n,K,K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of m,n,K,K) ) } is set
[:(dom ( the carrier of m,n,K,K)),(Seg (width ( the carrier of m,n,K,K))):] is Relation-like finite set
dom (K) is finite card K -element Element of bool NAT
(K) . P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(K) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
rng (K) is finite V212() V213() V214() V217() set
[((K) . P),((K) . Q)] is set
{((K) . P),((K) . Q)} is non empty finite V37() set
{((K) . P)} is non empty trivial finite V37() 1 -element set
{{((K) . P),((K) . Q)},{((K) . P)}} is non empty finite V37() without_zero V103() set
n * (((K) . P),((K) . Q)) is Element of the carrier of m
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal FinSequence of the carrier of m *
Indices n is set
dom n is finite Element of bool NAT
width n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width n) is finite width n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width n ) } is set
[:(dom n),(Seg (width n)):] is Relation-like finite set
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( [b₁,b₁] in Indices n & not n * (b₁,b₁) = 0. m ) } is set
(m,n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is finite without_zero Element of bool NAT
M is finite without_zero Element of bool NAT
[:K,M:] is Relation-like finite set
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of m,n,K,M) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card K, card K, the carrier of m
Det ( the carrier of m,n,K,M) is Element of the carrier of m
Permutations (card K) is non empty permutational set
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
FinOmega (Permutations (card K)) is Element of Fin (Permutations (card K))
Fin (Permutations (card K)) is preBoolean set
Path_product ( the carrier of m,n,K,M) is Relation-like Permutations (card K) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card K)), the carrier of m:]
[:(Permutations (card K)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card K)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card K))),(Path_product ( the carrier of m,n,K,M))) is Element of the carrier of m
R is set
card R is V26() V27() V28() cardinal set
P is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[Q,Q] is Element of [:NAT,NAT:]
{Q,Q} is non empty finite V37() set
{Q} is non empty trivial finite V37() 1 -element set
{{Q,Q},{Q}} is non empty finite V37() without_zero V103() set
n * (Q,Q) is Element of the carrier of m
P is finite without_zero Element of bool NAT
( the carrier of m,n,P,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
( the carrier of m,n,(card P),(card P),(P),(P)) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
S is set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[x1,x1] is Element of [:NAT,NAT:]
{x1,x1} is non empty finite V37() set
{x1} is non empty trivial finite V37() 1 -element set
{{x1,x1},{x1}} is non empty finite V37() without_zero V103() set
n * (x1,x1) is Element of the carrier of m
[:P,P:] is Relation-like finite set
S is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal upper_triangular lower_triangular Matrix of card P, card P, the carrier of m
diagonal_of_Matrix S is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
Det S is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product S is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product S)) is Element of the carrier of m
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
rng (P) is finite V212() V213() V214() V217() set
dom (diagonal_of_Matrix S) is finite Element of bool NAT
len (diagonal_of_Matrix S) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (diagonal_of_Matrix S)) is finite len (diagonal_of_Matrix S) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (diagonal_of_Matrix S) ) } is set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(diagonal_of_Matrix S) . i1 is set
S * (i1,i1) is Element of the carrier of m
dom (P) is finite card P -element Element of bool NAT
(P) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Indices S is set
dom S is finite Element of bool NAT
width S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width S) is finite width S -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width S ) } is set
[:(dom S),(Seg (width S)):] is Relation-like finite set
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
[:(Seg (card P)),(Seg (card P)):] is Relation-like finite set
[i1,i1] is Element of [:NAT,NAT:]
{i1,i1} is non empty finite V37() set
{i1} is non empty trivial finite V37() 1 -element set
{{i1,i1},{i1}} is non empty finite V37() without_zero V103() set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[i2,i2] is Element of [:NAT,NAT:]
{i2,i2} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,i2},{i2}} is non empty finite V37() without_zero V103() set
n * (i2,i2) is Element of the carrier of m
Det S is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product S is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product S)) is Element of the carrier of m
Product (diagonal_of_Matrix S) is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the multF of m $$ (diagonal_of_Matrix S) is Element of the carrier of m
Det S is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product S is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product S)) is Element of the carrier of m
Det S is Element of the carrier of m
Permutations (card P) is non empty permutational set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product S is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
[:(Permutations (card P)), the carrier of m:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of m:] is non empty set
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product S)) is Element of the carrier of m
( the carrier of m,n,P,P) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of m
Det ( the carrier of m,n,P,P) is Element of the carrier of m
Path_product ( the carrier of m,n,P,P) is Relation-like Permutations (card P) -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of m:]
the addF of m $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of m,n,P,P))) is Element of the carrier of m
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
m -VectSp_over n is non empty strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over n
the carrier of (m -VectSp_over n) is non empty set
m -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = m } is set
0. (m -VectSp_over n) is zero Element of the carrier of (m -VectSp_over n)
the ZeroF of (m -VectSp_over n) is Element of the carrier of (m -VectSp_over n)
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
m |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (0. n) is Relation-like Seg m -defined Seg m -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. n)}:] is Relation-like finite set
bool [:(Seg m),{(0. n)}:] is non empty finite V37() set
K is Element of the carrier of n
M is Element of the carrier of (m -VectSp_over n)
R is Element of the carrier of (m -VectSp_over n)
M + R is Element of the carrier of (m -VectSp_over n)
the addF of (m -VectSp_over n) is Relation-like [: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):] -defined the carrier of (m -VectSp_over n) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):]
[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):] is Relation-like non empty set
[:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is Relation-like non empty set
bool [:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is non empty set
the addF of (m -VectSp_over n) . (M,R) is Element of the carrier of (m -VectSp_over n)
P is Element of the carrier of (m -VectSp_over n)
K * P is Element of the carrier of (m -VectSp_over n)
the lmult of (m -VectSp_over n) is Relation-like [: the carrier of n, the carrier of (m -VectSp_over n):] -defined the carrier of (m -VectSp_over n) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):]
[: the carrier of n, the carrier of (m -VectSp_over n):] is Relation-like non empty set
[:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is non empty set
the lmult of (m -VectSp_over n) . (K,P) is Element of the carrier of (m -VectSp_over n)
Q is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
S is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
Q + S is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
K388( the carrier of n, the carrier of n, the carrier of n, the addF of n,Q,S) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
x1 is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
K * x1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,x1,(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
addLoopStr(# the carrier of (m -VectSp_over n), the addF of (m -VectSp_over n), the ZeroF of (m -VectSp_over n) #) is strict addLoopStr
m -Group_over n is non empty strict right_complementable V95() Abelian add-associative right_zeroed addLoopStr
product ( the addF of n,m) is Relation-like [:(m -tuples_on the carrier of n),(m -tuples_on the carrier of n):] -defined m -tuples_on the carrier of n -valued Function-like non empty total quasi_total Function-yielding V147() Element of bool [:[:(m -tuples_on the carrier of n),(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):]
[:(m -tuples_on the carrier of n),(m -tuples_on the carrier of n):] is Relation-like non empty set
[:[:(m -tuples_on the carrier of n),(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):] is Relation-like non empty set
bool [:[:(m -tuples_on the carrier of n),(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):] is non empty set
addLoopStr(# (m -tuples_on the carrier of n),(product ( the addF of n,m)),(m |-> (0. n)) #) is strict addLoopStr
rng x1 is finite set
(id the carrier of n) * x1 is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
m -Mult_over n is Relation-like [: the carrier of n,(m -tuples_on the carrier of n):] -defined m -tuples_on the carrier of n -valued Function-like non empty total quasi_total Function-yielding V147() Element of bool [:[: the carrier of n,(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):]
[: the carrier of n,(m -tuples_on the carrier of n):] is Relation-like non empty set
[:[: the carrier of n,(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):] is Relation-like non empty set
bool [:[: the carrier of n,(m -tuples_on the carrier of n):],(m -tuples_on the carrier of n):] is non empty set
(m -Mult_over n) . (K,P) is set
the multF of n [;] (K,x1) is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -VectSp_over m is non empty strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of (n -VectSp_over m) is non empty set
the carrier of m is non empty non trivial V103() set
n -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = n } is set
M is Element of the carrier of (n -VectSp_over m)
R is Element of the carrier of (n -VectSp_over m)
M + R is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) is Relation-like [: the carrier of (n -VectSp_over m), the carrier of (n -VectSp_over m):] -defined the carrier of (n -VectSp_over m) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (n -VectSp_over m), the carrier of (n -VectSp_over m):], the carrier of (n -VectSp_over m):]
[: the carrier of (n -VectSp_over m), the carrier of (n -VectSp_over m):] is Relation-like non empty set
[:[: the carrier of (n -VectSp_over m), the carrier of (n -VectSp_over m):], the carrier of (n -VectSp_over m):] is Relation-like non empty set
bool [:[: the carrier of (n -VectSp_over m), the carrier of (n -VectSp_over m):], the carrier of (n -VectSp_over m):] is non empty set
the addF of (n -VectSp_over m) . (M,R) is Element of the carrier of (n -VectSp_over m)
P is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
Q is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
P + Q is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,P,Q) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
Q + P is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,Q,P) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
R + M is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (R,M) is Element of the carrier of (n -VectSp_over m)
M is Element of the carrier of (n -VectSp_over m)
0. (n -VectSp_over m) is zero Element of the carrier of (n -VectSp_over m)
the ZeroF of (n -VectSp_over m) is Element of the carrier of (n -VectSp_over m)
M + (0. (n -VectSp_over m)) is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (M,(0. (n -VectSp_over m))) is Element of the carrier of (n -VectSp_over m)
R is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
P is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
R + P is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,R,P) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
n |-> (0. m) is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (0. m) is Relation-like Seg n -defined Seg n -defined the carrier of m -valued {(0. m)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. m)}:]
{(0. m)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. m)}:] is Relation-like finite set
bool [:(Seg n),{(0. m)}:] is non empty finite V37() set
R + (n |-> (0. m)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,R,(n |-> (0. m))) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
R is Element of the carrier of (n -VectSp_over m)
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = M } is set
P is Relation-like NAT -defined the carrier of m -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of m
- P is Relation-like NAT -defined the carrier of m -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of m
comp m is Relation-like the carrier of m -defined the carrier of m -valued Function-like non empty total quasi_total Element of bool [: the carrier of m, the carrier of m:]
bool [: the carrier of m, the carrier of m:] is non empty set
K391( the carrier of m, the carrier of m,P,(comp m)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
Q is Element of the carrier of (n -VectSp_over m)
R + Q is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (R,Q) is Element of the carrier of (n -VectSp_over m)
P + (- P) is Relation-like NAT -defined the carrier of m -valued Function-like finite M -element FinSequence-like FinSubsequence-like Element of M -tuples_on the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,P,(- P)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
R is Element of the carrier of (n -VectSp_over m)
M is Element of the carrier of (n -VectSp_over m)
P is Element of the carrier of (n -VectSp_over m)
M + R is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (M,R) is Element of the carrier of (n -VectSp_over m)
R + P is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (R,P) is Element of the carrier of (n -VectSp_over m)
(M + R) + P is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . ((M + R),P) is Element of the carrier of (n -VectSp_over m)
x2 is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
x2 + x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,x2,x1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
S is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
Q is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
S + Q is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,S,Q) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
(S + Q) + x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,(S + Q),x1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
Q + x1 is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,Q,x1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
S + (Q + x1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,S,(Q + x1)) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
i1 is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
S + i1 is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
K388( the carrier of m, the carrier of m, the carrier of m, the addF of m,S,i1) is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
M + (R + P) is Element of the carrier of (n -VectSp_over m)
the addF of (n -VectSp_over m) . (M,(R + P)) is Element of the carrier of (n -VectSp_over m)
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -VectSp_over m is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of (n -VectSp_over m) is non empty set
K is Element of the carrier of (n -VectSp_over m)
the carrier of m is non empty non trivial V103() set
n -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = n } is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
n -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is non empty set
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
rng M is finite set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is set
Q is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Q is V26() V27() V28() V32() finite cardinal V105() complex 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
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K, the carrier of m
rng M is finite set
K -VectSp_over m is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of (K -VectSp_over m) is non empty set
bool the carrier of (K -VectSp_over m) is non empty set
m is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
R is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of K,n, the carrier of M
(M,K,n,R) is finite Element of bool the carrier of (n -VectSp_over M)
n -VectSp_over M is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over M
the carrier of (n -VectSp_over M) is non empty set
bool the carrier of (n -VectSp_over M) is non empty set
dom R is finite Element of bool NAT
P is set
R . P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (R,Q) is Relation-like NAT -defined the carrier of M -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of M
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width R) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width R } is set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,P) is Relation-like NAT -defined the carrier of M -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of M
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width R) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width R } is set
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom R is finite Element of bool NAT
Seg (len R) is finite len R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
R . P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng R is finite set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
m -VectSp_over n is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over n
the carrier of (m -VectSp_over n) is non empty set
bool the carrier of (m -VectSp_over n) is non empty set
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is finite Element of bool the carrier of (m -VectSp_over n)
card K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card K) is finite card K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card K ) } is set
card (Seg (card K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is Relation-like Function-like set
dom R is set
rng R is set
Q is set
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng P is finite set
m -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 = m } is set
S is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of n *
len Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng Q is finite set
S is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -VectSp_over m is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of (n -VectSp_over m) is non empty set
K is Relation-like NAT -defined the carrier of (n -VectSp_over m) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over m)
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n -tuples_on the carrier of m is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
{ b₁ where b₁ is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of m * : len b₁ = n } is set
rng K is finite set
M is set
R is Relation-like NAT -defined the carrier of m -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of m
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of m *
rng M is finite set
R is Relation-like NAT -defined the carrier of m -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of m
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,K, the carrier of m
K -VectSp_over m is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of (K -VectSp_over m) is non empty set
(m,n,K,M) is finite Element of bool the carrier of (K -VectSp_over m)
bool the carrier of (K -VectSp_over m) is non empty set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom M is finite Element of bool NAT
R is set
P is set
M . R is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
M . P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Del (M,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
len (Del (M,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x1 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (M,S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width M) -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 M } is set
M . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
ReplaceLine (M,Q,(Line (M,S))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * (Line (M,S)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(1_ K) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((1_ K),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,S)),((1_ K) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine (M,Q,((1_ K) * (Line (M,S)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
DelLine (M,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
(K,(DelLine (M,Q))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of M is non empty non trivial V103() set
n -VectSp_over M is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over M
the carrier of (n -VectSp_over M) is non empty set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
R is Element of the carrier of M
P is Relation-like the carrier of (n -VectSp_over M) -defined the carrier of M -valued Function-like total quasi_total Linear_Combination of n -VectSp_over M
Q is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of M
len Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len Q) is finite len Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len Q ) } is set
Q . K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P . (Q . K) is set
(M,m,n,Q) is Relation-like NAT -defined the carrier of (n -VectSp_over M) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over M)
P (#) (M,m,n,Q) is Relation-like NAT -defined the carrier of (n -VectSp_over M) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over M)
(M,n,(P (#) (M,m,n,Q))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len (P (#) (M,m,n,Q)),n, the carrier of M
len (P (#) (M,m,n,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line ((M,n,(P (#) (M,m,n,Q))),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (M,n,(P (#) (M,m,n,Q))) -element FinSequence-like FinSubsequence-like Element of (width (M,n,(P (#) (M,m,n,Q)))) -tuples_on the carrier of M
