:: MATRIXJ2 semantic presentation

REAL is set
NAT is non empty non trivial V25() V26() V27() non finite cardinal limit_cardinal Element of bool REAL
bool REAL is set
K633() is strict algebraic-closed doubleLoopStr
the carrier of K633() is set
NAT is non empty non trivial V25() V26() V27() non finite cardinal limit_cardinal set
bool NAT is non trivial non finite set
bool NAT is non trivial non finite set
K95(NAT) is V24() set
INT is set
COMPLEX is set
RAT is set
{} is Relation-like non-empty empty-yielding RAT -valued functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() set
2 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
0 is Relation-like non-empty empty-yielding RAT -valued functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Element of NAT
K478(0,1,2) is finite set
[:K478(0,1,2),K478(0,1,2):] is Relation-like finite set
[:[:K478(0,1,2),K478(0,1,2):],K478(0,1,2):] is Relation-like finite set
bool [:[:K478(0,1,2),K478(0,1,2):],K478(0,1,2):] is finite V36() set
bool [:K478(0,1,2),K478(0,1,2):] is finite V36() set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is set
{{},1} is non empty finite V36() set
K705() is set
bool K705() is set
K706() is Element of bool K705()
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is set
[:NAT,NAT:] is Relation-like non trivial non finite set
[:[:NAT,NAT:],NAT:] is Relation-like non trivial non finite set
bool [:[:NAT,NAT:],NAT:] is non trivial non finite set
K843() is set
3 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
card {} is Relation-like non-empty empty-yielding RAT -valued functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() set
Seg 1 is non empty trivial finite 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V36() 1 -element set
Seg 2 is non empty finite 2 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V36() set
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of REAL
K667((<*> REAL)) is V46() ext-real V165() Element of REAL
n is set
K is set
{K} is non empty trivial finite 1 -element set
M is Relation-like Function-like set
dom M is set
M +* (n,K) is Relation-like Function-like set
rng (M +* (n,K)) is set
rng M is set
M . n is set
{(M . n)} is non empty trivial finite 1 -element set
(rng M) \ {(M . n)} is Element of bool (rng M)
bool (rng M) is set
((rng M) \ {(M . n)}) \/ {K} is non empty set
n .--> K is Relation-like {n} -defined Function-like one-to-one finite set
{n} is non empty trivial finite 1 -element set
{n} --> K is Relation-like {n} -defined {K} -valued Function-like constant non empty total V18({n},{K}) finite Element of bool [:{n},{K}:]
[:{n},{K}:] is Relation-like finite set
bool [:{n},{K}:] is finite V36() set
dom (n .--> K) is finite set
rng (n .--> K) is finite set
M +* (n .--> K) is Relation-like Function-like set
rng (M +* (n .--> K)) is set
(dom M) \ {n} is Element of bool (dom M)
bool (dom M) is set
M .: ((dom M) \ {n}) is set
(M .: ((dom M) \ {n})) \/ {K} is non empty set
M .: (dom M) is set
M .: {n} is finite set
(M .: (dom M)) \ (M .: {n}) is Element of bool (M .: (dom M))
bool (M .: (dom M)) is set
((M .: (dom M)) \ (M .: {n})) \/ {K} is non empty set
Im (M,n) is set
(rng M) \ (Im (M,n)) is Element of bool (rng M)
((rng M) \ (Im (M,n))) \/ {K} is non empty set
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is Element of the carrier of n
1_ n is Element of the carrier of n
1. n is V65(n) Element of the carrier of n
the OneF of n is Element of the carrier of n
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg V is finite V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= V ) } is set
[:(Seg V),(Seg V):] is Relation-like finite set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
B + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[B,M9] is set
{B,M9} is non empty finite V36() set
{B} is non empty trivial finite V36() 1 -element set
{{B,M9},{B}} is non empty finite V36() set
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of V,V, the carrier of n
Indices B is set
dom B is finite Element of bool NAT
width B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width B) is finite width B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[M9,T] is set
{M9,T} is non empty finite V36() set
{M9} is non empty trivial finite V36() 1 -element set
{{M9,T},{M9}} is non empty finite V36() set
B * (M9,T) is Element of the carrier of n
M9 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Indices V is set
dom V is finite Element of bool NAT
Seg (width V) is finite width V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width V ) } is set
[:(dom V),(Seg (width V)):] is Relation-like finite set
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Indices B is set
dom B is finite Element of bool NAT
Seg (width B) is finite width B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[M9,T] is set
{M9,T} is non empty finite V36() set
{M9} is non empty trivial finite V36() 1 -element set
{{M9,T},{M9}} is non empty finite V36() set
Seg M is finite M -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
[:(Seg M),(Seg M):] is Relation-like finite set
M9 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
V * (M9,T) is Element of the carrier of n
B * (M9,T) is Element of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
K is Element of the carrier of n
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,M) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
len (n,K,M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width (n,K,M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[B,M9] is set
{B,M9} is non empty finite V36() set
{B} is non empty trivial finite V36() 1 -element set
{{B,M9},{B}} is non empty finite V36() set
V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of M,M, the carrier of n
Indices V is set
dom V is finite Element of bool NAT
width V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width V) is finite width V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width V ) } is set
[:(dom V),(Seg (width V)):] is Relation-like finite set
B + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
V * (B,M9) is Element of the carrier of n
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
diagonal_of_Matrix (K,M,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
n |-> M 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
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
Indices (K,M,n) is set
dom (K,M,n) is finite Element of bool NAT
width (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (K,M,n)) is finite width (K,M,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (K,M,n) ) } is set
[:(dom (K,M,n)),(Seg (width (K,M,n))):] is Relation-like finite set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[B,B] is set
{B,B} is non empty finite V36() set
{B} is non empty trivial finite V36() 1 -element set
{{B,B},{B}} is non empty finite V36() set
(diagonal_of_Matrix (K,M,n)) . B is set
(K,M,n) * (B,B) is Element of the carrier of K
(n |-> M) . B is set
len (diagonal_of_Matrix (K,M,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (n |-> M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like total V18([: the carrier of K,NAT:], the carrier of K) Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial non finite set
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Det (K,M,n) is Element of the carrier of K
(power K) . (M,n) is Element of the carrier of K
diagonal_of_Matrix (K,M,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the multF of K $$ (diagonal_of_Matrix (K,M,n)) is Element of the carrier of K
n |-> M is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Product (n |-> M) is Element of the carrier of K
the multF of K $$ (n |-> M) is Element of the carrier of K
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Det (K,M,n) is Element of the carrier of K
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
n |-> M 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
dom (n |-> M) is finite n -element Element of bool NAT
(n |-> M) . n is set
Product (n |-> M) 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 total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the multF of K $$ (n |-> M) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like total V18([: the carrier of K,NAT:], the carrier of K) Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial non finite set
(power K) . (M,n) is Element of the carrier of K
Det (K,M,n) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like total V18([: the carrier of K,NAT:], the carrier of K) Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial non finite set
(power K) . (M,n) is Element of the carrier of K
n |-> M is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Product (n |-> M) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the multF of K $$ (n |-> M) is Element of the carrier of K
dom (n |-> M) is finite n -element Element of bool NAT
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n |-> M) . B is set
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
n + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
M is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of M is non empty non trivial set
1. (M,K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of K,K, the carrier of M
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
width (1. (M,K)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((1. (M,K)),n) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (1. (M,K)) -element FinSequence-like FinSubsequence-like Element of (width (1. (M,K))) -tuples_on the carrier of M
(width (1. (M,K))) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
Line ((1. (M,K)),(n + 1)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (1. (M,K)) -element FinSequence-like FinSubsequence-like Element of (width (1. (M,K))) -tuples_on the carrier of M
V is Element of the carrier of M
(M,V,K) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of K,K, the carrier of M
Line ((M,V,K),n) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (M,V,K) -element FinSequence-like FinSubsequence-like Element of (width (M,V,K)) -tuples_on the carrier of M
width (M,V,K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (M,V,K)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
V * (Line ((1. (M,K)),n)) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (1. (M,K)) -element FinSequence-like FinSubsequence-like Element of (width (1. (M,K))) -tuples_on the carrier of M
V multfield is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like set
bool [: the carrier of M, the carrier of M:] is set
the multF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
id the carrier of M is Relation-like the carrier of M -defined the carrier of M -valued non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
the multF of M [;] (V,(id the carrier of M)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
K440( the carrier of M, the carrier of M,(Line ((1. (M,K)),n)),(V multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(V * (Line ((1. (M,K)),n))) + (Line ((1. (M,K)),(n + 1))) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (1. (M,K)) -element FinSequence-like FinSubsequence-like Element of (width (1. (M,K))) -tuples_on 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
K437( the carrier of M, the carrier of M, the carrier of M, the addF of M,(V * (Line ((1. (M,K)),n))),(Line ((1. (M,K)),(n + 1)))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((V * (Line ((1. (M,K)),n))),(Line ((1. (M,K)),(n + 1)))) is Relation-like Function-like set
K685((V * (Line ((1. (M,K)),n))),(Line ((1. (M,K)),(n + 1)))) * the addF of M is Relation-like the carrier of M -valued set
Indices (1. (M,K)) is set
dom (1. (M,K)) is finite Element of bool NAT
Seg (width (1. (M,K))) is finite width (1. (M,K)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (M,K)) ) } is set
[:(dom (1. (M,K))),(Seg (width (1. (M,K)))):] is Relation-like finite set
Indices (M,V,K) is set
dom (M,V,K) is finite Element of bool NAT
Seg (width (M,V,K)) is finite width (M,V,K) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (M,V,K) ) } is set
[:(dom (M,V,K)),(Seg (width (M,V,K))):] is Relation-like finite set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
[:(Seg K),(Seg K):] is Relation-like finite set
Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM is Relation-like NAT -defined the carrier of M -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of M
IM . Jk is set
(1. (M,K)) * (n,Jk) is Element of the carrier of M
V * IM is Relation-like NAT -defined the carrier of M -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of M
K440( the carrier of M, the carrier of M,IM,(V multfield)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(V * IM) . Jk is set
V * ((1. (M,K)) * (n,Jk)) is Element of the carrier of M
[n,Jk] is set
{n,Jk} is non empty finite V36() set
{n} is non empty trivial finite V36() 1 -element set
{{n,Jk},{n}} is non empty finite V36() set
[(n + 1),Jk] is set
{(n + 1),Jk} is non empty finite V36() set
{(n + 1)} is non empty trivial finite V36() 1 -element set
{{(n + 1),Jk},{(n + 1)}} is non empty finite V36() set
KER is Relation-like NAT -defined the carrier of M -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of M
KER . Jk is set
(1. (M,K)) * ((n + 1),Jk) is Element of the carrier of M
(V * IM) + KER is Relation-like NAT -defined the carrier of M -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of M
K437( the carrier of M, the carrier of M, the carrier of M, the addF of M,(V * IM),KER) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((V * IM),KER) is Relation-like Function-like set
K685((V * IM),KER) * the addF of M is Relation-like the carrier of M -valued set
((V * IM) + KER) . Jk is set
(V * ((1. (M,K)) * (n,Jk))) + ((1. (M,K)) * ((n + 1),Jk)) is Element of the carrier of M
FK is Relation-like NAT -defined the carrier of M -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of M
FK . Jk is set
(M,V,K) * (n,Jk) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
V + (0. M) is Element of the carrier of M
1_ M is Element of the carrier of M
1. M is V65(M) Element of the carrier of M
the OneF of M is Element of the carrier of M
V * (1_ M) is Element of the carrier of M
(V * (1_ M)) + (0. M) is Element of the carrier of M
(V * ((1. (M,K)) * (n,Jk))) + (0. M) is Element of the carrier of M
1_ M is Element of the carrier of M
1. M is V65(M) Element of the carrier of M
the OneF of M is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
(0. M) + (1_ M) is Element of the carrier of M
V * (0. M) is Element of the carrier of M
(V * (0. M)) + (1_ M) is Element of the carrier of M
(V * ((1. (M,K)) * (n,Jk))) + (1_ M) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
(0. M) + (0. M) is Element of the carrier of M
V * (0. M) is Element of the carrier of M
(V * (0. M)) + (0. M) is Element of the carrier of M
(V * ((1. (M,K)) * (n,Jk))) + (0. M) is Element of the carrier of M
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
width (1. (K,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((1. (K,n)),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,n)) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,n))) -tuples_on the carrier of K
(width (1. (K,n))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
Line ((K,M,n),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,M,n) -element FinSequence-like FinSubsequence-like Element of (width (K,M,n)) -tuples_on the carrier of K
width (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (K,M,n)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M * (Line ((1. (K,n)),n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,n)) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,n))) -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 V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of K, the carrier of K:], the carrier of K) 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 set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued non empty total V18( the carrier of K, the carrier of K) 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 V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
K440( the carrier of K, the carrier of K,(Line ((1. (K,n)),n)),(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
Indices (1. (K,n)) is set
dom (1. (K,n)) is finite Element of bool NAT
Seg (width (1. (K,n))) is finite width (1. (K,n)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,n)) ) } is set
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like finite set
Indices (K,M,n) is set
dom (K,M,n) is finite Element of bool NAT
Seg (width (K,M,n)) is finite width (K,M,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (K,M,n) ) } is set
[:(dom (K,M,n)),(Seg (width (K,M,n))):] is Relation-like finite set
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[n,c₁₁] is set
{n,c₁₁} is non empty finite V36() set
{n} is non empty trivial finite V36() 1 -element set
{{n,c₁₁},{n}} is non empty finite V36() set
b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite J -element FinSequence-like FinSubsequence-like Element of J -tuples_on the carrier of K
b1 . c₁₁ is set
(1. (K,n)) * (n,c₁₁) is Element of the carrier of K
M * b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite J -element FinSequence-like FinSubsequence-like Element of J -tuples_on the carrier of K
K440( the carrier of K, the carrier of K,b1,(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 * b1) . c₁₁ is set
M * ((1. (K,n)) * (n,c₁₁)) is Element of the carrier of K
P is Relation-like NAT -defined the carrier of K -valued Function-like finite J -element FinSequence-like FinSubsequence-like Element of J -tuples_on the carrier of K
P . c₁₁ is set
(K,M,n) * (n,c₁₁) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
M * (1_ K) is Element of the carrier of K
n + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
n + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M * (0. K) is Element of the carrier of K
n + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of M is non empty non trivial set
n -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
V is Element of the carrier of M
(M,V,n) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of M
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
Line ((M,V,n),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite width (M,V,n) -element FinSequence-like FinSubsequence-like Element of (width (M,V,n)) -tuples_on the carrier of M
width (M,V,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (M,V,n)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
B 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
(Line ((M,V,n),K)) "*" B is Element of the carrier of M
mlt ((Line ((M,V,n),K)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,(Line ((M,V,n),K)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((Line ((M,V,n),K)),B) is Relation-like Function-like set
K685((Line ((M,V,n),K)),B) * the multF of M is Relation-like the carrier of M -valued set
Sum (mlt ((Line ((M,V,n),K)),B)) 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
the addF of M $$ (mlt ((Line ((M,V,n),K)),B)) is Element of the carrier of M
B /. K is Element of the carrier of M
V * (B /. K) is Element of the carrier of M
B /. (K + 1) is Element of the carrier of M
(V * (B /. K)) + (B /. (K + 1)) is Element of the carrier of M
(Line ((M,V,n),K)) . K is set
(M,V,n) * (K,K) is Element of the carrier of M
Indices (M,V,n) is set
dom (M,V,n) is finite Element of bool NAT
Seg (width (M,V,n)) is finite width (M,V,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (M,V,n) ) } is set
[:(dom (M,V,n)),(Seg (width (M,V,n))):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
[K,K] is set
{K,K} is non empty finite V36() set
{K} is non empty trivial finite V36() 1 -element set
{{K,K},{K}} is non empty finite V36() set
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
P is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
dom P is finite T -element Element of bool NAT
P . K is set
P /. K is Element of the carrier of M
b1 is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
mlt (b1,P) is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,b1,P) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685(b1,P) is Relation-like Function-like set
K685(b1,P) * the multF of M is Relation-like the carrier of M -valued set
dom (mlt (b1,P)) is finite T -element Element of bool NAT
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[K,c₁₁] is set
{K,c₁₁} is non empty finite V36() set
{{K,c₁₁},{K}} is non empty finite V36() set
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
(M,V,n) * (K,c₁₁) is Element of the carrier of M
b1 . c₁₁ is set
P . c₁₁ is set
P /. c₁₁ is Element of the carrier of M
(mlt (b1,P)) . c₁₁ is set
(0. M) * (P /. c₁₁) is Element of the carrier of M
[K,(K + 1)] is set
{K,(K + 1)} is non empty finite V36() set
{{K,(K + 1)},{K}} is non empty finite V36() set
b1 . K is set
(mlt (b1,P)) /. K is Element of the carrier of M
(mlt (b1,P)) . K is set
((M,V,n) * (K,K)) * (P /. K) is Element of the carrier of M
V * (P /. K) is Element of the carrier of M
P . (K + 1) is set
P /. (K + 1) is Element of the carrier of M
(M,V,n) * (K,(K + 1)) is Element of the carrier of M
b1 . (K + 1) is set
(mlt (b1,P)) /. (K + 1) is Element of the carrier of M
(mlt (b1,P)) . (K + 1) is set
((M,V,n) * (K,(K + 1))) * (P /. (K + 1)) is Element of the carrier of M
1_ M is Element of the carrier of M
1. M is V65(M) Element of the carrier of M
the OneF of M is Element of the carrier of M
(1_ M) * (P /. (K + 1)) is Element of the carrier of M
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
P . c₁₁ is set
P /. c₁₁ is Element of the carrier of M
[K,c₁₁] is set
{K,c₁₁} is non empty finite V36() set
{{K,c₁₁},{K}} is non empty finite V36() set
(Line ((M,V,n),K)) . c₁₁ is set
(M,V,n) * (K,c₁₁) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
mlt ((Line ((M,V,n),K)),P) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,(Line ((M,V,n),K)),P) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((Line ((M,V,n),K)),P) is Relation-like Function-like set
K685((Line ((M,V,n),K)),P) * the multF of M is Relation-like the carrier of M -valued set
(mlt ((Line ((M,V,n),K)),P)) . c₁₁ is set
(0. M) * (P /. c₁₁) is Element of the carrier of M
(mlt ((Line ((M,V,n),K)),P)) . K is set
((M,V,n) * (K,K)) * (P /. K) is Element of the carrier of M
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K - 1 is V46() ext-real V165() set
M is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of M is non empty non trivial set
n -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
V is Element of the carrier of M
(M,V,n) is Relation-like NAT -defined the carrier of M * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of M
the carrier of M * is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
Col ((M,V,n),K) is Relation-like NAT -defined the carrier of M -valued Function-like finite len (M,V,n) -element FinSequence-like FinSubsequence-like Element of (len (M,V,n)) -tuples_on the carrier of M
len (M,V,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (M,V,n)) -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