width (M,n,(P (#) (M,m,n,Q))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (M,n,(P (#) (M,m,n,Q)))) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width (M,n,(P (#) (M,m,n,Q))) } is set
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of M
(width Q) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width Q } is set
R * (Line (Q,K)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of M
R multfield is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
bool [: the carrier of M, the carrier of M:] is non empty set
the multF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
id the carrier of M is Relation-like the carrier of M -defined the carrier of M -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
the multF of M [;] (R,(id the carrier of M)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
K391( the carrier of M, the carrier of M,(Line (Q,K)),(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
dom Q is finite Element of bool NAT
(M,m,n,Q) /. K is Relation-like Function-like Element of the carrier of (n -VectSp_over M)
n -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = n } is set
dom (M,n,(P (#) (M,m,n,Q))) is finite Element of bool NAT
(M,n,(P (#) (M,m,n,Q))) . K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
R * ((M,m,n,Q) /. K) is Relation-like Function-like Element of the carrier of (n -VectSp_over M)
the lmult of (n -VectSp_over M) is Relation-like [: the carrier of M, the carrier of (n -VectSp_over M):] -defined the carrier of (n -VectSp_over M) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of M, the carrier of (n -VectSp_over M):], the carrier of (n -VectSp_over M):]
[: the carrier of M, the carrier of (n -VectSp_over M):] is Relation-like non empty set
[:[: the carrier of M, the carrier of (n -VectSp_over M):], the carrier of (n -VectSp_over M):] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of (n -VectSp_over M):], the carrier of (n -VectSp_over M):] is non empty set
the lmult of (n -VectSp_over M) . (R,((M,m,n,Q) /. K)) is Relation-like Function-like Element of the carrier of (n -VectSp_over M)
x2 is Relation-like NAT -defined the carrier of M -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of M
R * x2 is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K391( the carrier of M, the carrier of M,x2,(R multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
M is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
K -VectSp_over M is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over M
the carrier of (K -VectSp_over M) is non empty set
the carrier of M is non empty non trivial V103() set
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
R is Relation-like the carrier of (K -VectSp_over M) -defined the carrier of M -valued Function-like total quasi_total Linear_Combination of K -VectSp_over M
Carrier R is finite Element of bool the carrier of (K -VectSp_over M)
bool the carrier of (K -VectSp_over M) is non empty set
Sum R is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
(Sum R) . n is set
P is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,K, the carrier of M
(M,m,K,P) is finite Element of bool the carrier of (K -VectSp_over M)
(M,m,K,P) is Relation-like NAT -defined the carrier of (K -VectSp_over M) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over M)
R (#) (M,m,K,P) is Relation-like NAT -defined the carrier of (K -VectSp_over M) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over M)
(M,K,(R (#) (M,m,K,P))) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len (R (#) (M,m,K,P)),K, the carrier of M
len (R (#) (M,m,K,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col ((M,K,(R (#) (M,m,K,P))),n) is Relation-like NAT -defined the carrier of M -valued Function-like finite len (M,K,(R (#) (M,m,K,P))) -element FinSequence-like FinSubsequence-like Element of (len (M,K,(R (#) (M,m,K,P)))) -tuples_on the carrier of M
len (M,K,(R (#) (M,m,K,P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (M,K,(R (#) (M,m,K,P)))) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = len (M,K,(R (#) (M,m,K,P))) } is set
Sum (Col ((M,K,(R (#) (M,m,K,P))),n)) is Element of the carrier of M
the addF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like non empty set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is non empty set
the addF of M $$ (Col ((M,K,(R (#) (M,m,K,P))),n)) is Element of the carrier of M
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Col ((M,K,(R (#) (M,m,K,P))),n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:NAT, the carrier of M:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of M:] is non empty non trivial non finite V103() set
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
i2 is Relation-like NAT -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of M:]
i2 . (len P) is Element of the carrier of M
i2 . 0 is Element of the carrier of M
Sum (R (#) (M,m,K,P)) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
the addF of (K -VectSp_over M) is Relation-like [: the carrier of (K -VectSp_over M), the carrier of (K -VectSp_over M):] -defined the carrier of (K -VectSp_over M) -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of (K -VectSp_over M), the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):]
[: the carrier of (K -VectSp_over M), the carrier of (K -VectSp_over M):] is Relation-like non empty set
[:[: the carrier of (K -VectSp_over M), the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):] is Relation-like non empty set
bool [:[: the carrier of (K -VectSp_over M), the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):] is non empty set
the addF of (K -VectSp_over M) $$ (R (#) (M,m,K,P)) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
[:NAT, the carrier of (K -VectSp_over M):] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of (K -VectSp_over M):] is non empty non trivial non finite V103() set
0. (K -VectSp_over M) is Relation-like Function-like zero Element of the carrier of (K -VectSp_over M)
the ZeroF of (K -VectSp_over M) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
y1 is Relation-like NAT -defined the carrier of (K -VectSp_over M) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (K -VectSp_over M):]
y1 . (len P) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
y1 . 0 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y1 . Q is set
i2 . Q is set
Q + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
y1 . (Q + 1) is set
i2 . (Q + 1) is set
width (M,K,(R (#) (M,m,K,P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len P) is finite len P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len P ) } is set
dom (M,K,(R (#) (M,m,K,P))) is finite Element of bool NAT
(M,m,K,P) . (Q + 1) is set
Line (P,(Q + 1)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of M
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width P) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width P } is set
dom (M,m,K,P) is finite Element of bool NAT
(M,m,K,P) /. (Q + 1) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
(R (#) (M,m,K,P)) . (Q + 1) is set
P2 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
R . P2 is Element of the carrier of M
(R . P2) * P2 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
the lmult of (K -VectSp_over M) is Relation-like [: the carrier of M, the carrier of (K -VectSp_over M):] -defined the carrier of (K -VectSp_over M) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of M, the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):]
[: the carrier of M, the carrier of (K -VectSp_over M):] is Relation-like non empty set
[:[: the carrier of M, the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):] is Relation-like non empty set
bool [:[: the carrier of M, the carrier of (K -VectSp_over M):], the carrier of (K -VectSp_over M):] is non empty set
the lmult of (K -VectSp_over M) . ((R . P2),P2) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
i is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
i . n is set
y1 . (Q + 1) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
Q2 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 . Q1 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
Q2 + (y1 . Q1) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
the addF of (K -VectSp_over M) . (Q2,(y1 . Q1)) is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = y2 } is set
Q2i is Relation-like NAT -defined the carrier of M -valued Function-like finite y2 -element FinSequence-like FinSubsequence-like Element of y2 -tuples_on the carrier of M
(R . P2) * Q2i is Relation-like NAT -defined the carrier of M -valued Function-like finite y2 -element FinSequence-like FinSubsequence-like Element of y2 -tuples_on the carrier of M
(R . P2) multfield is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
bool [: the carrier of M, the carrier of M:] is non empty set
the multF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
id the carrier of M is Relation-like the carrier of M -defined the carrier of M -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
the multF of M [;] ((R . P2),(id the carrier of M)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of M:]
K391( the carrier of M, the carrier of M,Q2i,((R . P2) multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Line ((M,K,(R (#) (M,m,K,P))),(Q + 1)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (M,K,(R (#) (M,m,K,P))) -element FinSequence-like FinSubsequence-like Element of (width (M,K,(R (#) (M,m,K,P)))) -tuples_on the carrier of M
(width (M,K,(R (#) (M,m,K,P)))) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = width (M,K,(R (#) (M,m,K,P))) } is set
Q2 . n is set
(M,K,(R (#) (M,m,K,P))) * ((Q + 1),n) is Element of the carrier of M
m is Relation-like NAT -defined the carrier of M -valued Function-like finite y2 -element FinSequence-like FinSubsequence-like Element of y2 -tuples_on the carrier of M
dom m is finite y2 -element Element of bool NAT
m . n is set
rng m is finite set
(Col ((M,K,(R (#) (M,m,K,P))),n)) . (Q + 1) is set
i2 . (Q + 1) is Element of the carrier of M
P2m is Element of the carrier of M
i2 . Q1 is Element of the carrier of M
P2m + (i2 . Q1) is Element of the carrier of M
the addF of M . (P2m,(i2 . Q1)) is Element of the carrier of M
SQ2i is Relation-like NAT -defined the carrier of M -valued Function-like finite y2 -element FinSequence-like FinSubsequence-like Element of y2 -tuples_on the carrier of M
m + SQ2i is Relation-like NAT -defined the carrier of M -valued Function-like finite y2 -element FinSequence-like FinSubsequence-like Element of y2 -tuples_on the carrier of M
K388( the carrier of M, the carrier of M, the carrier of M, the addF of M,m,SQ2i) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
SQ2i . n is set
y1 . 0 is set
i2 . 0 is set
K |-> (0. M) is Relation-like NAT -defined the carrier of M -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of M
K -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
{ b₁ where b₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of M * : len b₁ = K } is set
(Seg K) --> (0. M) is Relation-like Seg K -defined Seg K -defined the carrier of M -valued {(0. M)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. M)}:]
{(0. M)} is non empty trivial finite 1 -element set
[:(Seg K),{(0. M)}:] is Relation-like finite set
bool [:(Seg K),{(0. M)}:] is non empty finite V37() set
y2 is Relation-like Function-like Element of the carrier of (K -VectSp_over M)
y2 . n is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K
the carrier of (m -VectSp_over K) is non empty set
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,n,m,M) is finite Element of bool the carrier of (m -VectSp_over K)
bool the carrier of (m -VectSp_over K) is non empty set
(K,n,m,M) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width R) -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 R } is set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
S is Element of the carrier of K
S * (Line (M,Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
S multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (S,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,Q)),(S multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,x1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
Line (M,x1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
Q is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom Q is finite Element of bool NAT
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
S is set
x1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
rng Q is finite set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
Q . x2 is set
i1 is Element of the carrier of K
i2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
dom M is finite Element of bool NAT
M . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
Q . y1 is set
M . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
x2 is zero Element of the carrier of K
i1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
Q . i2 is set
x1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
[: the carrier of (m -VectSp_over K), the carrier of K:] is Relation-like non empty set
bool [: the carrier of (m -VectSp_over K), the carrier of K:] is non empty set
S is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of (m -VectSp_over K), the carrier of K:]
Funcs ( the carrier of (m -VectSp_over K), the carrier of K) is functional non empty FUNCTION_DOMAIN of the carrier of (m -VectSp_over K), the carrier of K
x1 is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Element of Funcs ( the carrier of (m -VectSp_over K), the carrier of K)
x2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
x1 . x2 is Element of the carrier of K
x2 is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Linear_Combination of m -VectSp_over K
Carrier x2 is finite Element of bool the carrier of (m -VectSp_over K)
i1 is set
i2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
x2 . i2 is Element of the carrier of K
i2 is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Linear_Combination of (K,n,m,M)
i2 (#) (K,n,m,M) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
len (i2 (#) (K,n,m,M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q . y2 is set
Line (R,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width R) -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 R } is set
Line (M,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
Q is Element of the carrier of K
Q * (Line (M,y2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
Q multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (Q,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,y2)),(Q multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom (K,n,m,M) is finite Element of bool NAT
(K,n,m,M) /. y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
M . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m -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 = m } is set
P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
dom (i2 (#) (K,n,m,M)) is finite Element of bool NAT
(i2 (#) (K,n,m,M)) . y2 is set
i2 . ((K,n,m,M) /. y2) is Element of the carrier of K
(i2 . ((K,n,m,M) /. y2)) * ((K,n,m,M) /. y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) is Relation-like [: the carrier of K, the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of K, the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the lmult of (m -VectSp_over K) . ((i2 . ((K,n,m,M) /. y2)),((K,n,m,M) /. y2)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
i2 . P1 is set
Q * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K391( the carrier of K, the carrier of K,P1,(Q multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
R . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
m |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
m -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 = m } is set
(Seg m) --> (0. K) is Relation-like Seg m -defined Seg m -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. K)}:] is Relation-like finite set
bool [:(Seg m),{(0. K)}:] is non empty finite V37() set
0. (K,n,m) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
n |-> (m |-> (0. K)) is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (m -tuples_on the carrier of K)
n -tuples_on (m -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
(m -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (m -tuples_on the carrier of K) * : len b₁ = n } is set
(Seg n) --> (m |-> (0. K)) is Relation-like Seg n -defined Seg n -defined m -tuples_on the carrier of K -valued {(m |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(m |-> (0. K))}:]
{(m |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(m |-> (0. K))}:] is Relation-like finite set
bool [:(Seg n),{(m |-> (0. K))}:] is non empty finite V37() set
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K
len (m |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
width P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,n,m,P) is finite Element of bool the carrier of (m -VectSp_over K)
the carrier of (m -VectSp_over K) is non empty set
bool the carrier of (m -VectSp_over K) is non empty set
(K,n,m,P) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
x1 is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Linear_Combination of (K,n,m,P)
Sum x1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
0. (m -VectSp_over K) is Relation-like Function-like zero Element of the carrier of (m -VectSp_over K)
the ZeroF of (m -VectSp_over K) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Carrier x1 is finite Element of bool the carrier of (m -VectSp_over K)
x1 (#) (K,n,m,P) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
(K,m,(x1 (#) (K,n,m,P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len (x1 (#) (K,n,m,P)),m, the carrier of K
len (x1 (#) (K,n,m,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (i2,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width i2 -element FinSequence-like FinSubsequence-like Element of (width i2) -tuples_on the carrier of K
width i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width i2) -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 i2 } is set
Line (P,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
(width P) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width P } is set
y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
x1 . y2 is Element of the carrier of K
P . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Q is Element of the carrier of K
Q * (Line (P,y1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
Q multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (Q,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (P,y1)),(Q multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
y1 is set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (P,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
(width P) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width P } is set
Q is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
x1 . Q is Element of the carrier of K
P . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (i2,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width i2 -element FinSequence-like FinSubsequence-like Element of (width i2) -tuples_on the carrier of K
width i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width i2) -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 i2 } is set
P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
(x1 . Q) * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(x1 . Q) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((x1 . Q),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,P1,((x1 . Q) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(m |-> (0. K)) . Q1 is set
(Sum x1) . Q1 is set
Col (i2,Q1) is Relation-like NAT -defined the carrier of K -valued Function-like finite len i2 -element FinSequence-like FinSubsequence-like Element of (len i2) -tuples_on the carrier of K
len i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len i2) -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 i2 } is set
Sum (Col (i2,Q1)) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (Col (i2,Q1)) is Element of the carrier of K
i2 . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom P1 is finite m -element Element of bool NAT
len P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P1 . Q1 is set
(m |-> (0. K)) . Q1 is set
rng P1 is finite set
((x1 . Q) * P1) . Q1 is set
P2 is Element of the carrier of K
(x1 . Q) * P2 is Element of the carrier of K
the multF of K . ((x1 . Q),P2) is Element of the carrier of K
(Line (i2,y2)) . Q1 is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (P,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
(width P) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width P } is set
0. (m -VectSp_over K) is Relation-like Function-like zero Element of the carrier of (m -VectSp_over K)
the ZeroF of (m -VectSp_over K) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Q is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,P) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
S is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Linear_Combination of (K,n,m,P)
S (#) (K,n,m,P) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
Carrier S is finite Element of bool the carrier of (m -VectSp_over K)
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Sum S is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
(Sum S) . x1 is set
(K,m,(S (#) (K,n,m,P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len (S (#) (K,n,m,P)),m, the carrier of K
len (S (#) (K,n,m,P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col ((K,m,(S (#) (K,n,m,P))),x1) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (K,m,(S (#) (K,n,m,P))) -element FinSequence-like FinSubsequence-like Element of (len (K,m,(S (#) (K,n,m,P)))) -tuples_on the carrier of K
len (K,m,(S (#) (K,n,m,P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len (K,m,(S (#) (K,n,m,P)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len (K,m,(S (#) (K,n,m,P))) } is set
Sum (Col ((K,m,(S (#) (K,n,m,P))),x1)) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K $$ (Col ((K,m,(S (#) (K,n,m,P))),x1)) is Element of the carrier of K
(m |-> (0. K)) . x1 is set
x1 is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
len x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices Q is set
dom Q is finite Element of bool NAT
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width Q) is finite width Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width Q ) } is set
[:(dom Q),(Seg (width Q)):] is Relation-like finite set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[x2,i1] is set
{x2,i1} is non empty finite V37() set
{x2} is non empty trivial finite V37() 1 -element set
{{x2,i1},{x2}} is non empty finite V37() without_zero V103() set
Q * (x2,i1) is Element of the carrier of K
(0. (K,n,m)) * (x2,i1) is Element of the carrier of K
Indices (0. (K,n,m)) is set
dom (0. (K,n,m)) is finite Element of bool NAT
width (0. (K,n,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (0. (K,n,m))) is finite width (0. (K,n,m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,n,m)) ) } is set
[:(dom (0. (K,n,m))),(Seg (width (0. (K,n,m)))):] is Relation-like finite set
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len P) is finite len P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len P ) } is set
[:(Seg (len P)),(Seg (width P)):] is Relation-like finite set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (P,i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
P . i2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (P,i2)) . y1 is set
P * (i2,y1) is Element of the carrier of K
Line ((K,m,(S (#) (K,n,m,P))),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,m,(S (#) (K,n,m,P))) -element FinSequence-like FinSubsequence-like Element of (width (K,m,(S (#) (K,n,m,P)))) -tuples_on the carrier of K
width (K,m,(S (#) (K,n,m,P))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (K,m,(S (#) (K,n,m,P)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (K,m,(S (#) (K,n,m,P))) } is set
(Line ((K,m,(S (#) (K,n,m,P))),i2)) . y1 is set
y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
S . y2 is Element of the carrier of K
(S . y2) * (Line (P,i2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of K
(S . y2) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((S . y2),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (P,i2)),((S . y2) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((S . y2) * (Line (P,i2))) . y1 is set
(S . y2) * (P * (i2,y1)) is Element of the carrier of K
the multF of K . ((S . y2),(P * (i2,y1))) is Element of the carrier of K
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,n,m,M) is finite Element of bool the carrier of (m -VectSp_over K)
the carrier of (m -VectSp_over K) is non empty set
bool the carrier of (m -VectSp_over K) is non empty set
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
m |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
m -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 = m } is set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (0. K) is Relation-like Seg m -defined Seg m -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. K)}:] is Relation-like finite set
bool [:(Seg m),{(0. K)}:] is non empty finite V37() set
0. (K,n,m) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
n |-> (m |-> (0. K)) is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (m -tuples_on the carrier of K)
n -tuples_on (m -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
(m -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (m -tuples_on the carrier of K) * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (m |-> (0. K)) is Relation-like Seg n -defined Seg n -defined m -tuples_on the carrier of K -valued {(m |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(m |-> (0. K))}:]
{(m |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(m |-> (0. K))}:] is Relation-like finite set
bool [:(Seg n),{(m |-> (0. K))}:] is non empty finite V37() set
x1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
Indices x1 is set
dom x1 is finite Element of bool NAT
width x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width x1) is finite width x1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x1 ) } is set
[:(dom x1),(Seg (width x1)):] is Relation-like finite set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[x2,i1] is set
{x2,i1} is non empty finite V37() set
{x2} is non empty trivial finite V37() 1 -element set
{{x2,i1},{x2}} is non empty finite V37() without_zero V103() set
x1 * (x2,i1) is Element of the carrier of K
(0. (K,n,m)) * (x2,i1) is Element of the carrier of K
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices (0. (K,n,m)) is set
dom (0. (K,n,m)) is finite Element of bool NAT
width (0. (K,n,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (0. (K,n,m))) is finite width (0. (K,n,m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,n,m)) ) } is set
[:(dom (0. (K,n,m))),(Seg (width (0. (K,n,m)))):] is Relation-like finite set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 * (i2,y1) is Element of the carrier of K
Indices M is set
dom M is finite Element of bool NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(dom M),(Seg (width M)):] is Relation-like finite set
[:(Seg n),(Seg m):] is Relation-like finite set
Line (x1,i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x1 -element FinSequence-like FinSubsequence-like Element of (width x1) -tuples_on the carrier of K
(width x1) -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 x1 } is set
Line (M,i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
y2 is Element of the carrier of K
y2 * (Line (M,i2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
y2 multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (y2,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,i2)),(y2 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 (M,i2)) . y1 is set
M * (i2,y1) is Element of the carrier of K
ReplaceLine (M,i2,(y2 * (Line (M,i2)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,M) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
P1 is Relation-like the carrier of (m -VectSp_over K) -defined the carrier of K -valued Function-like total quasi_total Linear_Combination of (K,n,m,M)
P1 (#) (K,n,m,M) is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
len x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:NAT, the carrier of (m -VectSp_over K):] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of (m -VectSp_over K):] is non empty non trivial non finite V103() set
Sum (P1 (#) (K,n,m,M)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) is Relation-like [: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the addF of (m -VectSp_over K) $$ (P1 (#) (K,n,m,M)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
0. (m -VectSp_over K) is Relation-like Function-like zero Element of the carrier of (m -VectSp_over K)
the ZeroF of (m -VectSp_over K) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
P2 is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (m -VectSp_over K):]
P2 . n is set
P2 . 0 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,(ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) -tuples_on the carrier of K
width (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) -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 (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) } is set
len (Line (M,i2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (y2 * (Line (M,i2))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 . i is set
i + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
P2 . (i + 1) is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R -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 = R } is set
Line (x1,(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x1 -element FinSequence-like FinSubsequence-like Element of (width x1) -tuples_on the carrier of K
ES is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + ES is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)),ES) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + ES)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + ES)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . (i + 1) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . m is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Si is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)),Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
SQ2i is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
SQ2i + ES is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,SQ2i,ES) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
SQ2i + Si is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,SQ2i,Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Si is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
SQ2i + Si is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,SQ2i,Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (SQ2i + Si) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (SQ2i + Si) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
ReplaceLine ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),i2,(SQ2i + ES)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
mSi is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Replace ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
J is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,J) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
Replace ((Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,J)),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(SQ2i + ES)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
x1 . (i + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,n,m,x1) is finite Element of bool the carrier of (m -VectSp_over K)
(P1 (#) (K,n,m,M)) . (i + 1) is set
Line (M,(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
a is Element of the carrier of K
a * (Line (M,(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
a multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (a,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,(i + 1))),(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 ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) -tuples_on the carrier of K
Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) -tuples_on the carrier of K
(width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) -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 (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) } is set
RR is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
(P2 . m) + RR is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) . ((P2 . m),RR) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Si + (Line (x1,(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,Si,(Line (x1,(i + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P2m is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
(SQ2i + Si) + P2m is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(SQ2i + Si),P2m) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) -tuples_on the carrier of K
a * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),(i + 1))),(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 ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),i2)) + (a * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),(i + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),i2)),(a * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))),(i + 1))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + Si)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . 0 is set
i is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + i is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + i)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,((Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2)) + i)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 . i is set
i + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
P2 . (i + 1) is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R -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 = R } is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . m is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Line (x1,(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x1 -element FinSequence-like FinSubsequence-like Element of (width x1) -tuples_on the carrier of K
x1 . (i + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,n,m,x1) is finite Element of bool the carrier of (m -VectSp_over K)
(P1 (#) (K,n,m,M)) . (i + 1) is set
Si is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Si) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Si))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 . (i + 1) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Si is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
(P2 . m) + Si is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) . ((P2 . m),Si) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Q2i is Relation-like NAT -defined the carrier of K -valued Function-like finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on the carrier of K
ES is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
Q2i + ES is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,Q2i,ES) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Line (M,(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
LC is Element of the carrier of K
LC * (Line (M,(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
LC multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (LC,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,(i + 1))),(LC multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len ES is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2m is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
P2m + Q2i is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P2m,Q2i) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
len Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) -tuples_on the carrier of K
width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) -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 (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) } is set
Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) -tuples_on the carrier of K
LC * (Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (M,i2,(y2 * (Line (M,i2))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (M,i2,(y2 * (Line (M,i2)))))) -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1))),(LC 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 ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2)) + (LC * (Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2)),(LC * (Line ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),(i + 1))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),(i + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) -tuples_on the carrier of K
LC * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),(i + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),(i + 1))),(LC 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 ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2)) + (LC * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),(i + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2)),(LC * (Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),(i + 1))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
ReplaceLine ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2,Si) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
mSi is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Replace ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
J is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,J) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
Replace ((Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,J)),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
Replace ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,mSi) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,Q2i))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,i) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,i))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Carrier P1 is finite Element of bool the carrier of (m -VectSp_over K)
(K,m,(P1 (#) (K,n,m,M))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len (P1 (#) (K,n,m,M)),m, the carrier of K
len (P1 (#) (K,n,m,M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col (x1,m) is Relation-like NAT -defined the carrier of K -valued Function-like finite len x1 -element FinSequence-like FinSubsequence-like Element of (len x1) -tuples_on the carrier of K
(len x1) -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 x1 } is set
Sum (Col (x1,m)) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (Col (x1,m)) is Element of the carrier of K
Sum P1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
(Sum P1) . m is set
i is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
i . m is set
(m |-> (0. K)) . m is set
dom (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) is finite Element of bool NAT
Seg (len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K))))) is finite len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) ) } is set
Del ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))),i2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
len (Del ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))),i2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
m + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
len i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (x1,i2)) . y1 is set
y2 * (M * (i2,y1)) is Element of the carrier of K
the multF of K . (y2,(M * (i2,y1))) is Element of the carrier of K
len (m |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K))))) -tuples_on the carrier of K
(width (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K))))) -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 (ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))) } is set
(K,(ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
DelLine ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))),i2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
(K,(DelLine ((ReplaceLine ((ReplaceLine (M,i2,(y2 * (Line (M,i2))))),i2,(m |-> (0. K)))),i2))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Del (M,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
len (Del (M,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
DelLine (M,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence of the carrier of K *
(K,(DelLine (M,Q))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
m -VectSp_over n is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal upper_triangular lower_triangular Matrix of m,m, the carrier of n
(n,K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(n,m,m,K) is finite Element of bool the carrier of (m -VectSp_over n)
the carrier of (m -VectSp_over n) is non empty set
bool the carrier of (m -VectSp_over n) is non empty set
Lin (n,m,m,K) is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of m -VectSp_over n
Q is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
P is non empty right_complementable V95() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of m -VectSp_over n
m -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 = m } is set
width K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
S is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
x1 is Relation-like NAT -defined (width K) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of (width K) -tuples_on the carrier of n
len x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom x1 is finite Element of bool NAT
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
x2 is Relation-like NAT -defined the carrier of (m -VectSp_over n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over n)
(n,m,x2) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len x2,m, the carrier of n
len x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
Line (i1,i2) is Relation-like NAT -defined the carrier of n -valued Function-like finite width i1 -element FinSequence-like FinSubsequence-like Element of (width i1) -tuples_on the carrier of n
width i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width i1) -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 i1 } is set
i1 . i2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (K,i2) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
S /. i2 is Element of the carrier of n
K * (i2,i2) is Element of the carrier of n
(K * (i2,i2)) " is Element of the carrier of n
(S /. i2) * ((K * (i2,i2)) ") is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n . ((S /. i2),((K * (i2,i2)) ")) is Element of the carrier of n
((S /. i2) * ((K * (i2,i2)) ")) * (Line (K,i2)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
((S /. i2) * ((K * (i2,i2)) ")) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (((S /. i2) * ((K * (i2,i2)) ")),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (K,i2)),(((S /. i2) * ((K * (i2,i2)) ")) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,m,m,K) is Relation-like NAT -defined the carrier of (m -VectSp_over n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over n)
i2 is Relation-like the carrier of (m -VectSp_over n) -defined the carrier of n -valued Function-like total quasi_total Linear_Combination of (n,m,m,K)
i2 (#) (n,m,m,K) is Relation-like NAT -defined the carrier of (m -VectSp_over n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over n)
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Sum i2 is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
Carrier i2 is finite Element of bool the carrier of (m -VectSp_over n)
diagonal_of_Matrix K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(diagonal_of_Matrix K) . P1 is set
K * (P1,P1) is Element of the carrier of n
len (diagonal_of_Matrix K) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (diagonal_of_Matrix K) is finite Element of bool NAT
Line (K,P1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(Line (K,P1)) . P1 is set
Col (i1,P1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len i1 -element FinSequence-like FinSubsequence-like Element of (len i1) -tuples_on the carrier of n
len i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len i1) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len i1 } is set
dom S is finite m -element Element of bool NAT
len (Col (i1,P1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Col (i1,P1)) is finite len i1 -element Element of bool NAT
Seg (len i1) is finite len i1 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len i1 ) } is set
len K is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom K is finite Element of bool NAT
Det K is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product K is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product K)) is Element of the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
dom i1 is finite Element of bool NAT
Product (diagonal_of_Matrix K) is Element of the carrier of n
the multF of n $$ (diagonal_of_Matrix K) is Element of the carrier of n
Line (i1,P1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width i1 -element FinSequence-like FinSubsequence-like Element of (width i1) -tuples_on the carrier of n
i1 . P1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
S /. P1 is Element of the carrier of n
(K * (P1,P1)) " is Element of the carrier of n
(S /. P1) * ((K * (P1,P1)) ") is Element of the carrier of n
the multF of n . ((S /. P1),((K * (P1,P1)) ")) is Element of the carrier of n
((S /. P1) * ((K * (P1,P1)) ")) * (Line (K,P1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
((S /. P1) * ((K * (P1,P1)) ")) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (((S /. P1) * ((K * (P1,P1)) ")),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (K,P1)),(((S /. P1) * ((K * (P1,P1)) ")) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[P2,P1] is set
{P2,P1} is non empty finite V37() set
{P2} is non empty trivial finite V37() 1 -element set
{{P2,P1},{P2}} is non empty finite V37() without_zero V103() set