J 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
(Col ((M,V,n),K)) "*" J is Element of the carrier of M
mlt ((Col ((M,V,n),K)),J) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,(Col ((M,V,n),K)),J) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((Col ((M,V,n),K)),J) is Relation-like Function-like set
K685((Col ((M,V,n),K)),J) * the multF of M is Relation-like the carrier of M -valued set
Sum (mlt ((Col ((M,V,n),K)),J)) 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
the addF of M $$ (mlt ((Col ((M,V,n),K)),J)) is Element of the carrier of M
J /. K is Element of the carrier of M
V * (J /. K) is Element of the carrier of M
J /. (K - 1) is Element of the carrier of M
(V * (J /. K)) + (J /. (K - 1)) is Element of the carrier of M
dom (M,V,n) is finite Element of bool NAT
Seg (len (M,V,n)) is finite len (M,V,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (M,V,n) ) } is set
(Col ((M,V,n),K)) . K is set
(M,V,n) * (K,K) is Element of the carrier of M
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
Indices (M,V,n) is set
width (M,V,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (M,V,n)) is finite width (M,V,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (M,V,n) ) } is set
[:(dom (M,V,n)),(Seg (width (M,V,n))):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
[K,K] is set
{K,K} is non empty finite V36() set
{K} is non empty trivial finite V36() 1 -element set
{{K,K},{K}} is non empty finite V36() set
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T -tuples_on the carrier of M is functional non empty FinSequence-membered FinSequenceSet of the carrier of M
c₁₁ is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
dom c₁₁ is finite T -element Element of bool NAT
c₁₁ . K is set
c₁₁ /. K is Element of the carrier of M
P is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
mlt (P,c₁₁) is Relation-like NAT -defined the carrier of M -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on the carrier of M
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,P,c₁₁) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685(P,c₁₁) is Relation-like Function-like set
K685(P,c₁₁) * the multF of M is Relation-like the carrier of M -valued set
dom (mlt (P,c₁₁)) is finite T -element Element of bool NAT
(mlt (P,c₁₁)) /. K is Element of the carrier of M
(mlt (P,c₁₁)) . K is set
((M,V,n) * (K,K)) * (c₁₁ /. K) is Element of the carrier of M
V * (c₁₁ /. K) is Element of the carrier of M
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ . IM is set
c₁₁ /. IM is Element of the carrier of M
IM + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
[IM,K] is set
{IM,K} is non empty finite V36() set
{IM} is non empty trivial finite V36() 1 -element set
{{IM,K},{IM}} is non empty finite V36() set
(Col ((M,V,n),K)) . IM is set
(M,V,n) * (IM,K) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
mlt ((Col ((M,V,n),K)),c₁₁) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K437( the carrier of M, the carrier of M, the carrier of M, the multF of M,(Col ((M,V,n),K)),c₁₁) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
K685((Col ((M,V,n),K)),c₁₁) is Relation-like Function-like set
K685((Col ((M,V,n),K)),c₁₁) * the multF of M is Relation-like the carrier of M -valued set
(mlt ((Col ((M,V,n),K)),c₁₁)) . IM is set
(0. M) * (c₁₁ /. IM) is Element of the carrier of M
(mlt ((Col ((M,V,n),K)),c₁₁)) . K is set
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
[b1,K] is set
{b1,K} is non empty finite V36() set
{b1} is non empty trivial finite V36() 1 -element set
{{b1,K},{b1}} is non empty finite V36() set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[IM,K] is set
{IM,K} is non empty finite V36() set
{IM} is non empty trivial finite V36() 1 -element set
{{IM,K},{IM}} is non empty finite V36() set
IM + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
(M,V,n) * (IM,K) is Element of the carrier of M
P . IM is set
c₁₁ . IM is set
c₁₁ /. IM is Element of the carrier of M
(mlt (P,c₁₁)) . IM is set
(0. M) * (c₁₁ /. IM) is Element of the carrier of M
c₁₁ . b1 is set
c₁₁ /. b1 is Element of the carrier of M
(Col ((M,V,n),K)) . b1 is set
(M,V,n) * (b1,K) is Element of the carrier of M
(mlt (P,c₁₁)) /. b1 is Element of the carrier of M
(mlt (P,c₁₁)) . b1 is set
((M,V,n) * (b1,K)) * (c₁₁ /. b1) is Element of the carrier of M
1_ M is Element of the carrier of M
1. M is V65(M) Element of the carrier of M
the OneF of M is Element of the carrier of M
(1_ M) * (c₁₁ /. b1) is Element of the carrier of M
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like total V18([: the carrier of K,NAT:], the carrier of K) Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial non finite set
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
(K,M,n) ~ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
M " is Element of the carrier of K
- (M ") is Element of the carrier of K
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg V is finite V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= V ) } is set
[:(Seg V),(Seg V):] is Relation-like finite set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[B,M9] is set
{B,M9} is non empty finite V36() set
{B} is non empty trivial finite V36() 1 -element set
{{B,M9},{B}} is non empty finite V36() set
M9 -' B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(M9 -' B) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((M9 -' B) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((M9 -' B) + 1))) is Element of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of V,V, the carrier of K
Indices B is set
dom B is finite Element of bool NAT
width B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width B) is finite width B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
Indices (1. (K,n)) is set
dom (1. (K,n)) is finite Element of bool NAT
width (1. (K,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (1. (K,n))) is finite width (1. (K,n)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,n)) ) } is set
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like finite set
Seg n is finite n -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
J is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
J * (K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
Indices J is set
dom J is finite Element of bool NAT
width J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width J) is finite width J -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width J ) } is set
[:(dom J),(Seg (width J)):] is Relation-like finite set
Indices (K,M,n) is set
dom (K,M,n) is finite Element of bool NAT
width (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (K,M,n)) is finite width (K,M,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (K,M,n) ) } is set
[:(dom (K,M,n)),(Seg (width (K,M,n))):] is Relation-like finite set
Indices (J * (K,M,n)) is set
dom (J * (K,M,n)) is finite Element of bool NAT
width (J * (K,M,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (J * (K,M,n))) is finite width (J * (K,M,n)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (J * (K,M,n)) ) } is set
[:(dom (J * (K,M,n))),(Seg (width (J * (K,M,n)))):] is Relation-like finite set
len (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[P,c₁₁] is set
{P,c₁₁} is non empty finite V36() set
{P} is non empty trivial finite V36() 1 -element set
{{P,c₁₁},{P}} is non empty finite V36() set
Line (J,P) is Relation-like NAT -defined the carrier of K -valued Function-like finite width J -element FinSequence-like FinSubsequence-like Element of (width J) -tuples_on the carrier of K
(width J) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(J * (K,M,n)) * (P,c₁₁) is Element of the carrier of K
Col ((K,M,n),c₁₁) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (K,M,n) -element FinSequence-like FinSubsequence-like Element of (len (K,M,n)) -tuples_on the carrier of K
(len (K,M,n)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line (J,P)) "*" (Col ((K,M,n),c₁₁)) is Element of the carrier of K
mlt ((Line (J,P)),(Col ((K,M,n),c₁₁))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
K437( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (J,P)),(Col ((K,M,n),c₁₁))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K685((Line (J,P)),(Col ((K,M,n),c₁₁))) is Relation-like Function-like set
K685((Line (J,P)),(Col ((K,M,n),c₁₁))) * the multF of K is Relation-like the carrier of K -valued set
Sum (mlt ((Line (J,P)),(Col ((K,M,n),c₁₁)))) 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 total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (mlt ((Line (J,P)),(Col ((K,M,n),c₁₁)))) is Element of the carrier of K
(Col ((K,M,n),c₁₁)) "*" (Line (J,P)) is Element of the carrier of K
mlt ((Col ((K,M,n),c₁₁)),(Line (J,P))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K437( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Col ((K,M,n),c₁₁)),(Line (J,P))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K685((Col ((K,M,n),c₁₁)),(Line (J,P))) is Relation-like Function-like set
K685((Col ((K,M,n),c₁₁)),(Line (J,P))) * the multF of K is Relation-like the carrier of K -valued set
Sum (mlt ((Col ((K,M,n),c₁₁)),(Line (J,P)))) is Element of the carrier of K
the addF of K $$ (mlt ((Col ((K,M,n),c₁₁)),(Line (J,P)))) is Element of the carrier of K
(Line (J,P)) . c₁₁ is set
J * (P,c₁₁) is Element of the carrier of K
dom (Line (J,P)) is finite width J -element Element of bool NAT
(Line (J,P)) /. c₁₁ is Element of the carrier of K
M * (J * (P,c₁₁)) is Element of the carrier of K
P -' P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(P -' P) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((P -' P) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((P -' P) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((P -' P) + 1)))) is Element of the carrier of K
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),({} + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),({} + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),({} + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
M * (- (- (M "))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
M * (0. K) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
c₁₁ - 1 is V46() ext-real V165() set
KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
[P,KER] is set
{P,KER} is non empty finite V36() set
{{P,KER},{P}} is non empty finite V36() set
(Line (J,P)) /. KER is Element of the carrier of K
(Line (J,P)) . KER is set
J * (P,KER) is Element of the carrier of K
M * (J * (P,c₁₁)) is Element of the carrier of K
(M * (J * (P,c₁₁))) + (J * (P,KER)) is Element of the carrier of K
KER -' P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(KER -' P) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((KER -' P) + 1)) is Element of the carrier of K
c₁₁ -' P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ - P is V46() ext-real V165() set
KER - P is V46() ext-real V165() set
- ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
(M * (J * (P,c₁₁))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(c₁₁ -' P) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((c₁₁ -' P) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((c₁₁ -' P) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((c₁₁ -' P) + 1)))) is Element of the carrier of K
(M * (- ((power K) . ((- (M ")),((c₁₁ -' P) + 1))))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(- (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
- ((- (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
M * (- ((- (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1))))) is Element of the carrier of K
(M * (- ((- (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1)))))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
(- (- (M "))) * ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
M * ((- (- (M "))) * ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(M * ((- (- (M "))) * ((power K) . ((- (M ")),((KER -' P) + 1))))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(M ") * ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
M * ((M ") * ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(M * ((M ") * ((power K) . ((- (M ")),((KER -' P) + 1))))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
(M * (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
((M * (M ")) * ((power K) . ((- (M ")),((KER -' P) + 1)))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * ((power K) . ((- (M ")),((KER -' P) + 1))) is Element of the carrier of K
((1_ K) * ((power K) . ((- (M ")),((KER -' P) + 1)))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
((power K) . ((- (M ")),((KER -' P) + 1))) + (- ((power K) . ((- (M ")),((KER -' P) + 1)))) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(M * (J * (P,c₁₁))) + (0. K) is Element of the carrier of K
P -' P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(P -' P) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((P -' P) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((P -' P) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((P -' P) + 1)))) is Element of the carrier of K
(power K) . ((- (M ")),({} + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),({} + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),({} + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
M * (- (- (M "))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(M * (J * (P,c₁₁))) + (0. K) is Element of the carrier of K
M * (0. K) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(1. (K,n)) * (P,c₁₁) is Element of the carrier of K
(K,M,n) * J is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
len J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Indices ((K,M,n) * J) is set
dom ((K,M,n) * J) is finite Element of bool NAT
width ((K,M,n) * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width ((K,M,n) * J)) is finite width ((K,M,n) * J) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width ((K,M,n) * J) ) } is set
[:(dom ((K,M,n) * J)),(Seg (width ((K,M,n) * J))):] is Relation-like finite set
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[c₁₁,IM] is set
{c₁₁,IM} is non empty finite V36() set
{c₁₁} is non empty trivial finite V36() 1 -element set
{{c₁₁,IM},{c₁₁}} is non empty finite V36() set
c₁₁ + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
((K,M,n) * J) * (c₁₁,IM) is Element of the carrier of K
Line ((K,M,n),c₁₁) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,M,n) -element FinSequence-like FinSubsequence-like Element of (width (K,M,n)) -tuples_on the carrier of K
(width (K,M,n)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Col (J,IM) is Relation-like NAT -defined the carrier of K -valued Function-like finite len J -element FinSequence-like FinSubsequence-like Element of (len J) -tuples_on the carrier of K
(len J) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((K,M,n),c₁₁)) "*" (Col (J,IM)) is Element of the carrier of K
mlt ((Line ((K,M,n),c₁₁)),(Col (J,IM))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K437( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((K,M,n),c₁₁)),(Col (J,IM))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K685((Line ((K,M,n),c₁₁)),(Col (J,IM))) is Relation-like Function-like set
K685((Line ((K,M,n),c₁₁)),(Col (J,IM))) * the multF of K is Relation-like the carrier of K -valued set
Sum (mlt ((Line ((K,M,n),c₁₁)),(Col (J,IM)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line ((K,M,n),c₁₁)),(Col (J,IM)))) is Element of the carrier of K
(Col (J,IM)) . c₁₁ is set
J * (c₁₁,IM) is Element of the carrier of K
dom (Col (J,IM)) is finite len J -element Element of bool NAT
(Col (J,IM)) /. c₁₁ is Element of the carrier of K
M * (J * (c₁₁,IM)) is Element of the carrier of K
M * (0. K) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
n -' n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n -' n) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((n -' n) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((n -' n) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((n -' n) + 1)))) is Element of the carrier of K
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),({} + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),({} + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),({} + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
M * (- (- (M "))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
[(c₁₁ + 1),IM] is set
{(c₁₁ + 1),IM} is non empty finite V36() set
{(c₁₁ + 1)} is non empty trivial finite V36() 1 -element set
{{(c₁₁ + 1),IM},{(c₁₁ + 1)}} is non empty finite V36() set
(Col (J,IM)) /. (c₁₁ + 1) is Element of the carrier of K
(Col (J,IM)) . (c₁₁ + 1) is set
J * ((c₁₁ + 1),IM) is Element of the carrier of K
M * (J * (c₁₁,IM)) is Element of the carrier of K
(M * (J * (c₁₁,IM))) + (J * ((c₁₁ + 1),IM)) is Element of the carrier of K
(M * (J * (c₁₁,IM))) + (0. K) is Element of the carrier of K
M * (0. K) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(M * (J * (c₁₁,IM))) + (0. K) is Element of the carrier of K
J * (c₁₁,c₁₁) is Element of the carrier of K
M * (J * (c₁₁,c₁₁)) is Element of the carrier of K
c₁₁ -' c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(c₁₁ -' c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((c₁₁ -' c₁₁) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((c₁₁ -' c₁₁) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((c₁₁ -' c₁₁) + 1)))) is Element of the carrier of K
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),({} + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),({} + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),({} + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
M * (- (- (M "))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
IM -' (c₁₁ + 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(IM -' (c₁₁ + 1)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)) is Element of the carrier of K
IM -' c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
IM - c₁₁ is V46() ext-real V165() set
IM - (c₁₁ + 1) is V46() ext-real V165() set
- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
(M * (J * (c₁₁,IM))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(IM -' c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((IM -' c₁₁) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((IM -' c₁₁) + 1))) is Element of the carrier of K
M * (- ((power K) . ((- (M ")),((IM -' c₁₁) + 1)))) is Element of the carrier of K
(M * (- ((power K) . ((- (M ")),((IM -' c₁₁) + 1))))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(- (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
- ((- (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
M * (- ((- (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))))) is Element of the carrier of K
(M * (- ((- (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
- (- (M ")) is Element of the carrier of K
(- (- (M "))) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
M * ((- (- (M "))) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(M * ((- (- (M "))) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(M ") * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
M * ((M ") * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(M * ((M ") * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
M * (M ") is Element of the carrier of K
(M * (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
((M * (M ")) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) is Element of the carrier of K
((1_ K) * ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1))) + (- ((power K) . ((- (M ")),((IM -' (c₁₁ + 1)) + 1)))) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
(1. (K,n)) * (c₁₁,IM) is Element of the carrier of K
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[c₁₁,IM] is set
{c₁₁,IM} is non empty finite V36() set
{c₁₁} is non empty trivial finite V36() 1 -element set
{{c₁₁,IM},{c₁₁}} is non empty finite V36() set
J * (c₁₁,IM) is Element of the carrier of K
IM -' c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(IM -' c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(power K) . ((- (M ")),((IM -' c₁₁) + 1)) is Element of the carrier of K
- ((power K) . ((- (M ")),((IM -' c₁₁) + 1))) is Element of the carrier of K
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
V is Element of the carrier of K
V * (1. (K,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
(K,M,n) + (V * (1. (K,n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
M + V is Element of the carrier of K
(K,(M + V),n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
Indices (K,M,n) is set
dom (K,M,n) is finite Element of bool NAT
width (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (K,M,n)) is finite width (K,M,n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (K,M,n) ) } is set
[:(dom (K,M,n)),(Seg (width (K,M,n))):] is Relation-like finite set
Indices (K,(M + V),n) is set
dom (K,(M + V),n) is finite Element of bool NAT
width (K,(M + V),n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (K,(M + V),n)) is finite width (K,(M + V),n) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (K,(M + V),n) ) } is set
[:(dom (K,(M + V),n)),(Seg (width (K,(M + V),n))):] is Relation-like finite set
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[J,b1] is set
{J,b1} is non empty finite V36() set
{J} is non empty trivial finite V36() 1 -element set
{{J,b1},{J}} is non empty finite V36() set
Indices (1. (K,n)) is set
dom (1. (K,n)) is finite Element of bool NAT
width (1. (K,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (1. (K,n))) is finite width (1. (K,n)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,n)) ) } is set
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like finite set
(K,(M + V),n) * (J,b1) is Element of the carrier of K
(K,M,n) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + V is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
V * (1_ K) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * (1_ K)) is Element of the carrier of K
(1. (K,n)) * (J,b1) is Element of the carrier of K
V * ((1. (K,n)) * (J,b1)) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * ((1. (K,n)) * (J,b1))) is Element of the carrier of K
(V * (1. (K,n))) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + ((V * (1. (K,n))) * (J,b1)) is Element of the carrier of K
((K,M,n) + (V * (1. (K,n)))) * (J,b1) is Element of the carrier of K
J + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(K,(M + V),n) * (J,b1) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(1_ K) + (0. K) is Element of the carrier of K
(K,M,n) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + (0. K) is Element of the carrier of K
V * (0. K) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * (0. K)) is Element of the carrier of K
(1. (K,n)) * (J,b1) is Element of the carrier of K
V * ((1. (K,n)) * (J,b1)) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * ((1. (K,n)) * (J,b1))) is Element of the carrier of K
(V * (1. (K,n))) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + ((V * (1. (K,n))) * (J,b1)) is Element of the carrier of K
((K,M,n) + (V * (1. (K,n)))) * (J,b1) is Element of the carrier of K
J + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(K,(M + V),n) * (J,b1) is Element of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(0. K) + (0. K) is Element of the carrier of K
(K,M,n) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + (0. K) is Element of the carrier of K
V * (0. K) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * (0. K)) is Element of the carrier of K
(1. (K,n)) * (J,b1) is Element of the carrier of K
V * ((1. (K,n)) * (J,b1)) is Element of the carrier of K
((K,M,n) * (J,b1)) + (V * ((1. (K,n)) * (J,b1))) is Element of the carrier of K
(V * (1. (K,n))) * (J,b1) is Element of the carrier of K
((K,M,n) * (J,b1)) + ((V * (1. (K,n))) * (J,b1)) is Element of the carrier of K
((K,M,n) + (V * (1. (K,n)))) * (J,b1) is Element of the carrier of K
J + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(K,(M + V),n) * (J,b1) is Element of the carrier of K
((K,M,n) + (V * (1. (K,n)))) * (J,b1) is Element of the carrier of K
(K,(M + V),n) * (J,b1) is Element of the carrier of K
((K,M,n) + (V * (1. (K,n)))) * (J,b1) is Element of the carrier of K
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
<*> (( the carrier of n *) *) is Relation-like non-empty empty-yielding NAT -defined ( the carrier of n *) * -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() 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 V201() FinSequence of ( the carrier of n *) *
dom K is finite Element of bool NAT
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K . 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 unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( 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 V201() FinSequence of ( the carrier of n *) *
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom K is finite Element of bool NAT
K . M is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
V is Element of the carrier of n
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,V,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of B,B, the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
<*> (( the carrier of n *) *) is Relation-like non-empty empty-yielding NAT -defined ( the carrier of n *) * -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() 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 V201() FinSequence of ( the carrier of n *) *
dom K is finite Element of bool NAT
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K . 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 unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is Element of the carrier of n
<*> {} is Relation-like non-empty empty-yielding NAT -defined {} -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of {}
M is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
dom M is finite Element of bool NAT
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . V), len (M . V), the carrier of n
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is Element of the carrier of n
<*> {} is Relation-like non-empty empty-yielding NAT -defined {} -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of {}
M is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
dom M is finite Element of bool NAT
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . V), len (M . V), the carrier of n
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
<*(K,M,n)*> is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of K *) *
[1,(K,M,n)] is set
{1,(K,M,n)} is non empty finite V36() set
{{1,(K,M,n)},{1}} is non empty finite V36() set
{[1,(K,M,n)]} is Relation-like non empty trivial finite 1 -element set
dom <*(K,M,n)*> is non empty trivial finite 1 -element Element of bool NAT
{1} is non empty trivial finite V36() 1 -element Element of bool NAT
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
<*(K,M,n)*> . 1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (<*(K,M,n)*> . 1), len (<*(K,M,n)*> . 1), the carrier of K
<*(K,M,n)*> . 1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (<*(K,M,n)*> . 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*(K,M,n)*> . V is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (<*(K,M,n)*> . V), len (<*(K,M,n)*> . V), the carrier of K
<*(K,M,n)*> . V is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (<*(K,M,n)*> . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) FinSequence of ( the carrier of K *) *
dom V is finite Element of bool NAT
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . 1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . 1), len (V . 1), the carrier of K
V . 1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (V . 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V . B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . B), len (V . B), the carrier of K
V . B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (V . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is Element of the carrier of n
(n,K,1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of 1,1, the carrier of n
<*(n,K,1)*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
[1,(n,K,1)] is set
{1,(n,K,1)} is non empty finite V36() set
{{1,(n,K,1)},{1}} is non empty finite V36() set
{[1,(n,K,1)]} is Relation-like non empty trivial finite 1 -element set
M is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
dom M is finite Element of bool NAT
{1} is non empty trivial finite V36() 1 -element Element of bool NAT
V is set
M . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . 1), len (M . 1), the carrier of n
M . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . V), len (M . V), the carrier of n
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
1_ n is Element of the carrier of n
1. n is V65(n) Element of the carrier of n
the OneF of n is Element of the carrier of n
the Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 1_ n) is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 1_ n)
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is Element of the carrier of n
M is Element of the carrier of n
K + M is Element of the carrier of n
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
Len V is Relation-like NAT -defined NAT -valued Function-like finite len V -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len V) -tuples_on NAT
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len V) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
1. (n,(Len V)) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
(len V) |-> M is Relation-like NAT -defined the carrier of n -valued Function-like finite len V -element FinSequence-like FinSubsequence-like Element of (len V) -tuples_on the carrier of n
(len V) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
mlt (((len V) |-> M),(1. (n,(Len V)))) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
V (+) (mlt (((len V) |-> M),(1. (n,(Len V))))) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
dom (V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) is finite Element of bool NAT
dom (mlt (((len V) |-> M),(1. (n,(Len V))))) is finite Element of bool NAT
dom (1. (n,(Len V))) is finite Element of bool NAT
dom V is finite Element of bool NAT
Seg (len V) is finite len V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len V ) } is set
dom (Len V) is finite len V -element Element of bool NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9), len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9), the carrier of n
(V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . M9), len (V . M9), the carrier of n
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,T) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of T,T, the carrier of n
(n,(K + M),T) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of T,T, the carrier of n
len (V . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) . M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
len ((len V) |-> M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom ((len V) |-> M) is finite len V -element Element of bool NAT
((len V) |-> M) /. M9 is Element of the carrier of n
((len V) |-> M) . M9 is set
(mlt (((len V) |-> M),(1. (n,(Len V))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((mlt (((len V) |-> M),(1. (n,(Len V))))) . M9), len ((mlt (((len V) |-> M),(1. (n,(Len V))))) . M9), the carrier of n
(mlt (((len V) |-> M),(1. (n,(Len V))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((mlt (((len V) |-> M),(1. (n,(Len V))))) . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(1. (n,(Len V))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((1. (n,(Len V))) . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(1. (n,(Len V))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((1. (n,(Len V))) . M9), len ((1. (n,(Len V))) . M9), the carrier of n
M * ((1. (n,(Len V))) . M9) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((1. (n,(Len V))) . M9), len ((1. (n,(Len V))) . M9), the carrier of n
1. (n,T) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of T,T, the carrier of n
M * (1. (n,T)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of T,T, the carrier of n
(n,K,T) + (M * (1. (n,T))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of T,T, the carrier of n
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9), len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9), the carrier of n
(V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((V (+) (mlt (((len V) |-> M),(1. (n,(Len V)))))) . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,(K + M),T) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of T,T, the carrier of n
M9 is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom M9 is finite Element of bool NAT
M9 . T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M9 . T), len (M9 . T), the carrier of n
M9 . T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M9 . T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( 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 V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence 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 V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
K ^ M is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K ^ M is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom (K ^ M) is finite Element of bool NAT
(K ^ M) . V is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom K is finite Element of bool NAT
(K ^ M) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((K ^ M) . V), len ((K ^ M) . V), the carrier of n
(K ^ M) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((K ^ M) . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (K . V), len (K . V), the carrier of n
K . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (K . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M is finite Element of bool NAT
len K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len K) + B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len K) + B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(K ^ M) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((K ^ M) . V), len ((K ^ M) . V), the carrier of n
(K ^ M) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((K ^ M) . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . B), len (M . B), the carrier of n
M . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom K is finite Element of bool NAT
dom M is finite Element of bool NAT
len K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
M | K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
M | (Seg K) is Relation-like Seg K -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M | K is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
dom (M | K) is finite Element of bool NAT
dom M is finite Element of bool NAT
(M | K) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((M | K) . V), len ((M | K) . V), the carrier of n
(M | K) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((M | K) . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . V), len (M . V), the carrier of n
M . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M /^ K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M /^ K is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
dom (M /^ K) is finite Element of bool NAT
V + K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom M is finite Element of bool NAT
K + V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M . (K + V) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . (K + V)), len (M . (K + V)), the carrier of n
M . (K + V) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . (K + V)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M /. (K + V) is FinSequence-like Element of ( the carrier of n *) *
(M /^ K) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((M /^ K) . V), len ((M /^ K) . V), the carrier of n
(M /^ K) . V is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((M /^ K) . V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(M /^ K) /. V is FinSequence-like Element of ( the carrier of n *) *
B is Element of the carrier of n
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,B,M9) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of M9,M9, the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K 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 V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
M ^ V is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n,M,V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom (n,M,V) is finite Element of bool NAT
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,M,V) . B), len ((n,M,V) . B), the carrier of n
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,M,V) . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M is finite Element of bool NAT
M . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (M . B), len (M . B), the carrier of n
M . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (M . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
len M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len M) + M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len M) + M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . M9), len (V . M9), the carrier of n
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M is finite Element of bool NAT
dom V is finite Element of bool NAT
len M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
K is Element of the carrier of n
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
V | M is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg M is finite M -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
V | (Seg M) is Relation-like Seg M -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,M,V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
dom (n,M,V) is finite Element of bool NAT
dom V is finite Element of bool NAT
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,M,V) . B), len ((n,M,V) . B), the carrier of n
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,M,V) . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . B), len (V . B), the carrier of n
V . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V /^ M is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,M,V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
dom (n,M,V) is finite Element of bool NAT
B + M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom V is finite Element of bool NAT
M + B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . (M + B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . (M + B)), len (V . (M + B)), the carrier of n
V . (M + B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . (M + B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V /. (M + B) is FinSequence-like Element of ( the carrier of n *) *
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,M,V) . B), len ((n,M,V) . B), the carrier of n
(n,M,V) . B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,M,V) . B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,M,V) /. B is FinSequence-like Element of ( the carrier of n *) *
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,M9) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of M9,M9, the carrier of n
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ V is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
B is set
dom (M |^ V) is non empty set
(M |^ V) . B is set
M9 is Element of the carrier of K
(M |^ V) . M9 is Element of the carrier of K
(dom (M |^ V)) --> (0. K) is Relation-like dom (M |^ V) -defined the carrier of K -valued Function-like constant non empty total V18( dom (M |^ V), the carrier of K) Element of bool [:(dom (M |^ V)), the carrier of K:]
[:(dom (M |^ V)), the carrier of K:] is Relation-like set
bool [:(dom (M |^ V)), the carrier of K:] is set
{(0. K)} is non empty trivial finite 1 -element set
[:(dom (M |^ V)),{(0. K)}:] is Relation-like set
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K,{(0. K)}:] is Relation-like set
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ V is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
B is Element of the carrier of K
(M |^ V) . B is Element of the carrier of K
( the carrier of K --> (0. K)) . B is Element of the carrier of K
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
M |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
n is non empty right_complementable unital V131() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
M |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
M is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() VectSpStr over K
the carrier of M is non empty set
[: the carrier of M, the carrier of M:] is Relation-like set
bool [: the carrier of M, the carrier of M:] is set
V is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
V |^ n is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
ker (V |^ n) is non empty right_complementable strict V145(K) V146(K) V147(K) V148(K) V152() V153() V154() Subspace of M
V | (ker (V |^ n)) is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of M -valued Function-like non empty total V18( the carrier of (ker (V |^ n)), the carrier of M) additive homogeneous Element of bool [: the carrier of (ker (V |^ n)), the carrier of M:]
the carrier of (ker (V |^ n)) is non empty set
[: the carrier of (ker (V |^ n)), the carrier of M:] is Relation-like set
bool [: the carrier of (ker (V |^ n)), the carrier of M:] is set
V | the carrier of (ker (V |^ n)) is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of M -defined the carrier of M -valued set
[: the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n)):] is Relation-like set
bool [: the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n)):] is set
T is Element of the carrier of (ker (V |^ n))
J is Element of the carrier of M
M9 is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of (ker (V |^ n)) -valued Function-like non empty total V18( the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n))) additive homogeneous Element of bool [: the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n)):]
M9 |^ n is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of (ker (V |^ n)) -valued Function-like non empty total V18( the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n))) additive homogeneous Element of bool [: the carrier of (ker (V |^ n)), the carrier of (ker (V |^ n)):]
(M9 |^ n) . T is Element of the carrier of (ker (V |^ n))
(V |^ n) | (ker (V |^ n)) is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of M -valued Function-like non empty total V18( the carrier of (ker (V |^ n)), the carrier of M) additive homogeneous Element of bool [: the carrier of (ker (V |^ n)), the carrier of M:]
(V |^ n) | the carrier of (ker (V |^ n)) is Relation-like the carrier of (ker (V |^ n)) -defined the carrier of M -defined the carrier of M -valued set
((V |^ n) | (ker (V |^ n))) . T is Element of the carrier of M
(V |^ n) . J is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
0. (ker (V |^ n)) is V65( ker (V |^ n)) Element of the carrier of (ker (V |^ n))
the ZeroF of (ker (V |^ n)) is Element of the carrier of (ker (V |^ n))
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) (n,K) Element of bool [: the carrier of K, the carrier of K:]
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ V is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ B is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ V is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ B is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
[#] K is non empty non proper Element of bool the carrier of K
bool the carrier of K is set
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{(0. K)} is non empty trivial finite 1 -element Element of bool the carrier of K
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) (n,K) Element of bool [: the carrier of K, the carrier of K:]
(n,K,M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V is set
id K is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive Element of bool [: 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 non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
M |^ {} is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
( the carrier of K --> (0. K)) . V is set
V is set
M |^ {} is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
dom (M |^ {}) is non empty set
(M |^ {}) . V is set
id K is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive Element of bool [: 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 non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
B is Element of the carrier of K
(id K) . B is Element of the carrier of K
(dom (M |^ {})) --> (0. K) is Relation-like dom (M |^ {}) -defined the carrier of K -valued Function-like constant non empty total V18( dom (M |^ {}), the carrier of K) Element of bool [:(dom (M |^ {})), the carrier of K:]
[:(dom (M |^ {})), the carrier of K:] is Relation-like set
bool [:(dom (M |^ {})), the carrier of K:] is set
{(0. K)} is non empty trivial finite 1 -element set
[:(dom (M |^ {})),{(0. K)}:] is Relation-like set
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K,{(0. K)}:] is Relation-like set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
n is doubleLoopStr
K is non empty VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) (n,K) Element of bool [: the carrier of K, the carrier of K:]
(n,K,M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[#] K is non empty non proper Element of bool the carrier of K
bool the carrier of K is set
{(0. K)} is non empty trivial finite 1 -element Element of bool the carrier of K
B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ B is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ B) . (0. K) is Element of the carrier of K
B is set
M9 is Element of the carrier of K
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ T is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ T) . M9 is Element of the carrier of K
(n,K,M) - T is V46() ext-real V165() set
T - T is V46() ext-real V165() set
id K is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive Element of bool [: 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 non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(id K) . M9 is Element of the carrier of K
(n,K,M) - {} is V46() ext-real non negative V165() set