Indices K is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
Line (K,P2) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(Line (K,P2)) . P1 is set
K * (P2,P1) is Element of the carrier of n
(n,m,m,K) /. P2 is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
K . P2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (i1,P2) is Relation-like NAT -defined the carrier of n -valued Function-like finite width i1 -element FinSequence-like FinSubsequence-like Element of (width i1) -tuples_on the carrier of n
i1 . P2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i2 . ((n,m,m,K) /. P2) is Element of the carrier of n
(i2 . ((n,m,m,K) /. P2)) * ((n,m,m,K) /. P2) is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
the lmult of (m -VectSp_over n) is Relation-like [: the carrier of n, the carrier of (m -VectSp_over n):] -defined the carrier of (m -VectSp_over n) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):]
[: the carrier of n, the carrier of (m -VectSp_over n):] is Relation-like non empty set
[:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is non empty set
the lmult of (m -VectSp_over n) . ((i2 . ((n,m,m,K) /. P2)),((n,m,m,K) /. P2)) is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
(i2 . ((n,m,m,K) /. P2)) * (Line (K,P2)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(i2 . ((n,m,m,K) /. P2)) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] ((i2 . ((n,m,m,K) /. P2)),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K391( the carrier of n, the carrier of n,(Line (K,P2)),((i2 . ((n,m,m,K) /. P2)) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Line (i1,P2)) . P1 is set
(i2 . ((n,m,m,K) /. P2)) * (0. n) is Element of the carrier of n
the multF of n . ((i2 . ((n,m,m,K) /. P2)),(0. n)) is Element of the carrier of n
i1 * (P2,P1) is Element of the carrier of n
(Col (i1,P1)) . P2 is set
(Col (i1,P1)) . P1 is set
Sum (Col (i1,P1)) is Element of the carrier of n
the addF of n $$ (Col (i1,P1)) is Element of the carrier of n
y2 is Relation-like NAT -defined the carrier of n -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of n
y2 . P1 is set
i1 * (P1,P1) is Element of the carrier of n
(Line (i1,P1)) . P1 is set
((S /. P1) * ((K * (P1,P1)) ")) * (K * (P1,P1)) is Element of the carrier of n
the multF of n . (((S /. P1) * ((K * (P1,P1)) ")),(K * (P1,P1))) is Element of the carrier of n
((K * (P1,P1)) ") * (K * (P1,P1)) is Element of the carrier of n
the multF of n . (((K * (P1,P1)) "),(K * (P1,P1))) is Element of the carrier of n
(S /. P1) * (((K * (P1,P1)) ") * (K * (P1,P1))) is Element of the carrier of n
the multF of n . ((S /. P1),(((K * (P1,P1)) ") * (K * (P1,P1)))) is Element of the carrier of n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
the OneF of n is Element of the carrier of n
(S /. P1) * (1_ n) is Element of the carrier of n
the multF of n . ((S /. P1),(1_ n)) is Element of the carrier of n
S . P1 is set
len y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
the addF of (m -VectSp_over n) is Relation-like [: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):] -defined the carrier of (m -VectSp_over n) -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):]
[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):] is Relation-like non empty set
[:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is Relation-like non empty set
bool [:[: the carrier of (m -VectSp_over n), the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is non empty set
the ZeroF of (m -VectSp_over n) is Relation-like Function-like Element of the carrier of (m -VectSp_over n)
the lmult of (m -VectSp_over n) is Relation-like [: the carrier of n, the carrier of (m -VectSp_over n):] -defined the carrier of (m -VectSp_over n) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):]
[: the carrier of n, the carrier of (m -VectSp_over n):] is Relation-like non empty set
[:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of (m -VectSp_over n):], the carrier of (m -VectSp_over n):] is non empty set
VectSpStr(# the carrier of (m -VectSp_over n), the addF of (m -VectSp_over n), the ZeroF of (m -VectSp_over n), the lmult of (m -VectSp_over n) #) is non empty strict VectSpStr over n
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
1. (n,m) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(n,m,m,(1. (n,m))) is finite Element of bool the carrier of (m -VectSp_over n)
m -VectSp_over n is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over n
the carrier of (m -VectSp_over n) is non empty set
bool the carrier of (m -VectSp_over n) is non empty set
(n,(1. (n,m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Det (1. (n,m)) is Element of the carrier of n
Permutations m is non empty permutational set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
FinOmega (Permutations m) is Element of Fin (Permutations m)
Fin (Permutations m) is preBoolean set
Path_product (1. (n,m)) is Relation-like Permutations m -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:(Permutations m), the carrier of n:]
[:(Permutations m), the carrier of n:] is Relation-like non empty set
bool [:(Permutations m), the carrier of n:] is non empty set
the addF of n $$ ((FinOmega (Permutations m)),(Path_product (1. (n,m)))) is Element of the carrier of n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
the OneF of n is Element of the carrier of n
Indices (1. (n,m)) is set
dom (1. (n,m)) is finite Element of bool NAT
width (1. (n,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width (1. (n,m))) is finite width (1. (n,m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (n,m)) ) } is set
[:(dom (1. (n,m))),(Seg (width (1. (n,m)))):] is Relation-like finite set
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
[M,R] is set
{M,R} is non empty finite V37() set
{M} is non empty trivial finite V37() 1 -element set
{{M,R},{M}} is non empty finite V37() without_zero V103() set
(1. (n,m)) * (M,R) is Element of the carrier of n
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n -VectSp_over m is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
1. (m,n) is Relation-like NAT -defined the carrier of m * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,n, the carrier of m
the carrier of m is non empty non trivial V103() set
the carrier of m * is functional non empty FinSequence-membered FinSequenceSet of the carrier of m
(m,n,n,(1. (m,n))) is finite Element of bool the carrier of (n -VectSp_over m)
the carrier of (n -VectSp_over m) is non empty set
bool the carrier of (n -VectSp_over m) is non empty set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
m -VectSp_over n is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over n
dim (m -VectSp_over n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
1. (n,m) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,m, the carrier of n
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
len (1. (n,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (1. (n,m)) is finite Element of bool NAT
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(1. (n,m)) .: (Seg m) is finite set
(n,m,m,(1. (n,m))) is finite Element of bool the carrier of (m -VectSp_over n)
the carrier of (m -VectSp_over n) is non empty set
bool the carrier of (m -VectSp_over n) is non empty set
(n,(1. (n,m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (n,m,m,(1. (n,m))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(R) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (width M)) \ (R) is finite without_zero Element of bool NAT
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
card (Seg (len M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card ((Seg (width M)) \ (R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg (len M))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len M)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len M))) -tuples_on NAT
(card (Seg (len M))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len M)) } is set
(((Seg (width M)) \ (R))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width M)) \ (R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (width M)) \ (R))) -tuples_on NAT
(card ((Seg (width M)) \ (R))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width M)) \ (R)) } is set
( the carrier of K,M,(card (Seg (len M))),(card ((Seg (width M)) \ (R))),((Seg (len M))),(((Seg (width M)) \ (R)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
P is Element of the carrier of K
len ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is finite Element of bool NAT
dom M is finite Element of bool NAT
i1 is set
dom ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is finite set
i2 is set
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) . i1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) . i2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty finite V37() set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (M,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
(Line (M,y1)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of K:] is non empty non trivial non finite V103() set
((Seg (len M))) . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,(((Seg (len M))) . y1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,(((Seg (len M))) . y1))) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -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,M,(Seg (len M)),((Seg (width M)) \ (R))) } is set
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
((Seg (len M))) . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,(((Seg (len M))) . y2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,(((Seg (len M))) . y2))) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
Line (M,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,y2)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M * (y2,Q) is Element of the carrier of K
M * (y1,Q) is Element of the carrier of K
(Line (M,y1)) . Q is set
(Line (M,y2)) . Q is set
rng (((Seg (width M)) \ (R))) is finite V212() V213() V214() V217() set
dom (((Seg (width M)) \ (R))) is finite card ((Seg (width M)) \ (R)) -element Element of bool NAT
P1 is set
(((Seg (width M)) \ (R))) . P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Line (M,y1)) . Q is set
((Line (M,y1)) * (((Seg (width M)) \ (R)))) . P1 is set
(Line (M,y2)) . Q is set
len (Line (M,y2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Line (M,y1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
M . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty finite V37() set
i1 is set
dom M is finite set
i2 is set
M . i1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
M . i2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (M,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
M . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (M,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
M . y2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) . i1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -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,M,(Seg (len M)),((Seg (width M)) \ (R))) } is set
((Seg (len M))) . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,(((Seg (len M))) . y1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,(((Seg (len M))) . y1))) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of K:] is non empty non trivial non finite V103() set
(Line (M,y2)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
((Seg (len M))) . y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,(((Seg (len M))) . y2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,(((Seg (len M))) . y2))) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) . i2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,M) is finite Element of bool the carrier of (m -VectSp_over K)
the carrier of (m -VectSp_over K) is non empty set
bool the carrier of (m -VectSp_over K) is non empty set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
card (Seg (len M)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(R) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (width M)) \ (R) is finite without_zero Element of bool NAT
card ((Seg (width M)) \ (R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card ((Seg (width M)) \ (R))) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
((Seg (len M))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len M)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg (len M))) -tuples_on NAT
(card (Seg (len M))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len M)) } is set
(((Seg (width M)) \ (R))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width M)) \ (R)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (width M)) \ (R))) -tuples_on NAT
(card ((Seg (width M)) \ (R))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width M)) \ (R)) } is set
( the carrier of K,M,(card (Seg (len M))),(card ((Seg (width M)) \ (R))),((Seg (len M))),(((Seg (width M)) \ (R)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
(K,(card (Seg (len M))),(card ((Seg (width M)) \ (R))),( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) is finite Element of bool the carrier of ((card ((Seg (width M)) \ (R))) -VectSp_over K)
the carrier of ((card ((Seg (width M)) \ (R))) -VectSp_over K) is non empty set
bool the carrier of ((card ((Seg (width M)) \ (R))) -VectSp_over K) is non empty set
len ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
m -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 = m } is set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (0. K) is Relation-like Seg m -defined Seg m -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. K)}:] is Relation-like finite set
bool [:(Seg m),{(0. K)}:] is non empty finite V37() set
len (m |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (((Seg (width M)) \ (R))) is finite card ((Seg (width M)) \ (R)) -element Element of bool NAT
Seg (card ((Seg (width M)) \ (R))) is finite card ((Seg (width M)) \ (R)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card ((Seg (width M)) \ (R)) ) } is set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg (card (Seg (len M))) is finite card (Seg (len M)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (len M)) ) } is set
Line (M,x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
i1 is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
len i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 . i2 is set
(m |-> (0. K)) . i2 is set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty finite V37() set
((Seg (len M))) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
rng (((Seg (width M)) \ (R))) is finite V212() V213() V214() V217() set
M * (x2,i2) is Element of the carrier of K
y1 is set
(((Seg (width M)) \ (R))) . y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -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,M,(Seg (len M)),((Seg (width M)) \ (R))) } is set
(card ((Seg (width M)) \ (R))) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite card ((Seg (width M)) \ (R)) -element FinSequence-like FinSubsequence-like Element of (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K
(card ((Seg (width M)) \ (R))) -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 = card ((Seg (width M)) \ (R)) } is set
(Seg (card ((Seg (width M)) \ (R)))) --> (0. K) is Relation-like Seg (card ((Seg (width M)) \ (R))) -defined Seg (card ((Seg (width M)) \ (R))) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card ((Seg (width M)) \ (R)))),{(0. K)}:]
[:(Seg (card ((Seg (width M)) \ (R)))),{(0. K)}:] is Relation-like finite set
bool [:(Seg (card ((Seg (width M)) \ (R)))),{(0. K)}:] is non empty finite V37() set
Line (M,(((Seg (len M))) . x2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,(((Seg (len M))) . x2))) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of K:] is non empty non trivial non finite V103() set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),x2)) . y2 is set
(Line (M,(((Seg (len M))) . x2))) . i2 is set
0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ (R)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
(card (Seg (len M))) |-> ((card ((Seg (width M)) \ (R))) |-> (0. K)) is Relation-like NAT -defined (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K -valued Function-like finite card (Seg (len M)) -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (card (Seg (len M))) -tuples_on ((card ((Seg (width M)) \ (R))) -tuples_on the carrier of K)
(card (Seg (len M))) -tuples_on ((card ((Seg (width M)) \ (R))) -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K
((card ((Seg (width M)) \ (R))) -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((card ((Seg (width M)) \ (R))) -tuples_on the carrier of K) * : len b₁ = card (Seg (len M)) } is set
(Seg (card (Seg (len M)))) --> ((card ((Seg (width M)) \ (R))) |-> (0. K)) is Relation-like Seg (card (Seg (len M))) -defined Seg (card (Seg (len M))) -defined (card ((Seg (width M)) \ (R))) -tuples_on the carrier of K -valued {((card ((Seg (width M)) \ (R))) |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (card (Seg (len M)))),{((card ((Seg (width M)) \ (R))) |-> (0. K))}:]
{((card ((Seg (width M)) \ (R))) |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (card (Seg (len M)))),{((card ((Seg (width M)) \ (R))) |-> (0. K))}:] is Relation-like finite set
bool [:(Seg (card (Seg (len M)))),{((card ((Seg (width M)) \ (R))) |-> (0. K))}:] is non empty finite V37() set
x2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg (len M)), card ((Seg (width M)) \ (R)), the carrier of K
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (x2,i1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x2 -element FinSequence-like FinSubsequence-like Element of (width x2) -tuples_on the carrier of K
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width x2) -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 x2 } is set
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
i2 is Element of the carrier of K
i2 * (Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
i2 multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (i2,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i1)),(i2 multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (x2,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x2 -element FinSequence-like FinSubsequence-like Element of (width x2) -tuples_on the carrier of K
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
i1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom i1 is finite Element of bool NAT
i2 is Relation-like NAT -defined (width M) -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of (width M) -tuples_on the carrier of K
len i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom i2 is finite Element of bool NAT
y1 is Relation-like NAT -defined the carrier of (m -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (m -VectSp_over K)
(K,m,y1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of len y1,m, the carrier of K
len y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
len y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom y2 is finite Element of bool NAT
len x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len y2) is finite len y2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len y2 ) } is set
dom x2 is finite Element of bool NAT
Seg (len x2) is finite len x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len x2 ) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Col (y2,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite len y2 -element FinSequence-like FinSubsequence-like Element of (len y2) -tuples_on the carrier of K
(len y2) -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 y2 } is set
len (Col (y2,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n |-> (0. K) 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
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
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is non empty finite V37() set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
width y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y2 . P2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y2 /. P2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
M * (P2,R) is Element of the carrier of K
Line (M,P2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,P2)) . R is set
i1 /. P2 is Element of the carrier of K
(i1 /. P2) * (0. 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 non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the multF of K . ((i1 /. P2),(0. K)) is Element of the carrier of K
(i1 /. P2) * (Line (M,P2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(i1 /. P2) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((i1 /. P2),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,P2)),((i1 /. P2) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((i1 /. P2) * (Line (M,P2))) . R is set
(y2 /. P2) . R is set
Line (y2,P2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y2 -element FinSequence-like FinSubsequence-like Element of (width y2) -tuples_on the carrier of K
(width y2) -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 y2 } is set
(Line (y2,P2)) . R is set
y2 * (P2,R) is Element of the carrier of K
Col (y2,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite len y2 -element FinSequence-like FinSubsequence-like Element of (len y2) -tuples_on the carrier of K
(Col (y2,R)) . P2 is set
(Col (y2,Q)) . P2 is set
(n |-> (0. K)) . P2 is set
len (n |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Sum (Col (y2,Q)) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (Col (y2,Q)) is Element of the carrier of K
Q1 is set
(((Seg (width M)) \ (R))) . Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Col (x2,P2) is Relation-like NAT -defined the carrier of K -valued Function-like finite len x2 -element FinSequence-like FinSubsequence-like Element of (len x2) -tuples_on the carrier of K
(len x2) -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 x2 } is set
i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
width y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (M,i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(Line (M,i)) . Q is set
M * (i,Q) is Element of the carrier of K
i1 . i is set
Line (x2,i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x2 -element FinSequence-like FinSubsequence-like Element of (width x2) -tuples_on the carrier of K
(width x2) -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 x2 } is set
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
m is Element of the carrier of K
m * (Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
m multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (m,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i)),(m multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
y2 . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y2 /. i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(idseq n) . i is set
(Line (M,i)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
(Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i)) . P2 is set
(Col (y2,Q)) . i is set
y2 * (i,Q) is Element of the carrier of K