dom (M |^ T) is non empty set
(n,K,M) -' T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((n,K,M) -' T) + T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M |^ (n,K,M) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
(M |^ (n,K,M)) . M9 is Element of the carrier of K
M |^ ((n,K,M) -' T) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ ((n,K,M) -' T)) * (M |^ T) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
((M |^ ((n,K,M) -' T)) * (M |^ T)) . M9 is Element of the carrier of K
(M |^ ((n,K,M) -' T)) . (0. K) is Element of the carrier of K
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ J is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ J) . (0. K) is Element of the carrier of K
(n,K,M) -' J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,K,M) - J is V46() ext-real V165() set
b1 is set
M |^ ((n,K,M) -' J) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
dom (M |^ ((n,K,M) -' J)) is non empty set
P is Element of the carrier of K
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M |^ c₁₁ is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ c₁₁) . P is Element of the carrier of K
{} * J is Relation-like non-empty empty-yielding RAT -valued functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() set
c₁₁ + ({} * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + (KER * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + (KER * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + (KER * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
KER - 1 is V46() ext-real V165() set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FK * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + (FK * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(M |^ ((n,K,M) -' J)) . P is Element of the carrier of K
1 * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ + (1 * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,K,M) - IM is V46() ext-real V165() set
M |^ IM is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ IM) . P is Element of the carrier of K
dom (M |^ IM) is non empty set
IM - IM is V46() ext-real V165() set
(n,K,M) -' IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((n,K,M) -' IM) + IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(M |^ (n,K,M)) . P is Element of the carrier of K
M |^ ((n,K,M) -' IM) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(M |^ ((n,K,M) -' IM)) * (M |^ IM) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
((M |^ ((n,K,M) -' IM)) * (M |^ IM)) . P is Element of the carrier of K
(M |^ ((n,K,M) -' IM)) . (0. K) is Element of the carrier of K
IM + J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM + ((n,K,M) - IM) is V46() ext-real V165() set
KER + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(KER + 1) * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ + ((KER + 1) * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
IM + {} is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(M |^ ((n,K,M) -' J)) . P is Element of the carrier of K
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FK * J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + (FK * J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(M |^ ((n,K,M) -' J)) . P is Element of the carrier of K
(M |^ ((n,K,M) -' J)) . P is Element of the carrier of K
(M |^ ((n,K,M) -' J)) . b1 is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() Subspace of K
the carrier of M is non empty set
[: the carrier of M, the carrier of M:] is Relation-like set
bool [: the carrier of M, the carrier of M:] is set
V is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) (n,K) Element of bool [: the carrier of K, the carrier of K:]
V | M is Relation-like the carrier of M -defined the carrier of K -valued Function-like non empty total V18( the carrier of M, the carrier of K) Element of bool [: the carrier of M, the carrier of K:]
[: the carrier of M, the carrier of K:] is Relation-like set
bool [: the carrier of M, the carrier of K:] is set
V | the carrier of M is Relation-like the carrier of M -defined the carrier of K -defined the carrier of K -valued set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V |^ M9 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
[#] M is non empty non proper Element of bool the carrier of M
bool the carrier of M is set
[#] K is non empty non proper Element of bool the carrier of K
bool the carrier of K is set
([#] K) /\ ([#] M) is Element of bool the carrier of M
B is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
B |^ M9 is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
(ZeroMap (K,K)) | M is Relation-like the carrier of M -defined the carrier of K -valued Function-like non empty total V18( the carrier of M, the carrier of K) additive homogeneous Element of bool [: the carrier of M, the carrier of K:]
(ZeroMap (K,K)) | the carrier of M is Relation-like the carrier of M -defined the carrier of K -defined the carrier of K -valued set
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
( the carrier of K --> (0. K)) | ([#] M) is Relation-like the carrier of K -defined [#] M -defined the carrier of K -defined the carrier of K -valued Function-like constant Element of bool [: the carrier of K, the carrier of K:]
the carrier of K /\ ([#] M) is Element of bool the carrier of M
( the carrier of K /\ ([#] M)) --> (0. K) is Relation-like the carrier of K /\ ([#] M) -defined the carrier of K -valued Function-like constant total V18( the carrier of K /\ ([#] M), the carrier of K) Element of bool [:( the carrier of K /\ ([#] M)), the carrier of K:]
[:( the carrier of K /\ ([#] M)), the carrier of K:] is Relation-like set
bool [:( the carrier of K /\ ([#] M)), the carrier of K:] is set
[:( the carrier of K /\ ([#] M)),{(0. K)}:] is Relation-like set
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
the carrier of M --> (0. M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like constant non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
{(0. M)} is non empty trivial finite 1 -element set
[: the carrier of M,{(0. M)}:] is Relation-like set
ZeroMap (M,M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
M is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() VectSpStr over K
the carrier of M is non empty set
[: the carrier of M, the carrier of M:] is Relation-like set
bool [: the carrier of M, the carrier of M:] is set
B is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous (K,M) Element of bool [: the carrier of M, the carrier of M:]
B |^ n is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
im (B |^ n) is non empty right_complementable strict V145(K) V146(K) V147(K) V148(K) V152() V153() V154() Subspace of M
the carrier of (im (B |^ n)) is non empty set
[: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):] is Relation-like set
bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):] is set
B | (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of M) additive homogeneous Element of bool [: the carrier of (im (B |^ n)), the carrier of M:]
[: the carrier of (im (B |^ n)), the carrier of M:] is Relation-like set
bool [: the carrier of (im (B |^ n)), the carrier of M:] is set
B | the carrier of (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -defined the carrier of M -valued set
(K,M,B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
T is Relation-like the carrier of (im (B |^ n)) -defined the carrier of (im (B |^ n)) -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of (im (B |^ n))) (K, im (B |^ n)) Element of bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):]
(K,(im (B |^ n)),T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(K,(im (B |^ n)),T) + n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(K,M,B) - n is V46() ext-real V165() set
P is set
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T |^ b1 is Relation-like the carrier of (im (B |^ n)) -defined the carrier of (im (B |^ n)) -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of (im (B |^ n))) Element of bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):]
dom (T |^ b1) is non empty set
c₁₁ is Element of the carrier of (im (B |^ n))
dom (B |^ n) is non empty set
IM is Element of the carrier of M
KER is Element of the carrier of M
(B |^ n) . KER is Element of the carrier of M
B |^ (K,M,B) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(B |^ (K,M,B)) . KER is Element of the carrier of M
ZeroMap (M,M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(ZeroMap (M,M)) . KER is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
the carrier of M --> (0. M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like constant non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
{(0. M)} is non empty trivial finite 1 -element set
[: the carrier of M,{(0. M)}:] is Relation-like set
( the carrier of M --> (0. M)) . KER is Element of the carrier of M
0. (im (B |^ n)) is V65( im (B |^ n)) Element of the carrier of (im (B |^ n))
the ZeroF of (im (B |^ n)) is Element of the carrier of (im (B |^ n))
b1 + n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B |^ (b1 + n) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(B |^ (b1 + n)) . KER is Element of the carrier of M
B |^ b1 is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(B |^ b1) * (B |^ n) is Relation-like the carrier of M -defined the carrier of M -defined the carrier of M -valued the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
((B |^ b1) * (B |^ n)) . KER is Element of the carrier of M
(B |^ b1) . IM is Element of the carrier of M
(B |^ b1) | (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of M) additive homogeneous Element of bool [: the carrier of (im (B |^ n)), the carrier of M:]
(B |^ b1) | the carrier of (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -defined the carrier of M -valued set
((B |^ b1) | (im (B |^ n))) . c₁₁ is Element of the carrier of M
(T |^ b1) . P is set
[#] (im (B |^ n)) is non empty non proper Element of bool the carrier of (im (B |^ n))
bool the carrier of (im (B |^ n)) is set
the carrier of (im (B |^ n)) --> (0. (im (B |^ n))) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of (im (B |^ n)) -valued Function-like constant non empty total V18( the carrier of (im (B |^ n)), the carrier of (im (B |^ n))) Element of bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):]
{(0. (im (B |^ n)))} is non empty trivial finite 1 -element set
[: the carrier of (im (B |^ n)),{(0. (im (B |^ n)))}:] is Relation-like set
ZeroMap ((im (B |^ n)),(im (B |^ n))) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of (im (B |^ n)) -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of (im (B |^ n))) additive homogeneous Element of bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):]
c₁₁ is Element of the carrier of M
(B |^ n) . c₁₁ is Element of the carrier of M
T |^ (K,(im (B |^ n)),T) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of (im (B |^ n)) -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of (im (B |^ n))) Element of bool [: the carrier of (im (B |^ n)), the carrier of (im (B |^ n)):]
(T |^ (K,(im (B |^ n)),T)) . ((B |^ n) . c₁₁) is set
B |^ (K,(im (B |^ n)),T) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(B |^ (K,(im (B |^ n)),T)) | (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -valued Function-like non empty total V18( the carrier of (im (B |^ n)), the carrier of M) additive homogeneous Element of bool [: the carrier of (im (B |^ n)), the carrier of M:]
(B |^ (K,(im (B |^ n)),T)) | the carrier of (im (B |^ n)) is Relation-like the carrier of (im (B |^ n)) -defined the carrier of M -defined the carrier of M -valued set
((B |^ (K,(im (B |^ n)),T)) | (im (B |^ n))) . ((B |^ n) . c₁₁) is set
(B |^ (K,(im (B |^ n)),T)) . ((B |^ n) . c₁₁) is Element of the carrier of M
(B |^ (K,(im (B |^ n)),T)) * (B |^ n) is Relation-like the carrier of M -defined the carrier of M -defined the carrier of M -valued the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
((B |^ (K,(im (B |^ n)),T)) * (B |^ n)) . c₁₁ is Element of the carrier of M
B |^ ((K,(im (B |^ n)),T) + n) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(B |^ ((K,(im (B |^ n)),T) + n)) . c₁₁ is Element of the carrier of M
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of M is non empty set
V is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of V is non empty set
B is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional Subspace of M
the carrier of B is non empty set
M9 is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional Subspace of M
the carrier of M9 is non empty set
T is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional Subspace of V
the carrier of T is non empty set
J is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional Subspace of V
the carrier of J is non empty set
b1 is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M
len b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 /. n is Element of the carrier of M
P is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
len P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B
len c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom c₁₁ is finite Element of bool NAT
c₁₁ /. n is Element of the carrier of B
n + (len c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 /. (n + (len c₁₁)) is Element of the carrier of M
IM is Relation-like NAT -defined the carrier of M9 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M9
len IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ ^ IM is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom IM is finite Element of bool NAT
IM /. n is Element of the carrier of M9
KER is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
len KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FK is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
len FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER ^ FK is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len P, the carrier of K
Mx2Tran (Jk,b1,P) is Relation-like the carrier of M -defined the carrier of V -valued Function-like non empty total V18( the carrier of M, the carrier of V) additive homogeneous Element of bool [: the carrier of M, the carrier of V:]
[: the carrier of M, the carrier of V:] is Relation-like set
bool [: the carrier of M, the carrier of V:] is set
(Mx2Tran (Jk,b1,P)) . (b1 /. n) is Element of the carrier of V
(Mx2Tran (Jk,b1,P)) . (b1 /. (n + (len c₁₁))) is Element of the carrier of V
Bker is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len KER, the carrier of K
width Bker is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Mx2Tran (Bker,c₁₁,KER) is Relation-like the carrier of B -defined the carrier of T -valued Function-like non empty total V18( the carrier of B, the carrier of T) additive homogeneous Element of bool [: the carrier of B, the carrier of T:]
[: the carrier of B, the carrier of T:] is Relation-like set
bool [: the carrier of B, the carrier of T:] is set
(Mx2Tran (Bker,c₁₁,KER)) . (c₁₁ /. n) is Element of the carrier of T
FI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len IM, len FK, the carrier of K
<*Bker,FI*> is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding FinSequence of ( the carrier of K *) *
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
<*Bker*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,Bker] is set
{1,Bker} is non empty finite V36() set
{{1,Bker},{1}} is non empty finite V36() set
{[1,Bker]} is Relation-like non empty trivial finite 1 -element set
<*FI*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,FI] is set
{1,FI} is non empty finite V36() set
{{1,FI},{1}} is non empty finite V36() set
{[1,FI]} is Relation-like non empty trivial finite 1 -element set
<*Bker*> ^ <*FI*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
block_diagonal (<*Bker,FI*>,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len <*Bker,FI*>), Sum (Width <*Bker,FI*>), the carrier of K
Len <*Bker,FI*> is Relation-like NAT -defined NAT -valued Function-like finite len <*Bker,FI*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*Bker,FI*>) -tuples_on NAT
len <*Bker,FI*> is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len <*Bker,FI*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*Bker,FI*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width <*Bker,FI*> is Relation-like NAT -defined NAT -valued Function-like finite len <*Bker,FI*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*Bker,FI*>) -tuples_on NAT
Sum (Width <*Bker,FI*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width FI is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Mx2Tran (FI,IM,FK) is Relation-like the carrier of M9 -defined the carrier of J -valued Function-like non empty total V18( the carrier of M9, the carrier of J) additive homogeneous Element of bool [: the carrier of M9, the carrier of J:]
[: the carrier of M9, the carrier of J:] is Relation-like set
bool [: the carrier of M9, the carrier of J:] is set
(Mx2Tran (FI,IM,FK)) . (IM /. n) is Element of the carrier of J
dom b1 is finite Element of bool NAT
rng FK is finite set
rng KER is finite set
Seg (len b1) is finite len b1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len b1 ) } is set
1. (K,(len b1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len b1, the carrier of K
len (1. (K,(len b1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
1. (K,(len c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
dom (1. (K,(len c₁₁))) is finite Element of bool NAT
len (1. (K,(len c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len (1. (K,(len c₁₁)))) is finite len (1. (K,(len c₁₁))) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (1. (K,(len c₁₁))) ) } is set
dom (1. (K,(len b1))) is finite Element of bool NAT
Seg (len (1. (K,(len b1)))) is finite len (1. (K,(len b1))) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (1. (K,(len b1))) ) } is set
(len c₁₁) + n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len IM) is finite len IM -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len IM ) } is set
1. (K,(len IM)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len IM, len IM, the carrier of K
len (1. (K,(len IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width (1. (K,(len b1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line (FI,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width FI -element FinSequence-like FinSubsequence-like Element of (width FI) -tuples_on the carrier of K
(width FI) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Line (FI,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width Bker) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Bker -element FinSequence-like FinSubsequence-like Element of (width Bker) -tuples_on the carrier of K
(width Bker) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len ((width Bker) |-> (0. K)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len c₁₁) is finite len c₁₁ -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len c₁₁ ) } is set
len Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len Bker is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom Bker is finite Element of bool NAT
Line (Bker,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Bker -element FinSequence-like FinSubsequence-like Element of (width Bker) -tuples_on the carrier of K
len (Line (Bker,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width FI) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width FI -element FinSequence-like FinSubsequence-like Element of (width FI) -tuples_on the carrier of K
len ((width FI) |-> (0. K)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line (Jk,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Jk -element FinSequence-like FinSubsequence-like Element of (width Jk) -tuples_on the carrier of K
width Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width Jk) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line (Bker,n)) ^ ((width FI) |-> (0. K)) is Relation-like NAT -defined the carrier of K -valued Function-like finite (width Bker) + (width FI) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(width Bker) + (width FI) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(b1 /. n) |-- b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((b1 /. n) |-- b1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((b1 /. n) |-- b1), the carrier of K
len ((b1 /. n) |-- b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((b1 /. n) |-- b1)) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx ((b1 /. n) |-- b1)) * Jk),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((b1 /. n) |-- b1)) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((b1 /. n) |-- b1)) * Jk)) -tuples_on the carrier of K
width ((LineVec2Mx ((b1 /. n) |-- b1)) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((b1 /. n) |-- b1)) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((b1 /. n) |-- b1)) * Jk),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (((LineVec2Mx ((b1 /. n) |-- b1)) * Jk),1)),P)) is Element of the carrier of V
the addF of V is Relation-like [: the carrier of V, the carrier of V:] -defined the carrier of V -valued Function-like total V18([: the carrier of V, the carrier of V:], the carrier of V) Element of bool [:[: the carrier of V, the carrier of V:], the carrier of V:]
[: the carrier of V, the carrier of V:] is Relation-like set
[:[: the carrier of V, the carrier of V:], the carrier of V:] is Relation-like set
bool [:[: the carrier of V, the carrier of V:], the carrier of V:] is set
the addF of V $$ (lmlt ((Line (((LineVec2Mx ((b1 /. n) |-- b1)) * Jk),1)),P)) is Element of the carrier of V
Line ((1. (K,(len b1))),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len b1))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len b1)))) -tuples_on the carrier of K
(width (1. (K,(len b1)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len b1))),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (K,(len b1))),n)), the carrier of K
len (Line ((1. (K,(len b1))),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk)) -tuples_on the carrier of K
width ((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),n))) * Jk),1)),P)) is Element of the carrier of V
(1. (K,(len b1))) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((1. (K,(len b1))) * Jk),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len b1))) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len b1))) * Jk)) -tuples_on the carrier of K
width ((1. (K,(len b1))) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (K,(len b1))) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (K,(len b1))) * Jk),n)), the carrier of K
len (Line (((1. (K,(len b1))) * Jk),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),n))),1)),P)) is Element of the carrier of V
LineVec2Mx (Line (Jk,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (Jk,n)), the carrier of K
len (Line (Jk,n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (Jk,n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (Jk,n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (Jk,n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (Jk,n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (Jk,n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (Jk,n))),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((LineVec2Mx (Line (Jk,n))),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((LineVec2Mx (Line (Jk,n))),1)),P)) is Element of the carrier of V
lmlt ((Line (Jk,n)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (Jk,n)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (Jk,n)),P)) is Element of the carrier of V
Ji is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
lmlt ((Line (Bker,n)),Ji) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Bim is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
lmlt (((width FI) |-> (0. K)),Bim) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
(lmlt ((Line (Bker,n)),Ji)) ^ (lmlt (((width FI) |-> (0. K)),Bim)) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum ((lmlt ((Line (Bker,n)),Ji)) ^ (lmlt (((width FI) |-> (0. K)),Bim))) is Element of the carrier of V
the addF of V $$ ((lmlt ((Line (Bker,n)),Ji)) ^ (lmlt (((width FI) |-> (0. K)),Bim))) is Element of the carrier of V
Sum (lmlt ((Line (Bker,n)),Ji)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (Bker,n)),Ji)) is Element of the carrier of V
Sum (lmlt (((width FI) |-> (0. K)),Bim)) is Element of the carrier of V
the addF of V $$ (lmlt (((width FI) |-> (0. K)),Bim)) is Element of the carrier of V
(Sum (lmlt ((Line (Bker,n)),Ji))) + (Sum (lmlt (((width FI) |-> (0. K)),Bim))) is Element of the carrier of V
Sum Bim is Element of the carrier of V
the addF of V $$ Bim is Element of the carrier of V
(0. K) * (Sum Bim) is Element of the carrier of V
(Sum (lmlt ((Line (Bker,n)),Ji))) + ((0. K) * (Sum Bim)) is Element of the carrier of V
0. V is V65(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
(Sum (lmlt ((Line (Bker,n)),Ji))) + (0. V) is Element of the carrier of V
lmlt ((Line (Bker,n)),KER) is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
Sum (lmlt ((Line (Bker,n)),KER)) is Element of the carrier of T
the addF of T is Relation-like [: the carrier of T, the carrier of T:] -defined the carrier of T -valued Function-like total V18([: the carrier of T, the carrier of T:], the carrier of T) Element of bool [:[: the carrier of T, the carrier of T:], the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like set
[:[: the carrier of T, the carrier of T:], the carrier of T:] is Relation-like set
bool [:[: the carrier of T, the carrier of T:], the carrier of T:] is set
the addF of T $$ (lmlt ((Line (Bker,n)),KER)) is Element of the carrier of T
LineVec2Mx (Line (Bker,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (Bker,n)), the carrier of K
Line ((LineVec2Mx (Line (Bker,n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (Bker,n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (Bker,n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (Bker,n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (Bker,n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (Bker,n))),1)),KER) is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
Sum (lmlt ((Line ((LineVec2Mx (Line (Bker,n))),1)),KER)) is Element of the carrier of T
the addF of T $$ (lmlt ((Line ((LineVec2Mx (Line (Bker,n))),1)),KER)) is Element of the carrier of T
(1. (K,(len c₁₁))) * Bker is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((1. (K,(len c₁₁))) * Bker),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len c₁₁))) * Bker) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len c₁₁))) * Bker)) -tuples_on the carrier of K
width ((1. (K,(len c₁₁))) * Bker) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (K,(len c₁₁))) * Bker)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (K,(len c₁₁))) * Bker),n)), the carrier of K
len (Line (((1. (K,(len c₁₁))) * Bker),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))),1)),KER) is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))),1)),KER)) is Element of the carrier of T
the addF of T $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len c₁₁))) * Bker),n))),1)),KER)) is Element of the carrier of T
Line ((1. (K,(len c₁₁))),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len c₁₁))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len c₁₁)))) -tuples_on the carrier of K
width (1. (K,(len c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (1. (K,(len c₁₁)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len c₁₁))),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (K,(len c₁₁))),n)), the carrier of K
len (Line ((1. (K,(len c₁₁))),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker)) -tuples_on the carrier of K
width ((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker),1)),KER) is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker),1)),KER)) is Element of the carrier of T
the addF of T $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len c₁₁))),n))) * Bker),1)),KER)) is Element of the carrier of T
(c₁₁ /. n) |-- c₁₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((c₁₁ /. n) |-- c₁₁) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((c₁₁ /. n) |-- c₁₁), the carrier of K
len ((c₁₁ /. n) |-- c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker)) -tuples_on the carrier of K
width ((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker),1)),KER) is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of T
Sum (lmlt ((Line (((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker),1)),KER)) is Element of the carrier of T
the addF of T $$ (lmlt ((Line (((LineVec2Mx ((c₁₁ /. n) |-- c₁₁)) * Bker),1)),KER)) is Element of the carrier of T
dom (1. (K,(len IM))) is finite Element of bool NAT
Seg (len (1. (K,(len IM)))) is finite len (1. (K,(len IM))) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (1. (K,(len IM))) ) } is set
len FI is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom FI is finite Element of bool NAT
Line (Jk,((len c₁₁) + n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Jk -element FinSequence-like FinSubsequence-like Element of (width Jk) -tuples_on the carrier of K
width Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width Jk) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
((width Bker) |-> (0. K)) ^ (Line (FI,n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite (width Bker) + (width FI) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(width Bker) + (width FI) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 /. ((len c₁₁) + n) is Element of the carrier of M
(b1 /. ((len c₁₁) + n)) |-- b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((b1 /. ((len c₁₁) + n)) |-- b1), the carrier of K
len ((b1 /. ((len c₁₁) + n)) |-- b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk)) -tuples_on the carrier of K
width ((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk),1)),P)) is Element of the carrier of V
the addF of V is Relation-like [: the carrier of V, the carrier of V:] -defined the carrier of V -valued Function-like total V18([: the carrier of V, the carrier of V:], the carrier of V) Element of bool [:[: the carrier of V, the carrier of V:], the carrier of V:]
[: the carrier of V, the carrier of V:] is Relation-like set
[:[: the carrier of V, the carrier of V:], the carrier of V:] is Relation-like set
bool [:[: the carrier of V, the carrier of V:], the carrier of V:] is set
the addF of V $$ (lmlt ((Line (((LineVec2Mx ((b1 /. ((len c₁₁) + n)) |-- b1)) * Jk),1)),P)) is Element of the carrier of V
Line ((1. (K,(len b1))),((len c₁₁) + n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len b1))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len b1)))) -tuples_on the carrier of K
(width (1. (K,(len b1)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (K,(len b1))),((len c₁₁) + n))), the carrier of K
len (Line ((1. (K,(len b1))),((len c₁₁) + n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk)) -tuples_on the carrier of K
width ((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),((len c₁₁) + n)))) * Jk),1)),P)) is Element of the carrier of V
(1. (K,(len b1))) * Jk is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len b1))) * Jk) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len b1))) * Jk)) -tuples_on the carrier of K
width ((1. (K,(len b1))) * Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (K,(len b1))) * Jk)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n))), the carrier of K
len (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n))))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n))))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * Jk),((len c₁₁) + n)))),1)),P)) is Element of the carrier of V
LineVec2Mx (Line (Jk,((len c₁₁) + n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (Jk,((len c₁₁) + n))), the carrier of K
len (Line (Jk,((len c₁₁) + n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (Jk,((len c₁₁) + n)))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (Jk,((len c₁₁) + n)))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (Jk,((len c₁₁) + n))))) -tuples_on the carrier of K
width (LineVec2Mx (Line (Jk,((len c₁₁) + n)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (Jk,((len c₁₁) + n))))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (Jk,((len c₁₁) + n)))),1)),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((LineVec2Mx (Line (Jk,((len c₁₁) + n)))),1)),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((LineVec2Mx (Line (Jk,((len c₁₁) + n)))),1)),P)) is Element of the carrier of V
lmlt ((Line (Jk,((len c₁₁) + n))),P) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line (Jk,((len c₁₁) + n))),P)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (Jk,((len c₁₁) + n))),P)) is Element of the carrier of V
Ji is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
lmlt (((width Bker) |-> (0. K)),Ji) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Bim is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
lmlt ((Line (FI,n)),Bim) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
(lmlt (((width Bker) |-> (0. K)),Ji)) ^ (lmlt ((Line (FI,n)),Bim)) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum ((lmlt (((width Bker) |-> (0. K)),Ji)) ^ (lmlt ((Line (FI,n)),Bim))) is Element of the carrier of V
the addF of V $$ ((lmlt (((width Bker) |-> (0. K)),Ji)) ^ (lmlt ((Line (FI,n)),Bim))) is Element of the carrier of V
Sum (lmlt (((width Bker) |-> (0. K)),Ji)) is Element of the carrier of V
the addF of V $$ (lmlt (((width Bker) |-> (0. K)),Ji)) is Element of the carrier of V
Sum (lmlt ((Line (FI,n)),Bim)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line (FI,n)),Bim)) is Element of the carrier of V
(Sum (lmlt (((width Bker) |-> (0. K)),Ji))) + (Sum (lmlt ((Line (FI,n)),Bim))) is Element of the carrier of V
Sum Ji is Element of the carrier of V
the addF of V $$ Ji is Element of the carrier of V
(0. K) * (Sum Ji) is Element of the carrier of V
((0. K) * (Sum Ji)) + (Sum (lmlt ((Line (FI,n)),Bim))) is Element of the carrier of V
0. V is V65(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
(0. V) + (Sum (lmlt ((Line (FI,n)),Bim))) is Element of the carrier of V
lmlt ((Line (FI,n)),FK) is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
Sum (lmlt ((Line (FI,n)),FK)) is Element of the carrier of J
the addF of J is Relation-like [: the carrier of J, the carrier of J:] -defined the carrier of J -valued Function-like total V18([: the carrier of J, the carrier of J:], the carrier of J) Element of bool [:[: the carrier of J, the carrier of J:], the carrier of J:]
[: the carrier of J, the carrier of J:] is Relation-like set
[:[: the carrier of J, the carrier of J:], the carrier of J:] is Relation-like set
bool [:[: the carrier of J, the carrier of J:], the carrier of J:] is set
the addF of J $$ (lmlt ((Line (FI,n)),FK)) is Element of the carrier of J
LineVec2Mx (Line (FI,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (FI,n)), the carrier of K
Line ((LineVec2Mx (Line (FI,n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (FI,n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (FI,n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (FI,n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (FI,n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (FI,n))),1)),FK) is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
Sum (lmlt ((Line ((LineVec2Mx (Line (FI,n))),1)),FK)) is Element of the carrier of J
the addF of J $$ (lmlt ((Line ((LineVec2Mx (Line (FI,n))),1)),FK)) is Element of the carrier of J
(1. (K,(len IM))) * FI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((1. (K,(len IM))) * FI),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len IM))) * FI) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len IM))) * FI)) -tuples_on the carrier of K
width ((1. (K,(len IM))) * FI) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (K,(len IM))) * FI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len IM))) * FI),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (K,(len IM))) * FI),n)), the carrier of K
len (Line (((1. (K,(len IM))) * FI),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (K,(len IM))) * FI),n)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (K,(len IM))) * FI),n)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))),1)),FK) is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))),1)),FK)) is Element of the carrier of J
the addF of J $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len IM))) * FI),n))),1)),FK)) is Element of the carrier of J
Line ((1. (K,(len IM))),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len IM))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len IM)))) -tuples_on the carrier of K
width (1. (K,(len IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (1. (K,(len IM)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len IM))),n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (K,(len IM))),n)), the carrier of K
len (Line ((1. (K,(len IM))),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI)) -tuples_on the carrier of K
width ((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI),1)),FK) is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI),1)),FK)) is Element of the carrier of J
the addF of J $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len IM))),n))) * FI),1)),FK)) is Element of the carrier of J
(IM /. n) |-- IM is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((IM /. n) |-- IM) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((IM /. n) |-- IM), the carrier of K
len ((IM /. n) |-- IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((IM /. n) |-- IM)) * FI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx ((IM /. n) |-- IM)) * FI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((IM /. n) |-- IM)) * FI) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((IM /. n) |-- IM)) * FI)) -tuples_on the carrier of K
width ((LineVec2Mx ((IM /. n) |-- IM)) * FI) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((IM /. n) |-- IM)) * FI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((IM /. n) |-- IM)) * FI),1)),FK) is Relation-like NAT -defined the carrier of J -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of J
Sum (lmlt ((Line (((LineVec2Mx ((IM /. n) |-- IM)) * FI),1)),FK)) is Element of the carrier of J
the addF of J $$ (lmlt ((Line (((LineVec2Mx ((IM /. n) |-- IM)) * FI),1)),FK)) is Element of the carrier of J
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
V is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
1. (n,(len V)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len V, the carrier of n
T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len B, the carrier of n
Mx2Tran (T,V,B) is Relation-like the carrier of K -defined the carrier of M -valued Function-like non empty total V18( the carrier of K, the carrier of M) additive homogeneous Element of bool [: the carrier of K, the carrier of M:]
[: the carrier of K, the carrier of M:] is Relation-like set
bool [: the carrier of K, the carrier of M:] is set
J is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding FinSequence of ( the carrier of n *) *
block_diagonal (J,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len J), Sum (Width J), the carrier of n
Len J is Relation-like NAT -defined NAT -valued Function-like finite len J -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len J) -tuples_on NAT
len J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len J) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width J is Relation-like NAT -defined NAT -valued Function-like finite len J -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len J) -tuples_on NAT
Sum (Width J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (1. (n,(len V))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (1. (n,(len V))) is finite Element of bool NAT
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
min ((Len J),c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V /. c₁₁ is Element of the carrier of K
(Mx2Tran (T,V,B)) . (V /. c₁₁) is Element of the carrier of M
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
J . IM is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
IM -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J | (IM -' 1) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding FinSequence of ( the carrier of n *) *
Seg (IM -' 1) is finite IM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= IM -' 1 ) } is set
J | (Seg (IM -' 1)) is Relation-like Seg (IM -' 1) -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (J | (IM -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite len (J | (IM -' 1)) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (J | (IM -' 1))) -tuples_on NAT
len (J | (IM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (J | (IM -' 1))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (J | (IM -' 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ -' (Sum (Len (J | (IM -' 1)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (J . IM) -element FinSequence-like FinSubsequence-like Element of (width (J . IM)) -tuples_on the carrier of n
width (J . IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (J . IM)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
Width (J | (IM -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite len (J | (IM -' 1)) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (J | (IM -' 1))) -tuples_on NAT
Sum (Width (J | (IM -' 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J | IM is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding FinSequence of ( the carrier of n *) *
Seg IM is finite IM -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= IM ) } is set
J | (Seg IM) is Relation-like Seg IM -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Width (J | IM) is Relation-like NAT -defined NAT -valued Function-like finite len (J | IM) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (J | IM)) -tuples_on NAT
len (J | IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (J | IM)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Width (J | IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B | (Sum (Width (J | IM))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Seg (Sum (Width (J | IM))) is finite Sum (Width (J | IM)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Width (J | IM)) ) } is set
B | (Seg (Sum (Width (J | IM)))) is Relation-like Seg (Sum (Width (J | IM))) -defined NAT -defined the carrier of M -valued finite FinSubsequence-like set
(B | (Sum (Width (J | IM)))) /^ (Sum (Width (J | (IM -' 1)))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum (Width (J | IM)))) /^ (Sum (Width (J | (IM -' 1)))))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum (Width (J | IM)))) /^ (Sum (Width (J | (IM -' 1))))))) 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
the addF of M $$ (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum (Width (J | IM)))) /^ (Sum (Width (J | (IM -' 1))))))) is Element of the carrier of M
len ((B | (Sum (Width (J | IM)))) /^ (Sum (Width (J | (IM -' 1))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Width J) | (IM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(Width J) | (Seg (IM -' 1)) is Relation-like Seg (IM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Width J) | (IM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Width J) | (IM -' 1))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite Sum ((Width J) | (IM -' 1)) -element FinSequence-like FinSubsequence-like Element of (Sum ((Width J) | (IM -' 1))) -tuples_on the carrier of n
(Sum ((Width J) | (IM -' 1))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
len ((Sum ((Width J) | (IM -' 1))) |-> (0. n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Width J) | IM is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(Width J) | (Seg IM) is Relation-like Seg IM -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Width J) | IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Width J) /^ IM is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
((Width J) | IM) ^ ((Width J) /^ IM) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Sum ((Width J) /^ IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Width J) | IM)) + (Sum ((Width J) /^ IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Width J)) -' (Sum ((Width J) | IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Width J)) - (Sum ((Width J) | IM)) is V46() ext-real V165() set
((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite (Sum (Width J)) -' (Sum ((Width J) | IM)) -element FinSequence-like FinSubsequence-like Element of ((Sum (Width J)) -' (Sum ((Width J) | IM))) -tuples_on the carrier of n
((Sum (Width J)) -' (Sum ((Width J) | IM))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
len (((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len V) is finite len V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len V ) } is set
dom (Len J) is finite len J -element Element of bool NAT
dom J is finite Element of bool NAT
dom (Width J) is finite len J -element Element of bool NAT
len (Width J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Width J) . IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
<*(width (J . IM))*> is Relation-like NAT -defined NAT -valued Function-like one-to-one non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() V178() V179() V180() V181() Element of 1 -tuples_on NAT
1 -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
[1,(width (J . IM))] is set
{1,(width (J . IM))} is non empty finite V36() set
{{1,(width (J . IM))},{1}} is non empty finite V36() set
{[1,(width (J . IM))]} is Relation-like non empty trivial finite 1 -element set