Line (y2,i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y2 -element FinSequence-like FinSubsequence-like Element of (width y2) -tuples_on the carrier of K
(width y2) -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 y2 } is set
(Line (y2,i)) . Q is set
(y2 /. i) . Q is set
i1 /. i is Element of the carrier of K
(i1 /. i) * (Line (M,i)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(i1 /. i) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((i1 /. i),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,i)),((i1 /. i) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((i1 /. i) * (Line (M,i))) . Q is set
m * (Line (M,i)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(Line (M,i)),(m multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(m * (Line (M,i))) . Q is set
m * (M * (i,Q)) is Element of the carrier of K
the multF of K . (m,(M * (i,Q))) is Element of the carrier of K
(m * (Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),i))) . P2 is set
x2 * (i,P2) is Element of the carrier of K
(Col (x2,P2)) . i is set
len (Col (x2,P2)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Sum (Col (y2,Q)) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (Col (y2,Q)) is Element of the carrier of K
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 /. Q is Element of the carrier of K
Line (y2,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y2 -element FinSequence-like FinSubsequence-like Element of (width y2) -tuples_on the carrier of K
width y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width y2) -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 y2 } is set
y2 . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
P1 is Element of the carrier of K
P1 * (Line (M,Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
P1 multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (P1,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (M,Q)),(P1 multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. (K,n,m) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
n |-> (m |-> (0. K)) is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (m -tuples_on the carrier of K)
n -tuples_on (m -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
(m -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of m -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined m -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (m -tuples_on the carrier of K) * : len b₁ = n } is set
(Seg n) --> (m |-> (0. K)) is Relation-like Seg n -defined Seg n -defined m -tuples_on the carrier of K -valued {(m |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(m |-> (0. K))}:]
{(m |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(m |-> (0. K))}:] is Relation-like finite set
bool [:(Seg n),{(m |-> (0. K))}:] is non empty finite V37() set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 . Q is set
Line (x2,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x2 -element FinSequence-like FinSubsequence-like Element of (width x2) -tuples_on the carrier of K
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width x2) -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 x2 } is set
Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
P1 is Element of the carrier of K
P1 * (Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R)))) -tuples_on the carrier of K
P1 multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (P1,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (( the carrier of K,M,(Seg (len M)),((Seg (width M)) \ (R))),Q)),(P1 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 ((0. (K,n,m)),Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (0. (K,n,m)) -element FinSequence-like FinSubsequence-like Element of (width (0. (K,n,m))) -tuples_on the carrier of K
width (0. (K,n,m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (0. (K,n,m))) -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 (0. (K,n,m)) } is set
(0. (K,n,m)) . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i1 /. Q is Element of the carrier of K
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
P1 * (Line (M,Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(Line (M,Q)),(P1 multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
y2 . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (y2,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y2 -element FinSequence-like FinSubsequence-like Element of (width y2) -tuples_on the carrier of K
((Seg (len M))) . Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(Line (M,Q)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
(Seg m) /\ ((Seg (width M)) \ (R)) is finite Element of bool NAT
(((Seg (width M)) \ (R))) " (Seg m) is finite set
(((Seg (width M)) \ (R))) " (rng (((Seg (width M)) \ (R)))) is finite set
dom (Line (M,Q)) is finite width M -element Element of bool NAT
len (Line (M,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len (Line (M,Q))) is finite len (Line (M,Q)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Line (M,Q)) ) } is set
(Line ((0. (K,n,m)),Q)) * (((Seg (width M)) \ (R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
(0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ (R))))) . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x2 . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ (R))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of m is non empty non trivial V103() set
n is Element of the carrier of m
K is non empty right_complementable V95() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over m
the carrier of K is non empty set
bool the carrier of K is non empty set
M is finite Element of bool the carrier of K
R is Element of the carrier of K
P is Element of the carrier of K
{R} is non empty trivial finite 1 -element Element of bool the carrier of K
M \ {R} is finite Element of bool the carrier of K
n * P is Element of the carrier of K
the lmult of K is Relation-like [: the carrier of m, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of m, the carrier of K:], the carrier of K:]
[: the carrier of m, the carrier of K:] is Relation-like non empty set
[:[: the carrier of m, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of K:], the carrier of K:] is non empty set
the lmult of K . (n,P) is Element of the carrier of K
R + (n * P) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K . (R,(n * P)) is Element of the carrier of K
{(R + (n * P))} is non empty trivial finite 1 -element Element of bool the carrier of K
(M \ {R}) \/ {(R + (n * P))} is non empty finite Element of bool the carrier of K
x2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of (M \ {R}) \/ {(R + (n * P))}
Sum x2 is Element of the carrier of K
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
x2 . (R + (n * P)) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
Carrier x2 is finite Element of bool the carrier of K
i1 is set
i2 is Element of the carrier of K
x2 . i2 is Element of the carrier of m
i1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of M \ {R}
Sum i1 is Element of the carrier of K
x2 . (R + (n * P)) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
Carrier x2 is finite Element of bool the carrier of K
M \/ {(R + (n * P))} is non empty finite Element of bool the carrier of K
Lin {(R + (n * P))} is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of K
i1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {(R + (n * P))}
Sum i1 is Element of the carrier of K
(x2 . (R + (n * P))) * i1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
i2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {(R + (n * P))}
Carrier i2 is finite Element of bool the carrier of K
i2 . R is Element of the carrier of m
{P} is non empty trivial finite 1 -element Element of bool the carrier of K
Lin {P} is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of K
y1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {P}
Sum y1 is Element of the carrier of K
(x2 . (R + (n * P))) * n is Element of the carrier of m
the multF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
[: the carrier of m, the carrier of m:] is Relation-like non empty set
[:[: the carrier of m, the carrier of m:], the carrier of m:] is Relation-like non empty set
bool [:[: the carrier of m, the carrier of m:], the carrier of m:] is non empty set
the multF of m . ((x2 . (R + (n * P))),n) is Element of the carrier of m
((x2 . (R + (n * P))) * n) * y1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
y2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {P}
Carrier y2 is finite Element of bool the carrier of K
y2 . R is Element of the carrier of m
P - (n * P) is Element of the carrier of K
- (n * P) is Element of the carrier of K
P + (- (n * P)) is Element of the carrier of K
the addF of K . (P,(- (n * P))) is Element of the carrier of K
(n * P) - (n * P) is Element of the carrier of K
(n * P) + (- (n * P)) is Element of the carrier of K
the addF of K . ((n * P),(- (n * P))) is Element of the carrier of K
R + ((n * P) - (n * P)) is Element of the carrier of K
the addF of K . (R,((n * P) - (n * P))) is Element of the carrier of K
R + (0. K) is Element of the carrier of K
the addF of K . (R,(0. K)) is Element of the carrier of K
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
(1_ m) * P is Element of the carrier of K
the lmult of K . ((1_ m),P) is Element of the carrier of K
((1_ m) * P) + (- (n * P)) is Element of the carrier of K
the addF of K . (((1_ m) * P),(- (n * P))) is Element of the carrier of K
- n is Element of the carrier of m
(- n) * P is Element of the carrier of K
the lmult of K . ((- n),P) is Element of the carrier of K
((1_ m) * P) + ((- n) * P) is Element of the carrier of K
the addF of K . (((1_ m) * P),((- n) * P)) is Element of the carrier of K
(1_ m) - n is Element of the carrier of m
(1_ m) + (- n) is Element of the carrier of m
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the addF of m . ((1_ m),(- n)) is Element of the carrier of m
((1_ m) - n) * P is Element of the carrier of K
the lmult of K . (((1_ m) - n),P) is Element of the carrier of K
{P,R} is non empty finite Element of bool the carrier of K
y2 . (R + (n * P)) is Element of the carrier of m
{P,R} is non empty finite Element of bool the carrier of K
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
(1_ m) * R is Element of the carrier of K
the lmult of K . ((1_ m),R) is Element of the carrier of K
i1 . (R + (n * P)) is Element of the carrier of m
(i1 . (R + (n * P))) * (R + (n * P)) is Element of the carrier of K
the lmult of K . ((i1 . (R + (n * P))),(R + (n * P))) is Element of the carrier of K
1_ m is Element of the carrier of m
1. m is non zero Element of the carrier of m
the OneF of m is Element of the carrier of m
(1_ m) * (R + (n * P)) is Element of the carrier of K
the lmult of K . ((1_ m),(R + (n * P))) is Element of the carrier of K
Lin {R} is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of K
Q is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {R}
Sum Q is Element of the carrier of K
(x2 . (R + (n * P))) * Q is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
P1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of {R}
Carrier P1 is finite Element of bool the carrier of K
P1 . (R + (n * P)) is Element of the carrier of m
x2 + y2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(x2 + y2) + P1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
((x2 + y2) + P1) - i2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
y2 + P1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
x2 + (y2 + P1) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(x2 + (y2 + P1)) - i2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
- i2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(x2 + (y2 + P1)) + (- i2) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(y2 + P1) + (- i2) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
x2 + ((y2 + P1) + (- i2)) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(y2 + P1) - i2 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
x2 + ((y2 + P1) - i2) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
Carrier (((x2 + y2) + P1) - i2) is finite Element of bool the carrier of K
Carrier ((y2 + P1) - i2) is finite Element of bool the carrier of K
(Carrier x2) \/ (Carrier ((y2 + P1) - i2)) is finite Element of bool the carrier of K
Carrier (y2 + P1) is finite Element of bool the carrier of K
(Carrier (y2 + P1)) \/ (Carrier i2) is finite Element of bool the carrier of K
(Carrier y2) \/ (Carrier P1) is finite Element of bool the carrier of K
(((x2 + y2) + P1) - i2) . (R + (n * P)) is Element of the carrier of m
((x2 + y2) + P1) + (- i2) is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of K
(((x2 + y2) + P1) + (- i2)) . (R + (n * P)) is Element of the carrier of m
((x2 + y2) + P1) . (R + (n * P)) is Element of the carrier of m
(- i2) . (R + (n * P)) is Element of the carrier of m
(((x2 + y2) + P1) . (R + (n * P))) + ((- i2) . (R + (n * P))) is Element of the carrier of m
the addF of m is Relation-like [: the carrier of m, the carrier of m:] -defined the carrier of m -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of m, the carrier of m:], the carrier of m:]
the addF of m . ((((x2 + y2) + P1) . (R + (n * P))),((- i2) . (R + (n * P)))) is Element of the carrier of m
(x2 + y2) . (R + (n * P)) is Element of the carrier of m
((x2 + y2) . (R + (n * P))) + (P1 . (R + (n * P))) is Element of the carrier of m
the addF of m . (((x2 + y2) . (R + (n * P))),(P1 . (R + (n * P)))) is Element of the carrier of m
(((x2 + y2) . (R + (n * P))) + (P1 . (R + (n * P)))) + ((- i2) . (R + (n * P))) is Element of the carrier of m
the addF of m . ((((x2 + y2) . (R + (n * P))) + (P1 . (R + (n * P)))),((- i2) . (R + (n * P)))) is Element of the carrier of m
(x2 . (R + (n * P))) + (0. m) is Element of the carrier of m
the addF of m . ((x2 . (R + (n * P))),(0. m)) is Element of the carrier of m
((x2 . (R + (n * P))) + (0. m)) + (0. m) is Element of the carrier of m
the addF of m . (((x2 . (R + (n * P))) + (0. m)),(0. m)) is Element of the carrier of m
(((x2 . (R + (n * P))) + (0. m)) + (0. m)) + ((- i2) . (R + (n * P))) is Element of the carrier of m
the addF of m . ((((x2 . (R + (n * P))) + (0. m)) + (0. m)),((- i2) . (R + (n * P)))) is Element of the carrier of m
((x2 . (R + (n * P))) + (0. m)) + ((- i2) . (R + (n * P))) is Element of the carrier of m
the addF of m . (((x2 . (R + (n * P))) + (0. m)),((- i2) . (R + (n * P)))) is Element of the carrier of m
(x2 . (R + (n * P))) + ((- i2) . (R + (n * P))) is Element of the carrier of m
the addF of m . ((x2 . (R + (n * P))),((- i2) . (R + (n * P)))) is Element of the carrier of m
i2 . (R + (n * P)) is Element of the carrier of m
(x2 . (R + (n * P))) - (i2 . (R + (n * P))) is Element of the carrier of m
- (i2 . (R + (n * P))) is Element of the carrier of m
(x2 . (R + (n * P))) + (- (i2 . (R + (n * P)))) is Element of the carrier of m
the addF of m . ((x2 . (R + (n * P))),(- (i2 . (R + (n * P))))) is Element of the carrier of m
(x2 . (R + (n * P))) * (1_ m) is Element of the carrier of m
the multF of m . ((x2 . (R + (n * P))),(1_ m)) is Element of the carrier of m
(x2 . (R + (n * P))) - ((x2 . (R + (n * P))) * (1_ m)) is Element of the carrier of m
- ((x2 . (R + (n * P))) * (1_ m)) is Element of the carrier of m
(x2 . (R + (n * P))) + (- ((x2 . (R + (n * P))) * (1_ m))) is Element of the carrier of m
the addF of m . ((x2 . (R + (n * P))),(- ((x2 . (R + (n * P))) * (1_ m)))) is Element of the carrier of m
(x2 . (R + (n * P))) - (x2 . (R + (n * P))) is Element of the carrier of m
- (x2 . (R + (n * P))) is Element of the carrier of m
(x2 . (R + (n * P))) + (- (x2 . (R + (n * P)))) is Element of the carrier of m
the addF of m . ((x2 . (R + (n * P))),(- (x2 . (R + (n * P))))) is Element of the carrier of m
Q1 is set
P2 is Element of the carrier of K
(((x2 + y2) + P1) - i2) . P2 is Element of the carrier of m
x2 . R is Element of the carrier of m
Q . R is Element of the carrier of m
(Q . R) * R is Element of the carrier of K
the lmult of K . ((Q . R),R) is Element of the carrier of K
(1_ m) * R is Element of the carrier of K
the lmult of K . ((1_ m),R) is Element of the carrier of K
Q1 is Relation-like the carrier of K -defined the carrier of m -valued Function-like total quasi_total Linear_Combination of M
Q1 . R is Element of the carrier of m
(((x2 + y2) + P1) + (- i2)) . R is Element of the carrier of m
((x2 + y2) + P1) . R is Element of the carrier of m
(- i2) . R is Element of the carrier of m
(((x2 + y2) + P1) . R) + ((- i2) . R) is Element of the carrier of m
the addF of m . ((((x2 + y2) + P1) . R),((- i2) . R)) is Element of the carrier of m
(x2 + y2) . R is Element of the carrier of m
P1 . R is Element of the carrier of m
((x2 + y2) . R) + (P1 . R) is Element of the carrier of m
the addF of m . (((x2 + y2) . R),(P1 . R)) is Element of the carrier of m
(((x2 + y2) . R) + (P1 . R)) + ((- i2) . R) is Element of the carrier of m
the addF of m . ((((x2 + y2) . R) + (P1 . R)),((- i2) . R)) is Element of the carrier of m
(x2 . R) + (y2 . R) is Element of the carrier of m
the addF of m . ((x2 . R),(y2 . R)) is Element of the carrier of m
((x2 . R) + (y2 . R)) + (P1 . R) is Element of the carrier of m
the addF of m . (((x2 . R) + (y2 . R)),(P1 . R)) is Element of the carrier of m
(((x2 . R) + (y2 . R)) + (P1 . R)) + ((- i2) . R) is Element of the carrier of m
the addF of m . ((((x2 . R) + (y2 . R)) + (P1 . R)),((- i2) . R)) is Element of the carrier of m
(0. m) + (0. m) is Element of the carrier of m
the addF of m . ((0. m),(0. m)) is Element of the carrier of m
((0. m) + (0. m)) + (P1 . R) is Element of the carrier of m
the addF of m . (((0. m) + (0. m)),(P1 . R)) is Element of the carrier of m
(((0. m) + (0. m)) + (P1 . R)) - (0. m) is Element of the carrier of m
- (0. m) is Element of the carrier of m
(((0. m) + (0. m)) + (P1 . R)) + (- (0. m)) is Element of the carrier of m
the addF of m . ((((0. m) + (0. m)) + (P1 . R)),(- (0. m))) is Element of the carrier of m
(0. m) + (P1 . R) is Element of the carrier of m
the addF of m . ((0. m),(P1 . R)) is Element of the carrier of m
((0. m) + (P1 . R)) - (0. m) is Element of the carrier of m
((0. m) + (P1 . R)) + (- (0. m)) is Element of the carrier of m
the addF of m . (((0. m) + (P1 . R)),(- (0. m))) is Element of the carrier of m
(P1 . R) - (0. m) is Element of the carrier of m
(P1 . R) + (- (0. m)) is Element of the carrier of m
the addF of m . ((P1 . R),(- (0. m))) is Element of the carrier of m
Carrier Q1 is finite Element of bool the carrier of K
Sum (((x2 + y2) + P1) - i2) is Element of the carrier of K
Sum ((x2 + y2) + P1) is Element of the carrier of K
Sum i2 is Element of the carrier of K
(Sum ((x2 + y2) + P1)) - (Sum i2) is Element of the carrier of K
- (Sum i2) is Element of the carrier of K
(Sum ((x2 + y2) + P1)) + (- (Sum i2)) is Element of the carrier of K
the addF of K . ((Sum ((x2 + y2) + P1)),(- (Sum i2))) is Element of the carrier of K
Sum (x2 + y2) is Element of the carrier of K
Sum P1 is Element of the carrier of K
(Sum (x2 + y2)) + (Sum P1) is Element of the carrier of K
the addF of K . ((Sum (x2 + y2)),(Sum P1)) is Element of the carrier of K
((Sum (x2 + y2)) + (Sum P1)) - (Sum i2) is Element of the carrier of K
((Sum (x2 + y2)) + (Sum P1)) + (- (Sum i2)) is Element of the carrier of K
the addF of K . (((Sum (x2 + y2)) + (Sum P1)),(- (Sum i2))) is Element of the carrier of K
Sum y2 is Element of the carrier of K
(Sum x2) + (Sum y2) is Element of the carrier of K
the addF of K . ((Sum x2),(Sum y2)) is Element of the carrier of K
((Sum x2) + (Sum y2)) + (Sum P1) is Element of the carrier of K
the addF of K . (((Sum x2) + (Sum y2)),(Sum P1)) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + (Sum P1)) - (Sum i2) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + (Sum P1)) + (- (Sum i2)) is Element of the carrier of K
the addF of K . ((((Sum x2) + (Sum y2)) + (Sum P1)),(- (Sum i2))) is Element of the carrier of K
(x2 . (R + (n * P))) * (R + (n * P)) is Element of the carrier of K
the lmult of K . ((x2 . (R + (n * P))),(R + (n * P))) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + (Sum P1)) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
- ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + (Sum P1)) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . ((((Sum x2) + (Sum y2)) + (Sum P1)),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
(x2 . (R + (n * P))) * R is Element of the carrier of K
the lmult of K . ((x2 . (R + (n * P))),R) is Element of the carrier of K
((Sum x2) + (Sum y2)) + ((x2 . (R + (n * P))) * R) is Element of the carrier of K
the addF of K . (((Sum x2) + (Sum y2)),((x2 . (R + (n * P))) * R)) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + ((x2 . (R + (n * P))) * R)) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
(((Sum x2) + (Sum y2)) + ((x2 . (R + (n * P))) * R)) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . ((((Sum x2) + (Sum y2)) + ((x2 . (R + (n * P))) * R)),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
n * (x2 . (R + (n * P))) is Element of the carrier of m
the multF of m . (n,(x2 . (R + (n * P)))) is Element of the carrier of m
(n * (x2 . (R + (n * P)))) * P is Element of the carrier of K
the lmult of K . ((n * (x2 . (R + (n * P)))),P) is Element of the carrier of K
(Sum x2) + ((n * (x2 . (R + (n * P)))) * P) is Element of the carrier of K
the addF of K . ((Sum x2),((n * (x2 . (R + (n * P)))) * P)) is Element of the carrier of K
((Sum x2) + ((n * (x2 . (R + (n * P)))) * P)) + ((x2 . (R + (n * P))) * R) is Element of the carrier of K
the addF of K . (((Sum x2) + ((n * (x2 . (R + (n * P)))) * P)),((x2 . (R + (n * P))) * R)) is Element of the carrier of K
(((Sum x2) + ((n * (x2 . (R + (n * P)))) * P)) + ((x2 . (R + (n * P))) * R)) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
(((Sum x2) + ((n * (x2 . (R + (n * P)))) * P)) + ((x2 . (R + (n * P))) * R)) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . ((((Sum x2) + ((n * (x2 . (R + (n * P)))) * P)) + ((x2 . (R + (n * P))) * R)),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
(x2 . (R + (n * P))) * (n * P) is Element of the carrier of K
the lmult of K . ((x2 . (R + (n * P))),(n * P)) is Element of the carrier of K
(Sum x2) + ((x2 . (R + (n * P))) * (n * P)) is Element of the carrier of K
the addF of K . ((Sum x2),((x2 . (R + (n * P))) * (n * P))) is Element of the carrier of K
((Sum x2) + ((x2 . (R + (n * P))) * (n * P))) + ((x2 . (R + (n * P))) * R) is Element of the carrier of K
the addF of K . (((Sum x2) + ((x2 . (R + (n * P))) * (n * P))),((x2 . (R + (n * P))) * R)) is Element of the carrier of K
(((Sum x2) + ((x2 . (R + (n * P))) * (n * P))) + ((x2 . (R + (n * P))) * R)) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
(((Sum x2) + ((x2 . (R + (n * P))) * (n * P))) + ((x2 . (R + (n * P))) * R)) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . ((((Sum x2) + ((x2 . (R + (n * P))) * (n * P))) + ((x2 . (R + (n * P))) * R)),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R) is Element of the carrier of K