((Width J) | (IM -' 1)) ^ <*(width (J . IM))*> is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(((Width J) | (IM -' 1)) ^ <*(width (J . IM))*>) ^ ((Width J) /^ IM) is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Sum (((Width J) | (IM -' 1)) ^ <*(width (J . IM))*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (((Width J) | (IM -' 1)) ^ <*(width (J . IM))*>)) + (Sum ((Width J) /^ IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Width J) | (IM -' 1))) + (width (J . IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum ((Width J) | (IM -' 1))) + (width (J . IM))) + (Sum ((Width J) /^ IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite (Sum ((Width J) | (IM -' 1))) + (width (J . IM)) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Sum ((Width J) | (IM -' 1))) + (width (J . IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B | (Sum ((Width J) | IM)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Seg (Sum ((Width J) | IM)) is finite Sum ((Width J) | IM) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum ((Width J) | IM) ) } is set
B | (Seg (Sum ((Width J) | IM))) is Relation-like Seg (Sum ((Width J) | IM)) -defined NAT -defined the carrier of M -valued finite FinSubsequence-like set
(B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(B | (Sum (Width (J | IM)))) /^ (Sum ((Width J) | (IM -' 1))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len (B | (Sum ((Width J) | IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (J . IM)) + (Sum ((Width J) /^ IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Width J) | (IM -' 1))) + ((width (J . IM)) + (Sum ((Width J) /^ IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B | (Sum ((Width J) | (IM -' 1))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Seg (Sum ((Width J) | (IM -' 1))) is finite Sum ((Width J) | (IM -' 1)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum ((Width J) | (IM -' 1)) ) } is set
B | (Seg (Sum ((Width J) | (IM -' 1)))) is Relation-like Seg (Sum ((Width J) | (IM -' 1))) -defined NAT -defined the carrier of M -valued finite FinSubsequence-like set
len (B | (Sum ((Width J) | (IM -' 1)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(B | (Sum ((Width J) | IM))) | (Sum ((Width J) | (IM -' 1))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(B | (Sum ((Width J) | IM))) | (Seg (Sum ((Width J) | (IM -' 1)))) is Relation-like Seg (Sum ((Width J) | (IM -' 1))) -defined NAT -defined the carrier of M -valued finite FinSubsequence-like set
(B | (Sum ((Width J) | (IM -' 1)))) ^ ((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1)))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len ((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (B | (Sum ((Width J) | (IM -' 1))))) + (len ((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B /^ (Sum ((Width J) | IM)) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(B | (Sum ((Width J) | IM))) ^ (B /^ (Sum ((Width J) | IM))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len (B /^ (Sum ((Width J) | IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (B | (Sum ((Width J) | IM)))) + (len (B /^ (Sum ((Width J) | IM)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(V /. c₁₁) |-- V is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
LineVec2Mx ((V /. c₁₁) |-- V) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((V /. c₁₁) |-- V), the carrier of n
len ((V /. c₁₁) |-- V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((V /. c₁₁) |-- V)) * T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
Line (((LineVec2Mx ((V /. c₁₁) |-- V)) * T),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ((LineVec2Mx ((V /. c₁₁) |-- V)) * T) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((V /. c₁₁) |-- V)) * T)) -tuples_on the carrier of n
width ((LineVec2Mx ((V /. c₁₁) |-- V)) * T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((V /. c₁₁) |-- V)) * T)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line (((LineVec2Mx ((V /. c₁₁) |-- V)) * T),1)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line (((LineVec2Mx ((V /. c₁₁) |-- V)) * T),1)),B)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line (((LineVec2Mx ((V /. c₁₁) |-- V)) * T),1)),B)) is Element of the carrier of M
Line ((1. (n,(len V))),c₁₁) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (1. (n,(len V))) -element FinSequence-like FinSubsequence-like Element of (width (1. (n,(len V)))) -tuples_on the carrier of n
width (1. (n,(len V))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (1. (n,(len V)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
LineVec2Mx (Line ((1. (n,(len V))),c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (n,(len V))),c₁₁)), the carrier of n
len (Line ((1. (n,(len V))),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
Line (((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T)) -tuples_on the carrier of n
width ((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line (((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T),1)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T),1)),B)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (n,(len V))),c₁₁))) * T),1)),B)) is Element of the carrier of M
(1. (n,(len V))) * T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
Line (((1. (n,(len V))) * T),c₁₁) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ((1. (n,(len V))) * T) -element FinSequence-like FinSubsequence-like Element of (width ((1. (n,(len V))) * T)) -tuples_on the carrier of n
width ((1. (n,(len V))) * T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (n,(len V))) * T)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (n,(len V))) * T),c₁₁)), the carrier of n
len (Line (((1. (n,(len V))) * T),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁)))) -tuples_on the carrier of n
width (LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line ((LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))),1)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))),1)),B)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (n,(len V))) * T),c₁₁))),1)),B)) is Element of the carrier of M
Line (T,c₁₁) is Relation-like NAT -defined the carrier of n -valued Function-like finite width T -element FinSequence-like FinSubsequence-like Element of (width T) -tuples_on the carrier of n
(width T) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
LineVec2Mx (Line (T,c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (T,c₁₁)), the carrier of n
len (Line (T,c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (T,c₁₁))),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (LineVec2Mx (Line (T,c₁₁))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (T,c₁₁)))) -tuples_on the carrier of n
width (LineVec2Mx (Line (T,c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (T,c₁₁)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line ((LineVec2Mx (Line (T,c₁₁))),1)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line ((LineVec2Mx (Line (T,c₁₁))),1)),B)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line ((LineVec2Mx (Line (T,c₁₁))),1)),B)) is Element of the carrier of M
lmlt ((Line (T,c₁₁)),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line (T,c₁₁)),B)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line (T,c₁₁)),B)) is Element of the carrier of M
(Sum (Width (J | (IM -' 1)))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite Sum (Width (J | (IM -' 1))) -element FinSequence-like FinSubsequence-like Element of (Sum (Width (J | (IM -' 1)))) -tuples_on the carrier of n
(Sum (Width (J | (IM -' 1)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
((Sum (Width (J | (IM -' 1)))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite (Sum (Width (J | (IM -' 1)))) + (width (J . IM)) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Sum (Width (J | (IM -' 1)))) + (width (J . IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Width J)) -' (Sum (Width (J | IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite (Sum (Width J)) -' (Sum (Width (J | IM))) -element FinSequence-like FinSubsequence-like Element of ((Sum (Width J)) -' (Sum (Width (J | IM)))) -tuples_on the carrier of n
((Sum (Width J)) -' (Sum (Width (J | IM)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(((Sum (Width (J | (IM -' 1)))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n)) is Relation-like NAT -defined the carrier of n -valued Function-like finite ((Sum (Width (J | (IM -' 1)))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum (Width (J | IM)))) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((Sum (Width (J | (IM -' 1)))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum (Width (J | IM)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
lmlt (((((Sum (Width (J | (IM -' 1)))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt (((((Sum (Width (J | (IM -' 1)))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B)) is Element of the carrier of M
the addF of M $$ (lmlt (((((Sum (Width (J | (IM -' 1)))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B)) is Element of the carrier of M
(((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n)) is Relation-like NAT -defined the carrier of n -valued Function-like finite ((Sum ((Width J) | (IM -' 1))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum (Width (J | IM)))) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((Sum ((Width J) | (IM -' 1))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum (Width (J | IM)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B)) is Element of the carrier of M
the addF of M $$ (lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum (Width (J | IM)))) |-> (0. n))),B)) is Element of the carrier of M
(((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)) is Relation-like NAT -defined the carrier of n -valued Function-like finite ((Sum ((Width J) | (IM -' 1))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum ((Width J) | IM))) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((Sum ((Width J) | (IM -' 1))) + (width (J . IM))) + ((Sum (Width J)) -' (Sum ((Width J) | IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n))),B) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n))),B)) is Element of the carrier of M
the addF of M $$ (lmlt (((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))) ^ (((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n))),B)) is Element of the carrier of M
lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM)))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM)))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM))))) ^ (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM))))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum ((lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM))))) ^ (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM)))))) is Element of the carrier of M
the addF of M $$ ((lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM))))) ^ (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM)))))) is Element of the carrier of M
Sum (lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM))))) is Element of the carrier of M
the addF of M $$ (lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM))))) is Element of the carrier of M
Sum (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM))))) is Element of the carrier of M
the addF of M $$ (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM))))) is Element of the carrier of M
(Sum (lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM)))))) + (Sum (lmlt ((((Sum (Width J)) -' (Sum ((Width J) | IM))) |-> (0. n)),(B /^ (Sum ((Width J) | IM)))))) is Element of the carrier of M
Sum (B /^ (Sum ((Width J) | IM))) is Element of the carrier of M
the addF of M $$ (B /^ (Sum ((Width J) | IM))) is Element of the carrier of M
(0. n) * (Sum (B /^ (Sum ((Width J) | IM)))) is Element of the carrier of M
(Sum (lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM)))))) + ((0. n) * (Sum (B /^ (Sum ((Width J) | IM))))) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
(Sum (lmlt ((((Sum ((Width J) | (IM -' 1))) |-> (0. n)) ^ (Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1)))))))),(B | (Sum ((Width J) | IM)))))) + (0. M) is Element of the carrier of M
lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1))))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
(lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1)))))) ^ (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1)))))) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum ((lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1)))))) ^ (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))))) is Element of the carrier of M
the addF of M $$ ((lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1)))))) ^ (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))))) is Element of the carrier of M
Sum (lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1)))))) is Element of the carrier of M
the addF of M $$ (lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1)))))) is Element of the carrier of M
Sum (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1)))))) is Element of the carrier of M
the addF of M $$ (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1)))))) is Element of the carrier of M
(Sum (lmlt (((Sum ((Width J) | (IM -' 1))) |-> (0. n)),(B | (Sum ((Width J) | (IM -' 1))))))) + (Sum (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))))) is Element of the carrier of M
Sum (B | (Sum ((Width J) | (IM -' 1)))) is Element of the carrier of M
the addF of M $$ (B | (Sum ((Width J) | (IM -' 1)))) is Element of the carrier of M
(0. n) * (Sum (B | (Sum ((Width J) | (IM -' 1))))) is Element of the carrier of M
((0. n) * (Sum (B | (Sum ((Width J) | (IM -' 1)))))) + (Sum (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))))) is Element of the carrier of M
(0. M) + (Sum (lmlt ((Line ((J . IM),(c₁₁ -' (Sum (Len (J | (IM -' 1))))))),((B | (Sum ((Width J) | IM))) /^ (Sum ((Width J) | (IM -' 1))))))) is Element of the carrier of M
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
K is Element of the carrier of n
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
V is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
(n,K,(len V)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len V, len V, the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
Line ((n,K,(len V)),(len V)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (n,K,(len V)) -element FinSequence-like FinSubsequence-like Element of (width (n,K,(len V))) -tuples_on the carrier of n
width (n,K,(len V)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (n,K,(len V))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line ((n,K,(len V)),(len V))),V) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line ((n,K,(len V)),(len V))),V)) 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is Relation-like set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
the addF of M $$ (lmlt ((Line ((n,K,(len V)),(len V))),V)) is Element of the carrier of M
V /. (len V) is Element of the carrier of M
K * (V /. (len V)) is Element of the carrier of M
1. (n,(len V)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len V, the carrier of n
width (1. (n,(len V))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((1. (n,(len V))),(len V)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (1. (n,(len V))) -element FinSequence-like FinSubsequence-like Element of (width (1. (n,(len V)))) -tuples_on the carrier of n
(width (1. (n,(len V)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K * (Line ((1. (n,(len V))),(len V))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (1. (n,(len V))) -element FinSequence-like FinSubsequence-like Element of (width (1. (n,(len V)))) -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 V18( the carrier of n, the carrier of n) Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
bool [: the carrier of n, the carrier of n:] is set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued non empty total V18( the carrier of n, the carrier of n) 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 V18( the carrier of n, the carrier of n) Element of bool [: the carrier of n, the carrier of n:]
K440( the carrier of n, the carrier of n,(Line ((1. (n,(len V))),(len V))),(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
lmlt ((K * (Line ((1. (n,(len V))),(len V)))),V) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((K * (Line ((1. (n,(len V))),(len V)))),V)) is Element of the carrier of M
the addF of M $$ (lmlt ((K * (Line ((1. (n,(len V))),(len V)))),V)) is Element of the carrier of M
lmlt ((Line ((1. (n,(len V))),(len V))),V) is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of M
Sum (lmlt ((Line ((1. (n,(len V))),(len V))),V)) is Element of the carrier of M
the addF of M $$ (lmlt ((Line ((1. (n,(len V))),(len V))),V)) is Element of the carrier of M
K * (Sum (lmlt ((Line ((1. (n,(len V))),(len V))),V))) is Element of the carrier of M
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
n + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
M is Element of the carrier of K
V is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of V is non empty set
B is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
dom B is finite Element of bool NAT
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(K,M,(len B)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len B, len B, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Line ((K,M,(len B)),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,M,(len B)) -element FinSequence-like FinSubsequence-like Element of (width (K,M,(len B))) -tuples_on the carrier of K
width (K,M,(len B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (K,M,(len B))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((K,M,(len B)),n)),B) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((K,M,(len B)),n)),B)) is Element of the carrier of V
the addF of V is Relation-like [: the carrier of V, the carrier of V:] -defined the carrier of V -valued Function-like total V18([: the carrier of V, the carrier of V:], the carrier of V) Element of bool [:[: the carrier of V, the carrier of V:], the carrier of V:]
[: the carrier of V, the carrier of V:] is Relation-like set
[:[: the carrier of V, the carrier of V:], the carrier of V:] is Relation-like set
bool [:[: the carrier of V, the carrier of V:], the carrier of V:] is set
the addF of V $$ (lmlt ((Line ((K,M,(len B)),n)),B)) is Element of the carrier of V
B /. n is Element of the carrier of V
M * (B /. n) is Element of the carrier of V
B /. (n + 1) is Element of the carrier of V
(M * (B /. n)) + (B /. (n + 1)) is Element of the carrier of V
Seg (len B) is finite len B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
1. (K,(len B)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len B, len B, the carrier of K
Line ((1. (K,(len B))),(n + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len B))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len B)))) -tuples_on the carrier of K
width (1. (K,(len B))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (1. (K,(len B)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Line ((1. (K,(len B))),(n + 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (Line ((1. (K,(len B))),(n + 1))) is finite width (1. (K,(len B))) -element Element of bool NAT
Line ((1. (K,(len B))),n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len B))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len B)))) -tuples_on the carrier of K
M * (Line ((1. (K,(len B))),n)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len B))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len B)))) -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 V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of K, the carrier of K:], the carrier of K) 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 set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued non empty total V18( the carrier of K, the carrier of K) 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 V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
K440( the carrier of K, the carrier of K,(Line ((1. (K,(len B))),n)),(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
len (M * (Line ((1. (K,(len B))),n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Line ((1. (K,(len B))),n)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (M * (Line ((1. (K,(len B))),n))) is finite width (1. (K,(len B))) -element Element of bool NAT
lmlt ((M * (Line ((1. (K,(len B))),n))),B) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
dom (lmlt ((M * (Line ((1. (K,(len B))),n))),B)) is finite Element of bool NAT
lmlt ((Line ((1. (K,(len B))),(n + 1))),B) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
dom (lmlt ((Line ((1. (K,(len B))),(n + 1))),B)) is finite Element of bool NAT
len (lmlt ((M * (Line ((1. (K,(len B))),n))),B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (lmlt ((Line ((1. (K,(len B))),(n + 1))),B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(M * (Line ((1. (K,(len B))),n))) + (Line ((1. (K,(len B))),(n + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len B))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len B)))) -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 total V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K437( the carrier of K, the carrier of K, the carrier of K, the addF of K,(M * (Line ((1. (K,(len B))),n))),(Line ((1. (K,(len B))),(n + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K685((M * (Line ((1. (K,(len B))),n))),(Line ((1. (K,(len B))),(n + 1)))) is Relation-like Function-like set
K685((M * (Line ((1. (K,(len B))),n))),(Line ((1. (K,(len B))),(n + 1)))) * the addF of K is Relation-like the carrier of K -valued set
lmlt (((M * (Line ((1. (K,(len B))),n))) + (Line ((1. (K,(len B))),(n + 1)))),B) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt (((M * (Line ((1. (K,(len B))),n))) + (Line ((1. (K,(len B))),(n + 1)))),B)) is Element of the carrier of V
the addF of V $$ (lmlt (((M * (Line ((1. (K,(len B))),n))) + (Line ((1. (K,(len B))),(n + 1)))),B)) is Element of the carrier of V
(lmlt ((M * (Line ((1. (K,(len B))),n))),B)) + (lmlt ((Line ((1. (K,(len B))),(n + 1))),B)) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
K437( the carrier of V, the carrier of V, the carrier of V, the addF of V,(lmlt ((M * (Line ((1. (K,(len B))),n))),B)),(lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
K685((lmlt ((M * (Line ((1. (K,(len B))),n))),B)),(lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Relation-like Function-like set
K685((lmlt ((M * (Line ((1. (K,(len B))),n))),B)),(lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) * the addF of V is Relation-like the carrier of V -valued set
Sum ((lmlt ((M * (Line ((1. (K,(len B))),n))),B)) + (lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Element of the carrier of V
the addF of V $$ ((lmlt ((M * (Line ((1. (K,(len B))),n))),B)) + (lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Element of the carrier of V
Sum (lmlt ((M * (Line ((1. (K,(len B))),n))),B)) is Element of the carrier of V
the addF of V $$ (lmlt ((M * (Line ((1. (K,(len B))),n))),B)) is Element of the carrier of V
Sum (lmlt ((Line ((1. (K,(len B))),(n + 1))),B)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((1. (K,(len B))),(n + 1))),B)) is Element of the carrier of V
(Sum (lmlt ((M * (Line ((1. (K,(len B))),n))),B))) + (Sum (lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Element of the carrier of V
lmlt ((Line ((1. (K,(len B))),n)),B) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((1. (K,(len B))),n)),B)) is Element of the carrier of V
the addF of V $$ (lmlt ((Line ((1. (K,(len B))),n)),B)) is Element of the carrier of V
M * (Sum (lmlt ((Line ((1. (K,(len B))),n)),B))) is Element of the carrier of V
(M * (Sum (lmlt ((Line ((1. (K,(len B))),n)),B)))) + (Sum (lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Element of the carrier of V
(M * (B /. n)) + (Sum (lmlt ((Line ((1. (K,(len B))),(n + 1))),B))) is Element of the carrier of V
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
M is Element of the carrier of K
(K,M,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of n,n, the carrier of K
V is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of V is non empty set
B is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of B is non empty set
M9 is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V
len M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M9 is finite Element of bool NAT
T is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
1. (K,(len M9)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M9, len M9, the carrier of K
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M9, len T, the carrier of K
Mx2Tran (P,M9,T) is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) additive homogeneous Element of bool [: the carrier of V, the carrier of B:]
[: the carrier of V, the carrier of B:] is Relation-like set
bool [: the carrier of V, the carrier of B:] is set
len P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (1. (K,(len M9))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (1. (K,(len M9))) is finite Element of bool NAT
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M9 /. c₁₁ is Element of the carrier of V
(Mx2Tran (P,M9,T)) . (M9 /. c₁₁) is Element of the carrier of B
T /. c₁₁ is Element of the carrier of B
M * (T /. c₁₁) is Element of the carrier of B
c₁₁ + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
T /. (c₁₁ + 1) is Element of the carrier of B
(M * (T /. c₁₁)) + (T /. (c₁₁ + 1)) is Element of the carrier of B
(M9 /. c₁₁) |-- M9 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((M9 /. c₁₁) |-- M9) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len ((M9 /. c₁₁) |-- M9), the carrier of K
len ((M9 /. c₁₁) |-- M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P)) -tuples_on the carrier of K
width ((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P),1)),T) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
Sum (lmlt ((Line (((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P),1)),T)) is Element of the carrier of B
the addF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like total V18([: the carrier of B, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is Relation-like set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is Relation-like set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the addF of B $$ (lmlt ((Line (((LineVec2Mx ((M9 /. c₁₁) |-- M9)) * P),1)),T)) is Element of the carrier of B
Line ((1. (K,(len M9))),c₁₁) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len M9))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len M9)))) -tuples_on the carrier of K
width (1. (K,(len M9))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (1. (K,(len M9)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len M9))),c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line ((1. (K,(len M9))),c₁₁)), the carrier of K
len (Line ((1. (K,(len M9))),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P)) -tuples_on the carrier of K
width ((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P),1)),T) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P),1)),T)) is Element of the carrier of B
the addF of B $$ (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len M9))),c₁₁))) * P),1)),T)) is Element of the carrier of B
(1. (K,(len M9))) * P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
Line (((1. (K,(len M9))) * P),c₁₁) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len M9))) * P) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len M9))) * P)) -tuples_on the carrier of K
width ((1. (K,(len M9))) * P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width ((1. (K,(len M9))) * P)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (((1. (K,(len M9))) * P),c₁₁)), the carrier of K
len (Line (((1. (K,(len M9))) * P),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))),1)),T) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))),1)),T)) is Element of the carrier of B
the addF of B $$ (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len M9))) * P),c₁₁))),1)),T)) is Element of the carrier of B
Line (P,c₁₁) 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 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width P) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (P,c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of 1, len (Line (P,c₁₁)), the carrier of K
len (Line (P,c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Line ((LineVec2Mx (Line (P,c₁₁))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (P,c₁₁))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (P,c₁₁)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (P,c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (LineVec2Mx (Line (P,c₁₁)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (P,c₁₁))),1)),T) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
Sum (lmlt ((Line ((LineVec2Mx (Line (P,c₁₁))),1)),T)) is Element of the carrier of B
the addF of B $$ (lmlt ((Line ((LineVec2Mx (Line (P,c₁₁))),1)),T)) is Element of the carrier of B
lmlt ((Line (P,c₁₁)),T) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
Sum (lmlt ((Line (P,c₁₁)),T)) is Element of the carrier of B
the addF of B $$ (lmlt ((Line (P,c₁₁)),T)) is Element of the carrier of B
Seg (len M9) is finite len M9 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len M9 ) } is set
width (K,M,n) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom T is finite Element of bool NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Element of the carrier of n
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
V is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of V is non empty set
B is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom B is finite Element of bool NAT
M9 is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
len M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
block_diagonal (T,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len T), Sum (Len T), the carrier of n
Len T is Relation-like NAT -defined NAT -valued Function-like finite len T -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len T) -tuples_on NAT
len T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len T) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len B, len M9, the carrier of n
Mx2Tran (J,B,M9) is Relation-like the carrier of M -defined the carrier of V -valued Function-like non empty total V18( the carrier of M, the carrier of V) additive homogeneous Element of bool [: the carrier of M, the carrier of V:]
[: the carrier of M, the carrier of V:] is Relation-like set
bool [: the carrier of M, the carrier of V:] is set
Seg (len B) is finite len B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
len J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
min ((Len T),b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B /. b1 is Element of the carrier of M
(Mx2Tran (J,B,M9)) . (B /. b1) is Element of the carrier of V
M9 /. b1 is Element of the carrier of V
K * (M9 /. b1) is Element of the carrier of V
b1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M9 /. (b1 + 1) is Element of the carrier of V
(K * (M9 /. b1)) + (M9 /. (b1 + 1)) is Element of the carrier of V
P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,P,T) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
Seg P is finite P -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= P ) } is set
T | (Seg P) is Relation-like Seg P -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,K,P,T) is Relation-like NAT -defined NAT -valued Function-like finite len (n,K,P,T) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,K,P,T)) -tuples_on NAT
len (n,K,P,T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,K,P,T)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (n,K,P,T)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (Len T) is finite len T -element Element of bool NAT
(Len T) . P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
T . P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (T . P), len (T . P), the carrier of n
T . P is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (T . P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (T . P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,K,(P -' 1),T) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
Seg (P -' 1) is finite P -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= P -' 1 ) } is set
T | (Seg (P -' 1)) is Relation-like Seg (P -' 1) -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,K,(P -' 1),T) is Relation-like NAT -defined NAT -valued Function-like finite len (n,K,(P -' 1),T) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,K,(P -' 1),T)) -tuples_on NAT
len (n,K,(P -' 1),T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,K,(P -' 1),T)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (n,K,(P -' 1),T)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 -' (Sum (Len (n,K,(P -' 1),T))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width (n,K,(P -' 1),T) is Relation-like NAT -defined NAT -valued Function-like finite len (n,K,(P -' 1),T) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,K,(P -' 1),T)) -tuples_on NAT
Sum (Width (n,K,(P -' 1),T)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width (n,K,P,T) is Relation-like NAT -defined NAT -valued Function-like finite len (n,K,P,T) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,K,P,T)) -tuples_on NAT
Sum (Width (n,K,P,T)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 | (Sum (Width (n,K,P,T))) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Seg (Sum (Width (n,K,P,T))) is finite Sum (Width (n,K,P,T)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Width (n,K,P,T)) ) } is set
M9 | (Seg (Sum (Width (n,K,P,T)))) is Relation-like Seg (Sum (Width (n,K,P,T))) -defined NAT -defined the carrier of V -valued finite FinSubsequence-like set
(M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T))) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Line ((T . P),(b1 -' (Sum (Len (n,K,(P -' 1),T))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (T . P) -element FinSequence-like FinSubsequence-like Element of (width (T . P)) -tuples_on the carrier of n
width (T . P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width (T . P)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
lmlt ((Line ((T . P),(b1 -' (Sum (Len (n,K,(P -' 1),T)))))),((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T))))) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (lmlt ((Line ((T . P),(b1 -' (Sum (Len (n,K,(P -' 1),T)))))),((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))))) is Element of the carrier of V
the addF of V is Relation-like [: the carrier of V, the carrier of V:] -defined the carrier of V -valued Function-like total V18([: the carrier of V, the carrier of V:], the carrier of V) Element of bool [:[: the carrier of V, the carrier of V:], the carrier of V:]
[: the carrier of V, the carrier of V:] is Relation-like set
[:[: the carrier of V, the carrier of V:], the carrier of V:] is Relation-like set
bool [:[: the carrier of V, the carrier of V:], the carrier of V:] is set
the addF of V $$ (lmlt ((Line ((T . P),(b1 -' (Sum (Len (n,K,(P -' 1),T)))))),((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))))) is Element of the carrier of V
width J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width T is Relation-like NAT -defined NAT -valued Function-like finite len T -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len T) -tuples_on NAT
Sum (Width T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len ((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len T) | P is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(Len T) | (Seg P) is Relation-like Seg P -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
(Len T) | (P -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(Len T) | (Seg (P -' 1)) is Relation-like Seg (P -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
b1 - (Sum (Len (n,K,(P -' 1),T))) is V46() ext-real V165() set
P - 1 is V46() ext-real V165() set
(P -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len T) | ((P -' 1) + 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg ((P -' 1) + 1) is non empty finite (P -' 1) + 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (P -' 1) + 1 ) } is set
(Len T) | (Seg ((P -' 1) + 1)) is Relation-like Seg ((P -' 1) + 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
<*((Len T) . P)*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((Len T) . P)] is set
{1,((Len T) . P)} is non empty finite V36() set
{{1,((Len T) . P)},{1}} is non empty finite V36() set
{[1,((Len T) . P)]} is Relation-like non empty trivial finite 1 -element set
(Len (n,K,(P -' 1),T)) ^ <*((Len T) . P)*> is Relation-like NAT -defined Function-like non empty finite (len (n,K,(P -' 1),T)) + 1 -element FinSequence-like FinSubsequence-like set
(len (n,K,(P -' 1),T)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Sum (Len (n,K,(P -' 1),T))) + (len (T . P)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Len (n,K,(P -' 1),T))) + (b1 -' (Sum (Len (n,K,(P -' 1),T)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom T is finite Element of bool NAT
Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,Jk) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of Jk,Jk, the carrier of n
len (Len T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len T) | (len (Len T)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (len (Len T)) is finite len (Len T) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Len T) ) } is set
(Len T) | (Seg (len (Len T))) is Relation-like Seg (len (Len T)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len T) | (len (Len T))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 | (Sum (Len (n,K,P,T))) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Seg (Sum (Len (n,K,P,T))) is finite Sum (Len (n,K,P,T)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Len (n,K,P,T)) ) } is set
M9 | (Seg (Sum (Len (n,K,P,T)))) is Relation-like Seg (Sum (Len (n,K,P,T))) -defined NAT -defined the carrier of V -valued finite FinSubsequence-like set
len (M9 | (Sum (Len (n,K,P,T)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (M9 | (Sum (Len (n,K,P,T)))) is finite Element of bool NAT
dom ((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))) is finite Element of bool NAT
((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))) /. (b1 -' (Sum (Len (n,K,(P -' 1),T)))) is Element of the carrier of V
(Sum (Width (n,K,(P -' 1),T))) + (b1 -' (Sum (Len (n,K,(P -' 1),T)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(M9 | (Sum (Width (n,K,P,T)))) /. ((Sum (Width (n,K,(P -' 1),T))) + (b1 -' (Sum (Len (n,K,(P -' 1),T))))) is Element of the carrier of V
(M9 | (Sum (Width (n,K,P,T)))) /. ((Sum (Len (n,K,(P -' 1),T))) + (b1 -' (Sum (Len (n,K,(P -' 1),T))))) is Element of the carrier of V
Sum ((Len T) | P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 | (Sum ((Len T) | P)) is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Seg (Sum ((Len T) | P)) is finite Sum ((Len T) | P) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum ((Len T) | P) ) } is set
M9 | (Seg (Sum ((Len T) | P))) is Relation-like Seg (Sum ((Len T) | P)) -defined NAT -defined the carrier of V -valued finite FinSubsequence-like set
(M9 | (Sum ((Len T) | P))) /. b1 is Element of the carrier of V
(b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
((M9 | (Sum (Width (n,K,P,T)))) /^ (Sum (Width (n,K,(P -' 1),T)))) /. ((b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1) is Element of the carrier of V
(Sum (Width (n,K,(P -' 1),T))) + ((b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(M9 | (Sum (Width (n,K,P,T)))) /. ((Sum (Width (n,K,(P -' 1),T))) + ((b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1)) is Element of the carrier of V
Sum ((Len T) | (P -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Len T) | (P -' 1))) + ((b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(M9 | (Sum (Width (n,K,P,T)))) /. ((Sum ((Len T) | (P -' 1))) + ((b1 -' (Sum (Len (n,K,(P -' 1),T)))) + 1)) is Element of the carrier of V
(M9 | (Sum ((Len T) | P))) /. (b1 + 1) is Element of the carrier of V
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
len M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (V,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len V), Sum (Len V), the carrier of n
Len V is Relation-like NAT -defined NAT -valued Function-like finite len V -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len V) -tuples_on NAT
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len V) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M, len M, the carrier of n
Mx2Tran (B,M,M) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Mx2Tran (B,M,M)) |^ J is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
M9 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
dom M9 is non empty set
rng M is finite set
dom ((Mx2Tran (B,M,M)) |^ J) is non empty set
dim K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Omega). K is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace 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 total V18([: the carrier of K, the carrier of K:], the carrier of K) 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 set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the ZeroF of K 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 total V18([: the carrier of n, the carrier of K:], the carrier of K) 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 set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is set
VectSpStr(# the carrier of K, the addF of K, the ZeroF of K, the lmult of K #) is non empty strict VectSpStr over n
(0). K is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of K
0. K is V65(K) Element of the carrier of K
{(0. K)} is non empty trivial finite 1 -element Element of bool the carrier of K
bool the carrier of K is set
rng ((Mx2Tran (B,M,M)) |^ J) is non empty set
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
dom (Len V) is finite len V -element Element of bool NAT
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P is set
dom M is finite Element of bool NAT
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len V),c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (Sum (Len V)) is finite Sum (Len V) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Len V) ) } is set
(Len V) . (min ((Len V),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . (min ((Len V),c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . (min ((Len V),c₁₁))), len (V . (min ((Len V),c₁₁))), the carrier of n
V . (min ((Len V),c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . (min ((Len V),c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . (min ((Len V),c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) | (min ((Len V),c₁₁)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (min ((Len V),c₁₁)) is finite min ((Len V),c₁₁) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len V),c₁₁) ) } is set
(Len V) | (Seg (min ((Len V),c₁₁))) is Relation-like Seg (min ((Len V),c₁₁)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
(n,(0. n),(min ((Len V),c₁₁)),V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
V | (Seg (min ((Len V),c₁₁))) is Relation-like Seg (min ((Len V),c₁₁)) -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,(0. n),(min ((Len V),c₁₁)),V) is Relation-like NAT -defined NAT -valued Function-like finite len (n,(0. n),(min ((Len V),c₁₁)),V) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,(0. n),(min ((Len V),c₁₁)),V)) -tuples_on NAT
len (n,(0. n),(min ((Len V),c₁₁)),V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,(0. n),(min ((Len V),c₁₁)),V)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
len (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) | (len (Len V)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (len (Len V)) is finite len (Len V) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Len V) ) } is set
(Len V) | (Seg (len (Len V))) is Relation-like Seg (len (Len V)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len V) | (len (Len V))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M /. c₁₁ is Element of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(min ((Len V),c₁₁)) -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) | ((min ((Len V),c₁₁)) -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg ((min ((Len V),c₁₁)) -' 1) is finite (min ((Len V),c₁₁)) -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (min ((Len V),c₁₁)) -' 1 ) } is set
(Len V) | (Seg ((min ((Len V),c₁₁)) -' 1)) is Relation-like Seg ((min ((Len V),c₁₁)) -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) /. (min ((Len V),c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(min ((Len V),c₁₁)) - 1 is V46() ext-real V165() set
((min ((Len V),c₁₁)) -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
<*((Len V) . (min ((Len V),c₁₁)))*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((Len V) . (min ((Len V),c₁₁)))] is set
{1,((Len V) . (min ((Len V),c₁₁)))} is non empty finite V36() set
{{1,((Len V) . (min ((Len V),c₁₁)))},{1}} is non empty finite V36() set
{[1,((Len V) . (min ((Len V),c₁₁)))]} is Relation-like non empty trivial finite 1 -element set
((Len V) | ((min ((Len V),c₁₁)) -' 1)) ^ <*((Len V) . (min ((Len V),c₁₁)))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) + (len (V . (min ((Len V),c₁₁)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ -' (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ - (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) is V46() ext-real V165() set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(c₁₁ + FK) - (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) is V46() ext-real V165() set
((Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) + (len (V . (min ((Len V),c₁₁))))) - (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) is V46() ext-real V165() set
(c₁₁ -' (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1)))) + FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg ((Len V) /. (min ((Len V),c₁₁))) is finite (Len V) /. (min ((Len V),c₁₁)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (Len V) /. (min ((Len V),c₁₁)) ) } is set
((c₁₁ -' (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1)))) + FK) + (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len V),(((c₁₁ -' (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1)))) + FK) + (Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len V),(c₁₁ + FK)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