the addF of K . (((x2 . (R + (n * P))) * (n * P)),((x2 . (R + (n * P))) * R)) is Element of the carrier of K
(Sum x2) + (((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R)) is Element of the carrier of K
the addF of K . ((Sum x2),(((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R))) is Element of the carrier of K
((Sum x2) + (((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R))) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
((Sum x2) + (((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R))) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . (((Sum x2) + (((x2 . (R + (n * P))) * (n * P)) + ((x2 . (R + (n * P))) * R))),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
(n * P) + R is Element of the carrier of K
the addF of K . ((n * P),R) is Element of the carrier of K
(x2 . (R + (n * P))) * ((n * P) + R) is Element of the carrier of K
the lmult of K . ((x2 . (R + (n * P))),((n * P) + R)) is Element of the carrier of K
(Sum x2) + ((x2 . (R + (n * P))) * ((n * P) + R)) is Element of the carrier of K
the addF of K . ((Sum x2),((x2 . (R + (n * P))) * ((n * P) + R))) is Element of the carrier of K
((Sum x2) + ((x2 . (R + (n * P))) * ((n * P) + R))) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
((Sum x2) + ((x2 . (R + (n * P))) * ((n * P) + R))) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . (((Sum x2) + ((x2 . (R + (n * P))) * ((n * P) + R))),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
((x2 . (R + (n * P))) * (R + (n * P))) - ((x2 . (R + (n * P))) * (R + (n * P))) is Element of the carrier of K
((x2 . (R + (n * P))) * (R + (n * P))) + (- ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . (((x2 . (R + (n * P))) * (R + (n * P))),(- ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
(Sum x2) + (((x2 . (R + (n * P))) * (R + (n * P))) - ((x2 . (R + (n * P))) * (R + (n * P)))) is Element of the carrier of K
the addF of K . ((Sum x2),(((x2 . (R + (n * P))) * (R + (n * P))) - ((x2 . (R + (n * P))) * (R + (n * P))))) is Element of the carrier of K
(0. K) + (0. K) is Element of the carrier of K
the addF of K . ((0. K),(0. K)) is Element of the carrier of K
x2 . (R + (n * P)) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
m is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is non empty right_complementable V95() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over n
the carrier of K is non empty set
M is Element of the carrier of K
R is Element of the carrier of K
{M,R} is non empty finite Element of bool the carrier of K
bool the carrier of K is non empty set
Lin {M,R} is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of K
{M} is non empty trivial finite 1 -element Element of bool the carrier of K
P is Element of the carrier of n
P * M is Element of the carrier of K
the lmult of K is Relation-like [: the carrier of n, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of K:], the carrier of K:]
[: the carrier of n, the carrier of K:] is Relation-like non empty set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is non empty set
the lmult of K . (P,M) is Element of the carrier of K
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(P * M) + (0. K) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K . ((P * M),(0. K)) is Element of the carrier of K
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(0. n) * R is Element of the carrier of K
the lmult of K . ((0. n),R) is Element of the carrier of K
(P * M) + ((0. n) * R) is Element of the carrier of K
the addF of K . ((P * M),((0. n) * R)) is Element of the carrier of K
P is Element of the carrier of n
P * M is Element of the carrier of K
the lmult of K is Relation-like [: the carrier of n, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of K:], the carrier of K:]
[: the carrier of n, the carrier of K:] is Relation-like non empty set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is non empty set
the lmult of K . (P,M) is Element of the carrier of K
Q is Element of the carrier of n
Q * R is Element of the carrier of K
the lmult of K . (Q,R) is Element of the carrier of K
(P * M) + (Q * R) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K . ((P * M),(Q * R)) is Element of the carrier of K
P + Q is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the addF of n . (P,Q) is Element of the carrier of n
(P + Q) * M is Element of the carrier of K
the lmult of K . ((P + Q),M) is Element of the carrier of K
P is Relation-like the carrier of K -defined the carrier of n -valued Function-like total quasi_total Linear_Combination of {M,R}
Sum P is Element of the carrier of K
P . M is Element of the carrier of n
(P . M) * M is Element of the carrier of K
the lmult of K is Relation-like [: the carrier of n, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of K:], the carrier of K:]
[: the carrier of n, the carrier of K:] is Relation-like non empty set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is non empty set
the lmult of K . ((P . M),M) is Element of the carrier of K
P . R is Element of the carrier of n
(P . R) * R is Element of the carrier of K
the lmult of K . ((P . R),R) is Element of the carrier of K
((P . M) * M) + ((P . R) * R) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K . (((P . M) * M),((P . R) * R)) is Element of the carrier of K
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
P is Element of the carrier of n
P * M is Element of the carrier of K
the lmult of K is Relation-like [: the carrier of n, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of K:], the carrier of K:]
[: the carrier of n, the carrier of K:] is Relation-like non empty set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is non empty set
the lmult of K . (P,M) is Element of the carrier of K
Q is Element of the carrier of n
Q * R is Element of the carrier of K
the lmult of K . (Q,R) is Element of the carrier of K
(P * M) + (Q * R) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the addF of K . ((P * M),(Q * R)) is Element of the carrier of K
[: the carrier of K, the carrier of n:] is Relation-like non empty set
bool [: the carrier of K, the carrier of n:] is non empty set
S is Relation-like the carrier of K -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of n:]
S . M is Element of the carrier of n
S . R is Element of the carrier of n
Funcs ( the carrier of K, the carrier of n) is functional non empty FUNCTION_DOMAIN of the carrier of K, the carrier of n
x2 is Element of the carrier of K
x1 is Relation-like the carrier of K -defined the carrier of n -valued Function-like total quasi_total Element of Funcs ( the carrier of K, the carrier of n)
x1 . x2 is Element of the carrier of n
x2 is Relation-like the carrier of K -defined the carrier of n -valued Function-like total quasi_total Linear_Combination of K
Carrier x2 is finite Element of bool the carrier of K
i1 is set
x2 . i1 is set
i1 is Relation-like the carrier of K -defined the carrier of n -valued Function-like total quasi_total Linear_Combination of {M,R}
Sum i1 is Element of the carrier of K
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
M is Element of the carrier of K
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,R) is finite Element of bool the carrier of (m -VectSp_over K)
the carrier of (m -VectSp_over K) is non empty set
bool the carrier of (m -VectSp_over K) is non empty set
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len R) is finite len R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len R ) } is set
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
(width R) -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 R } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
M * (Line (R,S)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
M multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (M,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (R,S)),(M 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 (R,Q)) + (M * (Line (R,S))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line (R,Q)),(M * (Line (R,S)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,(ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) is finite Element of bool the carrier of (m -VectSp_over K)
dom R is finite Element of bool NAT
R . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
R . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 -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 = i2 } is set
y1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
M * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,P1,(M multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
M * y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) is Relation-like [: the carrier of K, the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of K, the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the lmult of (m -VectSp_over K) . (M,y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Q is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
Q + (M * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,Q,(M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
y1 + (M * y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) is Relation-like [: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the addF of (m -VectSp_over K) . (y1,(M * y2)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * y1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) . ((1_ K),y1) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
((1_ K) * y1) + (M * y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) . (((1_ K) * y1),(M * y2)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
{y1,y2} is functional non empty finite Element of bool the carrier of (m -VectSp_over K)
Lin {y1,y2} is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of m -VectSp_over K
Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line ((ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))),Q1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -tuples_on the carrier of K
width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -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 (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) } is set
Line (R,Q1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
- (1_ K) is Element of the carrier of K
(- (1_ K)) * M is Element of the carrier of K
the multF of K . ((- (1_ K)),M) is Element of the carrier of K
(1_ K) + ((- (1_ K)) * M) is Element of the carrier of K
the addF of K . ((1_ K),((- (1_ K)) * M)) is Element of the carrier of K
((1_ K) + ((- (1_ K)) * M)) * y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) . (((1_ K) + ((- (1_ K)) * M)),y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
((1_ K) + ((- (1_ K)) * M)) * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
((1_ K) + ((- (1_ K)) * M)) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (((1_ K) + ((- (1_ K)) * M)),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,P1,(((1_ K) + ((- (1_ K)) * M)) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- (1_ K)) * y1 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) . ((- (1_ K)),y1) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
(- (1_ K)) * Q is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
(- (1_ K)) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((- (1_ K)),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,Q,((- (1_ K)) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. (m -VectSp_over K) is Relation-like Function-like zero Element of the carrier of (m -VectSp_over K)
the ZeroF of (m -VectSp_over K) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
m |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite m -element FinSequence-like FinSubsequence-like Element of m -tuples_on the carrier of K
m -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 = m } is set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (0. K) is Relation-like Seg m -defined Seg m -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg m),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg m),{(0. K)}:] is Relation-like finite set
bool [:(Seg m),{(0. K)}:] is non empty finite V37() set
- (Q + (M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
comp K is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Q + (M * P1)),(comp K)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + (- (Q + (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,(- (Q + (M * P1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
- Q is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,Q,(comp K)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
- (M * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(M * P1),(comp K)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- Q) + (- (M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(- Q),(- (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + ((- Q) + (- (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,((- Q) + (- (M * P1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((- (1_ K)) * Q) + (- (M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,((- (1_ K)) * Q),(- (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + (((- (1_ K)) * Q) + (- (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,(((- (1_ K)) * Q) + (- (M * P1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- (1_ K)) * (M * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,(M * P1),((- (1_ K)) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((- (1_ K)) * Q) + ((- (1_ K)) * (M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,((- (1_ K)) * Q),((- (1_ K)) * (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + (((- (1_ K)) * Q) + ((- (1_ K)) * (M * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,(((- (1_ K)) * Q) + ((- (1_ K)) * (M * P1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((- (1_ K)) * M) * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
((- (1_ K)) * M) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (((- (1_ K)) * M),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,P1,(((- (1_ K)) * M) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((- (1_ K)) * Q) + (((- (1_ K)) * M) * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,((- (1_ K)) * Q),(((- (1_ K)) * M) * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + (((- (1_ K)) * Q) + (((- (1_ K)) * M) * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,(((- (1_ K)) * Q) + (((- (1_ K)) * M) * P1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((- (1_ K)) * M) * P1) + ((- (1_ K)) * Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(((- (1_ K)) * M) * P1),((- (1_ K)) * Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + ((((- (1_ K)) * M) * P1) + ((- (1_ K)) * Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,((((- (1_ K)) * M) * P1) + ((- (1_ K)) * Q))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
P1 + (((- (1_ K)) * M) * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,P1,(((- (1_ K)) * M) * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(P1 + (((- (1_ K)) * M) * P1)) + ((- (1_ K)) * Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(P1 + (((- (1_ K)) * M) * P1)),((- (1_ K)) * Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(1_ K) * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
(1_ K) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((1_ K),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,P1,((1_ K) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((1_ K) * P1) + (((- (1_ K)) * M) * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,((1_ K) * P1),(((- (1_ K)) * M) * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((1_ K) * P1) + (((- (1_ K)) * M) * P1)) + ((- (1_ K)) * Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(((1_ K) * P1) + (((- (1_ K)) * M) * P1)),((- (1_ K)) * Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((1_ K) + ((- (1_ K)) * M)) * P1) + ((- (1_ K)) * Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(((1_ K) + ((- (1_ K)) * M)) * P1),((- (1_ K)) * Q)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((1_ K) + ((- (1_ K)) * M)) * y2) + ((- (1_ K)) * y1) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) . ((((1_ K) + ((- (1_ K)) * M)) * y2),((- (1_ K)) * y1)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
Q2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
{Q2,y1,y2} is non empty finite Element of bool the carrier of (m -VectSp_over K)
i is Element of bool the carrier of (m -VectSp_over K)
R . Q1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{Q2} is functional non empty trivial finite 1 -element Element of bool the carrier of (m -VectSp_over K)
i \ {Q2} is Element of bool the carrier of (m -VectSp_over K)
Q is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
M * P1 is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K391( the carrier of K, the carrier of K,P1,(M multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Q + (M * P1) is Relation-like NAT -defined the carrier of K -valued Function-like finite i2 -element FinSequence-like FinSubsequence-like Element of i2 -tuples_on the carrier of K
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,Q,(M * P1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Q1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
P2 is set
Q .--> P2 is Relation-like {Q} -defined Function-like one-to-one finite set
{Q} is non empty trivial finite V37() 1 -element set
{Q} --> P2 is Relation-like {Q} -defined {P2} -valued Function-like constant non empty total quasi_total finite Element of bool [:{Q},{P2}:]
{P2} is non empty trivial finite 1 -element set
[:{Q},{P2}:] is Relation-like non empty finite set
bool [:{Q},{P2}:] is non empty finite V37() set
len (Q + (M * P1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
ReplaceLine (R,Q,(Q + (M * P1))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
Replace (R,Q,Q1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of K *
R +* (Q .--> P2) is Relation-like Function-like finite set
dom (Q .--> P2) is trivial finite V37() Element of bool {Q}
bool {Q} is non empty finite V37() set
(dom R) \ (dom (Q .--> P2)) is finite Element of bool NAT
R .: ((dom R) \ (dom (Q .--> P2))) is finite set
rng (Q .--> P2) is finite set
(R .: ((dom R) \ (dom (Q .--> P2)))) \/ (rng (Q .--> P2)) is finite set
(Q) is non empty trivial finite V37() 1 -element Element of bool NAT
(dom R) \ (Q) is finite Element of bool NAT
R .: ((dom R) \ (Q)) is finite set
(R .: ((dom R) \ (Q))) \/ (rng (Q .--> P2)) is finite set
(R .: ((dom R) \ (Q))) \/ {P2} is non empty finite set
R .: (dom R) is finite set
R .: (Q) is finite set
(R .: (dom R)) \ (R .: (Q)) is finite Element of bool (R .: (dom R))
bool (R .: (dom R)) is non empty finite V37() set
((R .: (dom R)) \ (R .: (Q))) \/ {P2} is non empty finite set
Im (R,Q) is set
R .: {Q} is finite set
(K,n,m,R) \ (Im (R,Q)) is finite Element of bool the carrier of (m -VectSp_over K)
((K,n,m,R) \ (Im (R,Q))) \/ {P2} is non empty finite set
{y1} is functional non empty trivial finite 1 -element Element of bool the carrier of (m -VectSp_over K)
(K,n,m,R) \ {y1} is finite Element of bool the carrier of (m -VectSp_over K)
{(Q + (M * P1))} is functional non empty trivial finite V37() 1 -element FinSequence-membered Element of bool (i2 -tuples_on the carrier of K)
bool (i2 -tuples_on the carrier of K) is non empty set
((K,n,m,R) \ {y1}) \/ {(Q + (M * P1))} is non empty finite set
Line ((ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))),Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -tuples_on the carrier of K
width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -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 (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) } is set
len (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i is set
dom (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) is finite Element of bool NAT
m is set
(ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) . m is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg (len (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) is finite len (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) ) } is set
Q2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) . Q2i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))),Q2i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -tuples_on the carrier of K
SQ2i is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) . SQ2i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))),SQ2i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (R,Q,((Line (R,Q)) + (M * (Line (R,S))))))) -tuples_on the carrier of K
Line (R,SQ2i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
Line (R,Q2i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
R . Q2i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
R . SQ2i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
M * y2 is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the lmult of (m -VectSp_over K) is Relation-like [: the carrier of K, the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of K, the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the lmult of (m -VectSp_over K) . (M,y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
y1 + (M * y2) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
the addF of (m -VectSp_over K) is Relation-like [: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] -defined the carrier of (m -VectSp_over K) -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):]
[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):] is Relation-like non empty set
[:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is Relation-like non empty set
bool [:[: the carrier of (m -VectSp_over K), the carrier of (m -VectSp_over K):], the carrier of (m -VectSp_over K):] is non empty set
the addF of (m -VectSp_over K) . (y1,(M * y2)) is Relation-like Function-like Element of the carrier of (m -VectSp_over K)
{(y1 + (M * y2))} is functional non empty trivial finite 1 -element Element of bool the carrier of (m -VectSp_over K)
((K,n,m,R) \ {y1}) \/ {(y1 + (M * y2))} is non empty finite Element of bool the carrier of (m -VectSp_over K)
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
( the carrier of K,R,M,(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
((Seg n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
( the carrier of K,R,(card M),(card (Seg n)),(M),((Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
(K,(card M),(card (Seg n)),( the carrier of K,R,M,(Seg n))) is finite Element of bool the carrier of ((card (Seg n)) -VectSp_over K)
(card (Seg n)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
(K,m,n,R) is finite Element of bool the carrier of (n -VectSp_over K)
n -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is non empty set
rng (M) is finite V212() V213() V214() V217() set
Q is set
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (( the carrier of K,R,M,(Seg n)),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,R,M,(Seg n)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,R,M,(Seg n))) -tuples_on the carrier of K
width ( the carrier of K,R,M,(Seg n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,R,M,(Seg 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 = width ( the carrier of K,R,M,(Seg n)) } is set
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) . S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,((M) . S)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
(width R) -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 R } is set
dom (M) is finite card M -element Element of bool NAT
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
n -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
(card (Seg n)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
M is finite without_zero Element of bool NAT
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,m,n,R) is finite Element of bool the carrier of (n -VectSp_over K)
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is non empty set
( the carrier of K,R,M,(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
((Seg n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
( the carrier of K,R,(card M),(card (Seg n)),(M),((Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
(K,(card M),(card (Seg n)),( the carrier of K,R,M,(Seg n))) is finite Element of bool the carrier of ((card (Seg n)) -VectSp_over K)
the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is finite without_zero Element of bool NAT
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
( the carrier of K,R,M,(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
card M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) is Relation-like NAT -defined NAT -valued Function-like finite card M -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card M) -tuples_on NAT
(card M) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card M } is set
((Seg n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
( the carrier of K,R,(card M),(card (Seg n)),(M),((Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card M, card (Seg n), the carrier of K
Q is set
dom ( the carrier of K,R,M,(Seg n)) is finite set
S is set
( the carrier of K,R,M,(Seg n)) . Q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
( the carrier of K,R,M,(Seg n)) . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom ( the carrier of K,R,M,(Seg n)) is finite Element of bool NAT
len ( the carrier of K,R,M,(Seg n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card M) is finite card M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card M ) } is set
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of K,R,M,(Seg n)),x1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,R,M,(Seg n)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,R,M,(Seg n))) -tuples_on the carrier of K
width ( the carrier of K,R,M,(Seg n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,R,M,(Seg 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 = width ( the carrier of K,R,M,(Seg n)) } is set
( the carrier of K,R,M,(Seg n)) . x1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of K,R,M,(Seg n)),x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,R,M,(Seg n)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,R,M,(Seg n))) -tuples_on the carrier of K
( the carrier of K,R,M,(Seg n)) . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (M) is finite card M -element Element of bool NAT
width R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(M) . x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,((M) . x1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
(width R) -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 R } is set
(M) . x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (R,((M) . x2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width R -element FinSequence-like FinSubsequence-like Element of (width R) -tuples_on the carrier of K
len R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
rng (M) is finite V212() V213() V214() V217() set
R . ((M) . x2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom R is finite Element of bool NAT
R . ((M) . x1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
n -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
M + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
P -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
Q is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
(K,R,P,Q) is finite Element of bool the carrier of (P -VectSp_over K)
the carrier of (P -VectSp_over K) is non empty set
bool the carrier of (P -VectSp_over K) is non empty set
(K,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
P |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite P -element FinSequence-like FinSubsequence-like Element of P -tuples_on the carrier of K
P -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = P } is set
Seg P is finite P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
(Seg P) --> (0. K) is Relation-like Seg P -defined Seg P -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg P),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg P),{(0. K)}:] is Relation-like finite set
bool [:(Seg P),{(0. K)}:] is non empty finite V37() set
len (P |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg M is finite M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
Seg R is finite R -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= R ) } is set
Line (Q,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of K
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width Q) -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 Q } is set
S is Relation-like NAT -defined the carrier of K -valued Function-like finite P -element FinSequence-like FinSubsequence-like Element of P -tuples_on the carrier of K
len S is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
S . x1 is set
(P |-> (0. K)) . x1 is set
len (Line (Q,R)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Q * (R,x1) is Element of the carrier of K
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i1 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
y1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
Line (y1,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
width y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width y1) -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 y1 } is set
(K,R,P,y1) is finite Element of bool the carrier of (P -VectSp_over K)
(K,y1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
Line (y1,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
width y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width y1) -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 y1 } is set
(K,R,P,y1) is finite Element of bool the carrier of (P -VectSp_over K)
(K,y1) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
y1 * ((i1 + 1),x1) is Element of the carrier of K
Line (y1,(i1 + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
(Q * (R,x1)) " is Element of the carrier of K
((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the multF of K . (((Q * (R,x1)) "),(y1 * ((i1 + 1),x1))) is Element of the carrier of K
- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
(- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
(- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty 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 total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K391( the carrier of K, the carrier of K,(Line (y1,R)),((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) 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 (y1,(i1 + 1))) + ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K388( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line (y1,(i1 + 1))),((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine (y1,(i1 + 1),((Line (y1,(i1 + 1))) + ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
Q is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
Line (Q,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of K
width Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width Q) -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 Q } is set
(K,R,P,Q) is finite Element of bool the carrier of (P -VectSp_over K)
(K,Q) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q * (P2,x1) is Element of the carrier of K
(Line (y1,(i1 + 1))) . x1 is set
0 + 1 is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
len ((Line (y1,(i1 + 1))) + ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (Q,(i1 + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of K
(Line (y1,R)) . x1 is set
((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Line (y1,R))) . x1 is set
(- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1)) is Element of the carrier of K
the multF of K . ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))),(Q * (R,x1))) is Element of the carrier of K
(Line (Q,(i1 + 1))) . x1 is set
(y1 * ((i1 + 1),x1)) + ((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1)))) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the multF of K . ((1_ K),(((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
- ((1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
(- ((1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))))) * (Q * (R,x1)) is Element of the carrier of K
the multF of K . ((- ((1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))))),(Q * (R,x1))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + ((- ((1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))))) * (Q * (R,x1))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- ((1_ K) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))))) * (Q * (R,x1)))) is Element of the carrier of K
- (1_ K) is Element of the carrier of K
(- (1_ K)) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the multF of K . ((- (1_ K)),(((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
((- (1_ K)) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1)) is Element of the carrier of K
the multF of K . (((- (1_ K)) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))),(Q * (R,x1))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + (((- (1_ K)) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),(((- (1_ K)) * (((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1)))) * (Q * (R,x1)))) is Element of the carrier of K
(((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) * (Q * (R,x1)) is Element of the carrier of K
the multF of K . ((((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))),(Q * (R,x1))) is Element of the carrier of K
(- (1_ K)) * ((((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) * (Q * (R,x1))) is Element of the carrier of K
the multF of K . ((- (1_ K)),((((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) * (Q * (R,x1)))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + ((- (1_ K)) * ((((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) * (Q * (R,x1)))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- (1_ K)) * ((((Q * (R,x1)) ") * (y1 * ((i1 + 1),x1))) * (Q * (R,x1))))) is Element of the carrier of K
((Q * (R,x1)) ") * (Q * (R,x1)) is Element of the carrier of K
the multF of K . (((Q * (R,x1)) "),(Q * (R,x1))) is Element of the carrier of K
(((Q * (R,x1)) ") * (Q * (R,x1))) * (y1 * ((i1 + 1),x1)) is Element of the carrier of K
the multF of K . ((((Q * (R,x1)) ") * (Q * (R,x1))),(y1 * ((i1 + 1),x1))) is Element of the carrier of K
(- (1_ K)) * ((((Q * (R,x1)) ") * (Q * (R,x1))) * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the multF of K . ((- (1_ K)),((((Q * (R,x1)) ") * (Q * (R,x1))) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + ((- (1_ K)) * ((((Q * (R,x1)) ") * (Q * (R,x1))) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- (1_ K)) * ((((Q * (R,x1)) ") * (Q * (R,x1))) * (y1 * ((i1 + 1),x1))))) is Element of the carrier of K
(1_ K) * (y1 * ((i1 + 1),x1)) is Element of the carrier of K
the multF of K . ((1_ K),(y1 * ((i1 + 1),x1))) is Element of the carrier of K
(- (1_ K)) * ((1_ K) * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the multF of K . ((- (1_ K)),((1_ K) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + ((- (1_ K)) * ((1_ K) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- (1_ K)) * ((1_ K) * (y1 * ((i1 + 1),x1))))) is Element of the carrier of K
(- (1_ K)) * (y1 * ((i1 + 1),x1)) is Element of the carrier of K
the multF of K . ((- (1_ K)),(y1 * ((i1 + 1),x1))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + ((- (1_ K)) * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),((- (1_ K)) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
- ((1_ K) * (y1 * ((i1 + 1),x1))) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + (- ((1_ K) * (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),(- ((1_ K) * (y1 * ((i1 + 1),x1))))) is Element of the carrier of K
- (y1 * ((i1 + 1),x1)) is Element of the carrier of K
(y1 * ((i1 + 1),x1)) + (- (y1 * ((i1 + 1),x1))) is Element of the carrier of K
the addF of K . ((y1 * ((i1 + 1),x1)),(- (y1 * ((i1 + 1),x1)))) is Element of the carrier of K
y1 * (P2,x1) is Element of the carrier of K
Line (y1,P2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width y1 -element FinSequence-like FinSubsequence-like Element of (width y1) -tuples_on the carrier of K
(Line (y1,P2)) . x1 is set
Line (Q,P2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of K
(Line (Q,P2)) . x1 is set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Q * (x2,x1) is Element of the carrier of K
x2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of R,P, the carrier of K
Line (x2,R) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x2 -element FinSequence-like FinSubsequence-like Element of (width x2) -tuples_on the carrier of K
width x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width x2) -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 x2 } is set
(K,R,P,x2) is finite Element of bool the carrier of (P -VectSp_over K)
(K,x2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,x2,(Seg M),(Seg P)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg M), card (Seg P), the carrier of K
card (Seg M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (Seg P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg M)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg M) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg M)) -tuples_on NAT
(card (Seg M)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg M) } is set
((Seg P)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg P) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg P)) -tuples_on NAT
(card (Seg P)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg P) } is set
( the carrier of K,x2,(card (Seg M)),(card (Seg P)),((Seg M)),((Seg P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg M), card (Seg P), the carrier of K
len ( the carrier of K,x2,(Seg M),(Seg P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
((Seg P)) . x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
idseq P is Relation-like NAT -defined Function-like finite P -element FinSequence-like FinSubsequence-like set
id (Seg P) is Relation-like Seg P -defined Seg P -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg P),(Seg P):]
[:(Seg P),(Seg P):] is Relation-like finite set
bool [:(Seg P),(Seg P):] is non empty finite V37() set
(idseq P) . x1 is set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
((Seg M)) . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
idseq M is Relation-like NAT -defined Function-like finite M -element FinSequence-like FinSubsequence-like set
id (Seg M) is Relation-like Seg M -defined Seg M -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg M),(Seg M):]
[:(Seg M),(Seg M):] is Relation-like finite set
bool [:(Seg M),(Seg M):] is non empty finite V37() set
(idseq M) . i2 is set
width ( the carrier of K,x2,(Seg M),(Seg P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[i2,x1] is set
{i2,x1} is non empty finite V37() set
{i2} is non empty trivial finite V37() 1 -element set
{{i2,x1},{i2}} is non empty finite V37() without_zero V103() set
Seg (width ( the carrier of K,x2,(Seg M),(Seg P))) is finite width ( the carrier of K,x2,(Seg M),(Seg P)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,x2,(Seg M),(Seg P)) ) } is set
[:(Seg M),(Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))):] is Relation-like finite set
Indices ( the carrier of K,x2,(Seg M),(Seg P)) is set
dom ( the carrier of K,x2,(Seg M),(Seg P)) is finite Element of bool NAT
[:(dom ( the carrier of K,x2,(Seg M),(Seg P))),(Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))):] is Relation-like finite set
( the carrier of K,x2,(Seg M),(Seg P)) * (i2,x1) is Element of the carrier of K
x2 * ((((Seg M)) . i2),(((Seg P)) . x1)) is Element of the carrier of K
(x1) is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1) is finite without_zero Element of bool NAT
( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg M), card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)), the carrier of K
card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -tuples_on NAT
(card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)) } is set
( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(card (Seg M)),(card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),((Seg M)),(((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (Seg M), card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)), the carrier of K
(card (Seg P)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
(K,(card (Seg M)),(card (Seg P)),( the carrier of K,x2,(Seg M),(Seg P))) is finite Element of bool the carrier of ((card (Seg P)) -VectSp_over K)
the carrier of ((card (Seg P)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg P)) -VectSp_over K) is non empty set
(card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
(K,(card (Seg M)),(card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)))) is finite Element of bool the carrier of ((card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -VectSp_over K)
the carrier of ((card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -VectSp_over K) is non empty set
bool the carrier of ((card ((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -VectSp_over K) is non empty set
(K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Indices ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) is set
dom ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) is finite Element of bool NAT
width ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)))) is finite width ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) ) } is set
[:(dom ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1)))),(Seg (width ( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))))):] is Relation-like finite set
y2 is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
[:y2,Q:] is Relation-like finite set
card y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),y2,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y2, card y2, the carrier of K
Det ( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),y2,Q) is Element of the carrier of K
Permutations (card y2) is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
FinOmega (Permutations (card y2)) is Element of Fin (Permutations (card y2))
Fin (Permutations (card y2)) is preBoolean set
Path_product ( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),y2,Q) is Relation-like Permutations (card y2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card y2)), the carrier of K:]
[:(Permutations (card y2)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card y2)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card y2))),(Path_product ( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),y2,Q))) is Element of the carrier of K
((Seg M)) .: y2 is finite V212() V213() V214() V217() set
(((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))) .: Q is finite V212() V213() V214() V217() set
( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),y2,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y2, card Q, the carrier of K
(y2) is Relation-like NAT -defined NAT -valued Function-like finite card y2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card y2) -tuples_on NAT
(card y2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card y2 } is set
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
( the carrier of K,( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(Seg M),((Seg (width ( the carrier of K,x2,(Seg M),(Seg P)))) \ (x1))),(card y2),(card Q),(y2),(Q)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card y2, card Q, the carrier of K
P1 is finite without_zero Element of bool NAT
Q1 is finite without_zero Element of bool NAT
card P1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),P1,Q1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of K
(P1) is Relation-like NAT -defined NAT -valued Function-like finite card P1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P1) -tuples_on NAT
(card P1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P1 } is set
(Q1) is Relation-like NAT -defined NAT -valued Function-like finite card Q1 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q1) -tuples_on NAT
(card Q1) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q1 } is set
( the carrier of K,( the carrier of K,x2,(Seg M),(Seg P)),(card P1),(card Q1),(P1),(Q1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card Q1, the carrier of K
[:P1,Q1:] is Relation-like finite set
((Seg M)) .: P1 is finite V212() V213() V214() V217() set
((Seg P)) .: Q1 is finite V212() V213() V214() V217() set
P2 is finite without_zero Element of bool NAT
Q2 is finite without_zero Element of bool NAT
card P2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,x2,P2,Q2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of K
(P2) is Relation-like NAT -defined NAT -valued Function-like finite card P2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P2) -tuples_on NAT
(card P2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P2 } is set
(Q2) is Relation-like NAT -defined NAT -valued Function-like finite card Q2 -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q2) -tuples_on NAT
(card Q2) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q2 } is set
( the carrier of K,x2,(card P2),(card Q2),(P2),(Q2)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card Q2, the carrier of K
(idseq P) .: Q1 is finite set
i is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(i) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
Q2 \/ (i) is non empty finite without_zero V103() Element of bool NAT
((Q2 \/ (i))) is Relation-like NAT -defined NAT -valued Function-like finite card (Q2 \/ (i)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Q2 \/ (i))) -tuples_on NAT
card (Q2 \/ (i)) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(card (Q2 \/ (i))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Q2 \/ (i)) } is set
rng ((Q2 \/ (i))) is finite V212() V213() V214() V217() set
(idseq M) .: P1 is finite set
( the carrier of K,x2,P1,Q1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P1, card P1, the carrier of K
( the carrier of K,x2,P2,Q2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of K
len ( the carrier of K,x2,P2,Q2) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
(m) is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
P2 \/ (m) is non empty finite without_zero V103() Element of bool NAT
( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P2 \/ (m)), card (P2 \/ (m)), the carrier of K
card (P2 \/ (m)) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
Seg m is non empty finite m -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom ((Q2 \/ (i))) is finite card (Q2 \/ (i)) -element Element of bool NAT
Si is set
((Q2 \/ (i))) . Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card (P2 \/ (m))) -' 1,(card (P2 \/ (m))) -' 1, the carrier of K
(card (P2 \/ (m))) -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
m -' 1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:(P2 \/ (m)),(Q2 \/ (i)):] is Relation-like non empty finite set
[:(Seg m),(Seg P):] is Relation-like finite set
Indices x2 is set
dom x2 is finite Element of bool NAT
Seg (width x2) is finite width x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x2 ) } is set
[:(dom x2),(Seg (width x2)):] is Relation-like finite set
card (Seg m) is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
dom ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is finite Element of bool NAT
len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i)))) is finite len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) ) } is set
Seg (len x2) is finite len x2 -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len x2 ) } is set
LC is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Col (x2,i) is Relation-like NAT -defined the carrier of K -valued Function-like finite len x2 -element FinSequence-like FinSubsequence-like Element of (len x2) -tuples_on the carrier of K
(len x2) -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 x2 } is set
( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P2 \/ (m)), card (Q2 \/ (i)), the carrier of K
((P2 \/ (m))) is Relation-like NAT -defined NAT -valued Function-like finite card (P2 \/ (m)) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (P2 \/ (m))) -tuples_on NAT
(card (P2 \/ (m))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (P2 \/ (m)) } is set
( the carrier of K,x2,(card (P2 \/ (m))),(card (Q2 \/ (i))),((P2 \/ (m))),((Q2 \/ (i)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card (P2 \/ (m)), card (Q2 \/ (i)), the carrier of K
Col (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i)))) -tuples_on the carrier of K
len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i)))) -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,x2,(P2 \/ (m)),(Q2 \/ (i))) } is set
Col (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i)))) -tuples_on the carrier of K
(len ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i)))) -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,x2,(P2 \/ (m)),(Q2 \/ (i))) } is set
( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (LC,Si) is Element of the carrier of K
(Col (x2,i)) . LC is set
x2 * (LC,i) is Element of the carrier of K
( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (m,Si) is Element of the carrier of K
x2 * (m,i) is Element of the carrier of K
Line (Q,m) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Q -element FinSequence-like FinSubsequence-like Element of (width Q) -tuples_on the carrier of K
(Line (Q,m)) . i is set
LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) is finite Element of bool NAT
mSi is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
x2 * (mSi,i) is Element of the carrier of K
( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (mSi,Si) is Element of the carrier of K
J is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si) 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 non empty total quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non empty non trivial non finite V103() set
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
the OneF of K is Element of the carrier of K
- (1_ K) is Element of the carrier of K
J + Si is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
(power K) . ((- (1_ K)),(J + Si)) is Element of the carrier of K
Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si) is Element of the carrier of K
Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of (card (P2 \/ (m))) -' 1,(card (P2 \/ (m))) -' 1, the carrier of K
Det (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si)) is Element of the carrier of K
Permutations ((card (P2 \/ (m))) -' 1) is non empty permutational set
FinOmega (Permutations ((card (P2 \/ (m))) -' 1)) is Element of Fin (Permutations ((card (P2 \/ (m))) -' 1))
Fin (Permutations ((card (P2 \/ (m))) -' 1)) is preBoolean set
Path_product (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si)) is Relation-like Permutations ((card (P2 \/ (m))) -' 1) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card (P2 \/ (m))) -' 1)), the carrier of K:]
[:(Permutations ((card (P2 \/ (m))) -' 1)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations ((card (P2 \/ (m))) -' 1)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations ((card (P2 \/ (m))) -' 1))),(Path_product (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si)))) is Element of the carrier of K
((power K) . ((- (1_ K)),(J + Si))) * (Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si)) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the multF of K . (((power K) . ((- (1_ K)),(J + Si))),(Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si))) is Element of the carrier of K
(( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (mSi,Si)) * (Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si)) is Element of the carrier of K
the multF of K . ((( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (mSi,Si)),(Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),J,Si))) is Element of the carrier of K
(LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) . mSi is set
(LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) . m is set
Sum (LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) is Element of the carrier of K
the addF of K $$ (LaplaceExpC (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),Si)) is Element of the carrier of K
Det ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is Element of the carrier of K
Permutations (card (P2 \/ (m))) is non empty permutational set
FinOmega (Permutations (card (P2 \/ (m)))) is Element of Fin (Permutations (card (P2 \/ (m))))
Fin (Permutations (card (P2 \/ (m)))) is preBoolean set
Path_product ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) is Relation-like Permutations (card (P2 \/ (m))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card (P2 \/ (m)))), the carrier of K:]
[:(Permutations (card (P2 \/ (m)))), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card (P2 \/ (m)))), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card (P2 \/ (m))))),(Path_product ( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))))) is Element of the carrier of K
m + Si is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative Element of NAT
mSi is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(power K) . ((- (1_ K)),mSi) is Element of the carrier of K
((P2 \/ (m))) . m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
id (Seg m) is Relation-like Seg m -defined Seg m -valued Function-like one-to-one non empty total quasi_total finite Element of bool [:(Seg m),(Seg m):]
[:(Seg m),(Seg m):] is Relation-like non empty finite set
bool [:(Seg m),(Seg m):] is non empty finite V37() set
(idseq m) . m is set
(P2 \/ (m)) \ (m) is finite without_zero Element of bool NAT
(Q2 \/ (i)) \ (i) is finite without_zero Element of bool NAT
( the carrier of K,x2,((P2 \/ (m)) \ (m)),((Q2 \/ (i)) \ (i))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card ((P2 \/ (m)) \ (m)), card ((P2 \/ (m)) \ (m)), the carrier of K
card ((P2 \/ (m)) \ (m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,x2,P2,((Q2 \/ (i)) \ (i))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P2, card P2, the carrier of K
Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si) is Element of the carrier of K
Det (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is Element of the carrier of K
Path_product (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is Relation-like Permutations ((card (P2 \/ (m))) -' 1) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations ((card (P2 \/ (m))) -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations ((card (P2 \/ (m))) -' 1))),(Path_product (Delete (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)))) is Element of the carrier of K
((power K) . ((- (1_ K)),mSi)) * (Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is Element of the carrier of K
the multF of K . (((power K) . ((- (1_ K)),mSi)),(Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si))) is Element of the carrier of K
Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si) is Element of the carrier of K
m + Si is non empty V26() V27() V28() V32() finite cardinal V105() complex ext-real positive non negative set
(power K) . ((- (1_ K)),(m + Si)) is Element of the carrier of K
((power K) . ((- (1_ K)),(m + Si))) * (Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is Element of the carrier of K
the multF of K . (((power K) . ((- (1_ K)),(m + Si))),(Minor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si))) is Element of the carrier of K
(( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (m,Si)) * (Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si)) is Element of the carrier of K
the multF of K . ((( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))) * (m,Si)),(Cofactor (( the carrier of K,x2,(P2 \/ (m)),(Q2 \/ (i))),m,Si))) is Element of the carrier of K
M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
R -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of M,R, the carrier of K
(K,M,R,P) is finite Element of bool the carrier of (R -VectSp_over K)
the carrier of (R -VectSp_over K) is non empty set
bool the carrier of (R -VectSp_over K) is non empty set
(K,P) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,m,n,M) is finite Element of bool the carrier of (n -VectSp_over K)
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is non empty set
(K,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
R is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,R) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,m,n,R) is finite Element of bool the carrier of (n -VectSp_over K)
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
card (Seg m) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex 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 Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
m -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of (m -VectSp_over K) is non empty set
bool the carrier of (m -VectSp_over K) is non empty set
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of n,m, the carrier of K
(K,n,m,M) is finite Element of bool the carrier of (m -VectSp_over K)
R is Element of bool the carrier of (m -VectSp_over K)
P is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width M) -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 M } is set
[:R,(Seg n):] is Relation-like set
bool [:R,(Seg n):] is non empty set
P is Relation-like R -defined Seg n -valued Function-like quasi_total Element of bool [:R,(Seg n):]
rng P is set
Q is finite without_zero Element of bool NAT
( the carrier of K,M,Q,(Seg m)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card (Seg m), the carrier of K
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
((Seg m)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg m) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg m)) -tuples_on NAT
(card (Seg m)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg m) } is set
( the carrier of K,M,(card Q),(card (Seg m)),(Q),((Seg m))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card (Seg m), the carrier of K
rng (Q) is finite V212() V213() V214() V217() set
(K,(card Q),(card (Seg m)),( the carrier of K,M,Q,(Seg m))) is finite Element of bool the carrier of ((card (Seg m)) -VectSp_over K)
(card (Seg m)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of ((card (Seg m)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg m)) -VectSp_over K) is non empty set
x1 is set
dom ( the carrier of K,M,Q,(Seg m)) is finite Element of bool NAT
len ( the carrier of K,M,Q,(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of K,M,Q,(Seg m))) is finite len ( the carrier of K,M,Q,(Seg m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,M,Q,(Seg m)) ) } is set
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
x2 is set
( the carrier of K,M,Q,(Seg m)) . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
dom (Q) is finite card Q -element Element of bool NAT
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Q) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom P is Element of bool R
bool R is non empty set
i2 is set
P . i2 is set
Line (( the carrier of K,M,Q,(Seg m)),i1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,Q,(Seg m)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,Q,(Seg m))) -tuples_on the carrier of K
width ( the carrier of K,M,Q,(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,Q,(Seg m))) -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,M,Q,(Seg m)) } is set
y1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,y1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
x1 is set
dom (Q) is finite card Q -element Element of bool NAT
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
P . x1 is set
x2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width M) -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 M } is set
dom P is Element of bool R
bool R is non empty set
i1 is set
(Q) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Line (( the carrier of K,M,Q,(Seg m)),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,Q,(Seg m)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,Q,(Seg m))) -tuples_on the carrier of K
width ( the carrier of K,M,Q,(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,Q,(Seg m))) -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,M,Q,(Seg m)) } is set
x1 is set
dom ( the carrier of K,M,Q,(Seg m)) is finite set
x2 is set
( the carrier of K,M,Q,(Seg m)) . x1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
( the carrier of K,M,Q,(Seg m)) . x2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom ( the carrier of K,M,Q,(Seg m)) is finite Element of bool NAT
len ( the carrier of K,M,Q,(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of K,M,Q,(Seg m))) is finite len ( the carrier of K,M,Q,(Seg m)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,M,Q,(Seg m)) ) } is set
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
dom (Q) is finite card Q -element Element of bool NAT
i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Q) . i1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
dom P is Element of bool R
bool R is non empty set
y1 is set
P . y1 is set
y2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,y2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width M) -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 M } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(Q) . i2 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
y2 is set
P . y2 is set
Line (( the carrier of K,M,Q,(Seg m)),i2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,Q,(Seg m)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,Q,(Seg m))) -tuples_on the carrier of K
width ( the carrier of K,M,Q,(Seg m)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(width ( the carrier of K,M,Q,(Seg m))) -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,M,Q,(Seg m)) } is set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
Line (( the carrier of K,M,Q,(Seg m)),i1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,M,Q,(Seg m)) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,M,Q,(Seg m))) -tuples_on the carrier of K
( the carrier of K,M,Q,(Seg m)) . i1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
Line (M,Q) is Relation-like NAT -defined the carrier of K -valued Function-like finite width M -element FinSequence-like FinSubsequence-like Element of (width M) -tuples_on the carrier of K
(width M) -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 M } is set
m is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V167() left_unital doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
n -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is non empty set
M is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of m,n, the carrier of K
(K,M) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,m,n,M) is finite Element of bool the carrier of (n -VectSp_over K)
R is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
P is finite Element of bool the carrier of (n -VectSp_over K)
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg m is finite m -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
Q is finite without_zero Element of bool NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,M,Q,(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card (Seg n), the carrier of K
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
((Seg n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
( the carrier of K,M,(card Q),(card (Seg n)),(Q),((Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card Q, card (Seg n), the carrier of K
(K,(card Q),(card (Seg n)),( the carrier of K,M,Q,(Seg n))) is finite Element of bool the carrier of ((card (Seg n)) -VectSp_over K)
(card (Seg n)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
dom ( the carrier of K,M,Q,(Seg n)) is finite Element of bool NAT
( the carrier of K,M,Q,(Seg n)) .: (dom ( the carrier of K,M,Q,(Seg n))) is finite set
card (dom ( the carrier of K,M,Q,(Seg n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
len ( the carrier of K,M,Q,(Seg n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of K,M,Q,(Seg n))) is finite len ( the carrier of K,M,Q,(Seg n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,M,Q,(Seg n)) ) } is set
card (Seg (len ( the carrier of K,M,Q,(Seg n)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card Q) is finite card Q -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card Q ) } is set
card (Seg (card Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
[:Q,(Seg n):] is Relation-like finite set
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
[:(Seg (len M)),(Seg (width M)):] is Relation-like finite set
Indices M is set
dom M is finite Element of bool NAT
[:(dom M),(Seg (width M)):] is Relation-like finite set
(K,( the carrier of K,M,Q,(Seg n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
width M is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (width M) is finite width M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width M ) } is set
Indices M is set
dom M is finite Element of bool NAT
[:(dom M),(Seg (width M)):] is Relation-like finite set
0. K is zero Element of the carrier of K
the ZeroF of K is Element of the carrier of K
P is finite without_zero Element of bool NAT
Q is finite without_zero Element of bool NAT
[:P,Q:] is Relation-like finite set
card P is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card Q is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,M,P,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card P, the carrier of K
Det ( the carrier of K,M,P,Q) is Element of the carrier of K
Permutations (card P) is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
FinOmega (Permutations (card P)) is Element of Fin (Permutations (card P))
Fin (Permutations (card P)) is preBoolean set
Path_product ( the carrier of K,M,P,Q) is Relation-like Permutations (card P) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (card P)), the carrier of K:]
[:(Permutations (card P)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (card P)), the carrier of K:] is non empty set
the addF of K $$ ((FinOmega (Permutations (card P))),(Path_product ( the carrier of K,M,P,Q))) is Element of the carrier of K
( the carrier of K,M,P,(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card (Seg n), the carrier of K
(P) is Relation-like NAT -defined NAT -valued Function-like finite card P -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card P) -tuples_on NAT
(card P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card P } is set
((Seg n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
( the carrier of K,M,(card P),(card (Seg n)),(P),((Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card (Seg n), the carrier of K
len ( the carrier of K,M,P,(Seg n)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
( the carrier of K,M,P,Q) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of K
(Q) is Relation-like NAT -defined NAT -valued Function-like finite card Q -element FinSequence-like FinSubsequence-like V202() V203() V204() V205() Element of (card Q) -tuples_on NAT
(card Q) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card Q } is set
( the carrier of K,M,(card P),(card Q),(P),(Q)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular Matrix of card P, card Q, the carrier of K
(K,( the carrier of K,M,P,Q)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(K,( the carrier of K,M,P,(Seg n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
(card (Seg n)) -VectSp_over K is non empty right_complementable V95() Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over K
(K,(card P),(card (Seg n)),( the carrier of K,M,P,(Seg n))) is finite Element of bool the carrier of ((card (Seg n)) -VectSp_over K)
the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
bool the carrier of ((card (Seg n)) -VectSp_over K) is non empty set
dom ( the carrier of K,M,P,(Seg n)) is finite Element of bool NAT
( the carrier of K,M,P,(Seg n)) .: (dom ( the carrier of K,M,P,(Seg n))) is finite set
card (K,(card P),(card (Seg n)),( the carrier of K,M,P,(Seg n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
card (dom ( the carrier of K,M,P,(Seg n))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len ( the carrier of K,M,P,(Seg n))) is finite len ( the carrier of K,M,P,(Seg n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,M,P,(Seg n)) ) } is set
card (Seg (len ( the carrier of K,M,P,(Seg n)))) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (card P) is finite card P -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= card P ) } is set
card (Seg (card P)) is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
Seg (len M) is finite len M -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
x1 is finite Element of bool the carrier of (n -VectSp_over K)
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT
x1 is finite Element of bool the carrier of (n -VectSp_over K)
card x1 is V26() V27() V28() V32() finite cardinal V105() complex ext-real non negative Element of NAT