c₁₁ + FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FK + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ (FK + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (FK + 1)) . (M /. c₁₁) is Element of the carrier of K
(c₁₁ + FK) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. ((c₁₁ + FK) + 1) is Element of the carrier of K
c₁₁ + (FK + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(FK + 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ ((FK + 1) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ ((FK + 1) + 1)) . (M /. c₁₁) is Element of the carrier of K
(c₁₁ + (FK + 1)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. ((c₁₁ + (FK + 1)) + 1) is Element of the carrier of K
dom ((Mx2Tran (B,M,M)) |^ (FK + 1)) is non empty set
min ((Len V),(c₁₁ + (FK + 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
Seg (min ((Len V),(c₁₁ + (FK + 1)))) is finite min ((Len V),(c₁₁ + (FK + 1))) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len V),(c₁₁ + (FK + 1))) ) } is set
V | (Seg (min ((Len V),(c₁₁ + (FK + 1))))) is Relation-like Seg (min ((Len V),(c₁₁ + (FK + 1)))) -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V) is Relation-like NAT -defined NAT -valued Function-like finite len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V)) -tuples_on NAT
len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (n,(0. n),(min ((Len V),(c₁₁ + (FK + 1)))),V)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M /. (c₁₁ + (FK + 1)) is Element of the carrier of K
(Mx2Tran (B,M,M)) . (M /. (c₁₁ + (FK + 1))) is Element of the carrier of K
(0. n) * (M /. (c₁₁ + (FK + 1))) is Element of the carrier of K
((0. n) * (M /. (c₁₁ + (FK + 1)))) + (M /. ((c₁₁ + (FK + 1)) + 1)) is Element of the carrier of K
(0. K) + (M /. ((c₁₁ + (FK + 1)) + 1)) is Element of the carrier of K
(FK + 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ ((FK + 1) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ ((FK + 1) + 1)) . (M /. c₁₁) is Element of the carrier of K
(Mx2Tran (B,M,M)) |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ (FK + 1)) is Relation-like the carrier of K -defined the carrier of K -defined the carrier of K -valued the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
(((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ (FK + 1))) . (M /. c₁₁) is Element of the carrier of K
((Mx2Tran (B,M,M)) |^ 1) . (M /. (c₁₁ + (FK + 1))) is Element of the carrier of K
Sum ((Len V) | (min ((Len V),c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) - c₁₁ is V46() ext-real V165() set
c₁₁ + {} is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
{} + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ ({} + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ ({} + 1)) . (M /. c₁₁) is Element of the carrier of K
(c₁₁ + {}) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. ((c₁₁ + {}) + 1) is Element of the carrier of K
(Mx2Tran (B,M,M)) . (M /. c₁₁) is Element of the carrier of K
(0. n) * (M /. c₁₁) is Element of the carrier of K
c₁₁ + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (c₁₁ + 1) is Element of the carrier of K
((0. n) * (M /. c₁₁)) + (M /. (c₁₁ + 1)) is Element of the carrier of K
(0. K) + (M /. (c₁₁ + 1)) is Element of the carrier of K
((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ (((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1)) . (M /. c₁₁) is Element of the carrier of K
(Mx2Tran (B,M,M)) . (M /. c₁₁) is Element of the carrier of K
(0. n) * (M /. c₁₁) is Element of the carrier of K
((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) - 1 is V46() ext-real V165() set
(Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
dom ((Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁)) is non empty set
(Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) - {} is V46() ext-real non negative V165() set
(Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) - 1 is V46() ext-real V165() set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FK + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ (FK + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (FK + 1)) . (M /. c₁₁) is Element of the carrier of K
c₁₁ + FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(c₁₁ + FK) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. ((c₁₁ + FK) + 1) is Element of the carrier of K
M /. (Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) is Element of the carrier of K
((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len V),(Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Mx2Tran (B,M,M)) . (M /. (Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V)))) is Element of the carrier of K
(0. n) * (M /. (Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V)))) is Element of the carrier of K
((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ (((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁) + 1)) . (M /. c₁₁) is Element of the carrier of K
(Mx2Tran (B,M,M)) |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁)) is Relation-like the carrier of K -defined the carrier of K -defined the carrier of K -valued the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
(((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁))) . (M /. c₁₁) is Element of the carrier of K
((Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁)) . (M /. c₁₁) is Element of the carrier of K
((Mx2Tran (B,M,M)) |^ 1) . (((Mx2Tran (B,M,M)) |^ ((Sum (Len (n,(0. n),(min ((Len V),c₁₁)),V))) -' c₁₁)) . (M /. c₁₁)) is Element of the carrier of K
c₁₁ - c₁₁ is V46() ext-real V165() set
(Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) - c₁₁ is V46() ext-real V165() set
(len (V . (min ((Len V),c₁₁)))) + {} is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (V . (min ((Len V),c₁₁)))) + ((Sum ((Len V) | ((min ((Len V),c₁₁)) -' 1))) - c₁₁) is V46() ext-real V165() set
J - 1 is V46() ext-real V165() set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FK + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Jk + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ (Jk + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (Jk + 1)) . (M /. c₁₁) is Element of the carrier of K
(Jk + 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ ((Jk + 1) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ ((Jk + 1) + 1)) . (M /. c₁₁) is Element of the carrier of K
dom ((Mx2Tran (B,M,M)) |^ (Jk + 1)) is non empty set
(Jk + 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (B,M,M)) |^ ((Jk + 1) + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ ((Jk + 1) + 1)) . (M /. c₁₁) is Element of the carrier of K
(Mx2Tran (B,M,M)) |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ (Jk + 1)) is Relation-like the carrier of K -defined the carrier of K -defined the carrier of K -valued the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
(((Mx2Tran (B,M,M)) |^ 1) * ((Mx2Tran (B,M,M)) |^ (Jk + 1))) . (M /. c₁₁) is Element of the carrier of K
((Mx2Tran (B,M,M)) |^ 1) . (((Mx2Tran (B,M,M)) |^ (Jk + 1)) . (M /. c₁₁)) is Element of the carrier of K
(Mx2Tran (B,M,M)) . (0. K) is Element of the carrier of K
(0. n) * (0. K) is Element of the carrier of K
(Mx2Tran (B,M,M)) . ((0. n) * (0. K)) is Element of the carrier of K
(0. n) * ((Mx2Tran (B,M,M)) . (0. K)) is Element of the carrier of K
(Mx2Tran (B,M,M)) |^ (FK + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
((Mx2Tran (B,M,M)) |^ (FK + 1)) . (M /. c₁₁) is Element of the carrier of K
M9 * M 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 trivial non finite set
bool [:NAT, the carrier of K:] is non trivial non finite set
(M9 * M) . P is set
M . P is set
M9 . (M . P) is set
M /. P is Element of the carrier of K
M9 . (M /. P) is Element of the carrier of K
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
( the carrier of K --> (0. K)) . (M /. P) is Element of the carrier of K
M . c₁₁ is set
((Mx2Tran (B,M,M)) |^ J) . (M . c₁₁) is set
((Mx2Tran (B,M,M)) |^ J) * M is Relation-like NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
(((Mx2Tran (B,M,M)) |^ J) * M) . P is set
dom (M9 * M) is finite set
dom (((Mx2Tran (B,M,M)) |^ J) * M) is finite set
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
0. K is V65(K) Element of the carrier of K
the ZeroF of K 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 total V18([: the carrier of K,NAT:], the carrier of K) Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial non finite set
M is Element of the carrier of K
(power K) . (M,n) is Element of the carrier of K
V is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of V is non empty set
B is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom B is finite Element of bool NAT
M9 is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) (K,M)
block_diagonal (M9,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len M9), Sum (Len M9), the carrier of K
Len M9 is Relation-like NAT -defined NAT -valued Function-like finite len M9 -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len M9) -tuples_on NAT
len M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len M9) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len M9),(len B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(K,M,(min ((Len M9),(len B))),M9) is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) (K,M)
Seg (min ((Len M9),(len B))) is finite min ((Len M9),(len B)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len M9),(len B)) ) } is set
M9 | (Seg (min ((Len M9),(len B)))) is Relation-like Seg (min ((Len M9),(len B))) -defined NAT -defined ( the carrier of K *) * -valued finite FinSubsequence-like Function-yielding set
Len (K,M,(min ((Len M9),(len B))),M9) is Relation-like NAT -defined NAT -valued Function-like finite len (K,M,(min ((Len M9),(len B))),M9) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (K,M,(min ((Len M9),(len B))),M9)) -tuples_on NAT
len (K,M,(min ((Len M9),(len B))),M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (K,M,(min ((Len M9),(len B))),M9)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (K,M,(min ((Len M9),(len B))),M9)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9))) is Element of the carrier of V
((power K) . (M,n)) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
Seg (len B) is finite len B -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len B, len B, the carrier of K
Mx2Tran (P,B,B) is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
[: the carrier of V, the carrier of V:] is Relation-like set
bool [: the carrier of V, the carrier of V:] is set
(Mx2Tran (P,B,B)) |^ n is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) |^ n) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
len P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Mx2Tran (P,B,B)) |^ {} is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) |^ {}) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
id V is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
id the carrier of V is Relation-like the carrier of V -defined the carrier of V -valued non empty total V18( the carrier of V, the carrier of V) Element of bool [: the carrier of V, the carrier of V:]
(id V) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(1_ K) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(power K) . (M,{}) is Element of the carrier of K
((power K) . (M,{})) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(Len M9) | (min ((Len M9),(len B))) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
(Len M9) | (Seg (min ((Len M9),(len B)))) is Relation-like Seg (min ((Len M9),(len B))) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len M9) | (min ((Len M9),(len B)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Len M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len M9) | (len (Len M9)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (len (Len M9)) is finite len (Len M9) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Len M9) ) } is set
(Len M9) | (Seg (len (Len M9))) is Relation-like Seg (len (Len M9)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
dom (Len M9) is finite len M9 -element Element of bool NAT
Sum ((Len M9) | (len (Len M9))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Mx2Tran (P,B,B)) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
M * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Mx2Tran (P,B,B)) |^ IM is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) |^ IM) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(power K) . (M,IM) is Element of the carrier of K
((power K) . (M,IM)) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
IM + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Mx2Tran (P,B,B)) |^ (IM + 1) is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) |^ (IM + 1)) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(power K) . (M,(IM + 1)) is Element of the carrier of K
((power K) . (M,(IM + 1))) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
dom ((Mx2Tran (P,B,B)) |^ IM) is non empty set
(Mx2Tran (P,B,B)) |^ 1 is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) |^ 1) * ((Mx2Tran (P,B,B)) |^ IM) is Relation-like the carrier of V -defined the carrier of V -defined the carrier of V -valued the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
(((Mx2Tran (P,B,B)) |^ 1) * ((Mx2Tran (P,B,B)) |^ IM)) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(Mx2Tran (P,B,B)) * ((Mx2Tran (P,B,B)) |^ IM) is Relation-like the carrier of V -defined the carrier of V -defined the carrier of V -valued the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
((Mx2Tran (P,B,B)) * ((Mx2Tran (P,B,B)) |^ IM)) . (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
(Mx2Tran (P,B,B)) . (((power K) . (M,IM)) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9))))) is Element of the carrier of V
((power K) . (M,IM)) * (M * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9))))) is Element of the carrier of V
((power K) . (M,IM)) * M is Element of the carrier of K
(((power K) . (M,IM)) * M) * (B /. (Sum (Len (K,M,(min ((Len M9),(len B))),M9)))) is Element of the carrier of V
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is Element of the carrier of n
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
V is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
block_diagonal (B,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len B), Sum (Len B), the carrier of n
Len B is Relation-like NAT -defined NAT -valued Function-like finite len B -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len B) -tuples_on NAT
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len V, the carrier of n
Mx2Tran (M9,V,V) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M, the carrier of M:] is Relation-like set
bool [: the carrier of M, the carrier of M:] is set
min ((Len B),(len V)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,K,(min ((Len B),(len V))),B) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,K)
Seg (min ((Len B),(len V))) is finite min ((Len B),(len V)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len B),(len V)) ) } is set
B | (Seg (min ((Len B),(len V)))) is Relation-like Seg (min ((Len B),(len V))) -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,K,(min ((Len B),(len V))),B) is Relation-like NAT -defined NAT -valued Function-like finite len (n,K,(min ((Len B),(len V))),B) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,K,(min ((Len B),(len V))),B)) -tuples_on NAT
len (n,K,(min ((Len B),(len V))),B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,K,(min ((Len B),(len V))),B)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (n,K,(min ((Len B),(len V))),B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng V is finite set
bool the carrier of M is set
dom V is finite Element of bool NAT
V . (Sum (Len (n,K,(min ((Len B),(len V))),B))) is set
V /. (Sum (Len (n,K,(min ((Len B),(len V))),B))) is Element of the carrier of M
0. M is V65(M) Element of the carrier of M
the ZeroF of M is Element of the carrier of M
power n is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like total V18([: the carrier of n,NAT:], the carrier of n) Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non trivial non finite set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non trivial non finite set
c₁₁ is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous (n,M) Element of bool [: the carrier of M, the carrier of M:]
(n,M,c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(power n) . (K,(n,M,c₁₁)) is Element of the carrier of n
((power n) . (K,(n,M,c₁₁))) * (V /. (Sum (Len (n,K,(min ((Len B),(len V))),B)))) is Element of the carrier of M
c₁₁ |^ (n,M,c₁₁) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(c₁₁ |^ (n,M,c₁₁)) . (V /. (Sum (Len (n,K,(min ((Len B),(len V))),B)))) is Element of the carrier of M
ZeroMap (M,M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
(ZeroMap (M,M)) . (V /. (Sum (Len (n,K,(min ((Len B),(len V))),B)))) is Element of the carrier of M
the carrier of M --> (0. M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like constant non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
{(0. M)} is non empty trivial finite 1 -element set
[: the carrier of M,{(0. M)}:] is Relation-like set
( the carrier of M --> (0. M)) . (V /. (Sum (Len (n,K,(min ((Len B),(len V))),B)))) is Element of the carrier of M
(0. n) * (V /. (Sum (Len (n,K,(min ((Len B),(len V))),B)))) is Element of the carrier of M
(n,M,c₁₁) |-> K is Relation-like NAT -defined the carrier of n -valued Function-like finite (n,M,c₁₁) -element FinSequence-like FinSubsequence-like Element of (n,M,c₁₁) -tuples_on the carrier of n
(n,M,c₁₁) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
Product ((n,M,c₁₁) |-> 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 total V18([: the carrier of n, the carrier of n:], the carrier of n) 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 set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n $$ ((n,M,c₁₁) |-> K) is Element of the carrier of n
dom ((n,M,c₁₁) |-> K) is finite (n,M,c₁₁) -element Element of bool NAT
Seg (n,M,c₁₁) is finite (n,M,c₁₁) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (n,M,c₁₁) ) } is set
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((n,M,c₁₁) |-> K) . IM is set
dim M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Omega). M is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace 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 total V18([: the carrier of M, the carrier of M:], the carrier of M) 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 set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
the ZeroF of M is Element of the carrier of M
the lmult of M is Relation-like [: the carrier of n, the carrier of M:] -defined the carrier of M -valued Function-like total V18([: the carrier of n, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of n, the carrier of M:], the carrier of M:]
[: the carrier of n, the carrier of M:] is Relation-like set
[:[: the carrier of n, the carrier of M:], the carrier of M:] is Relation-like set
bool [:[: the carrier of n, the carrier of M:], the carrier of M:] is set
VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) is non empty strict VectSpStr over n
(0). M is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M
0. M is V65(M) Element of the carrier of M
{(0. M)} is non empty trivial finite 1 -element Element of bool the carrier of M
bool the carrier of M is set
(Mx2Tran (M9,V,V)) |^ 1 is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
rng ((Mx2Tran (M9,V,V)) |^ 1) is non empty set
dom ((Mx2Tran (M9,V,V)) |^ 1) is non empty set
(dom ((Mx2Tran (M9,V,V)) |^ 1)) --> (0. M) is Relation-like dom ((Mx2Tran (M9,V,V)) |^ 1) -defined the carrier of M -valued Function-like constant non empty total V18( dom ((Mx2Tran (M9,V,V)) |^ 1), the carrier of M) Element of bool [:(dom ((Mx2Tran (M9,V,V)) |^ 1)), the carrier of M:]
[:(dom ((Mx2Tran (M9,V,V)) |^ 1)), the carrier of M:] is Relation-like set
bool [:(dom ((Mx2Tran (M9,V,V)) |^ 1)), the carrier of M:] is set
{(0. M)} is non empty trivial finite 1 -element set
[:(dom ((Mx2Tran (M9,V,V)) |^ 1)),{(0. M)}:] is Relation-like set
the carrier of M --> (0. M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like constant non empty total V18( the carrier of M, the carrier of M) Element of bool [: the carrier of M, the carrier of M:]
[: the carrier of M,{(0. M)}:] is Relation-like set
ZeroMap (M,M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
dom B is finite Element of bool NAT
dom (Len B) is finite len B -element Element of bool NAT
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
B . J is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (B . J), len (B . J), the carrier of n
B . J is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (B . J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (B . J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len B) . J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Mx2Tran (M9,V,V)) |^ (Sum (Len B)) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
ZeroMap (M,M) is Relation-like the carrier of M -defined the carrier of M -valued Function-like non empty total V18( the carrier of M, the carrier of M) additive homogeneous Element of bool [: the carrier of M, the carrier of M:]
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
len M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (V,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len V), Sum (Len V), the carrier of n
Len V is Relation-like NAT -defined NAT -valued Function-like finite len V -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len V) -tuples_on NAT
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len V) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M, len M, the carrier of n
Mx2Tran (B,M,M) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len V),(len M)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (Len V) is finite len V -element Element of bool NAT
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . T), len (V . T), the carrier of n
V . T is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) . T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Seg (len M) is finite len M -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len M ) } is set
V . (min ((Len V),(len M))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . (min ((Len V),(len M)))), len (V . (min ((Len V),(len M)))), the carrier of n
V . (min ((Len V),(len M))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . (min ((Len V),(len M)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . (min ((Len V),(len M)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T - 1 is V46() ext-real V165() set
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . b1), len (V . b1), the carrier of n
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
len (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) | (len (Len V)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (len (Len V)) is finite len (Len V) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Len V) ) } is set
(Len V) | (Seg (len (Len V))) is Relation-like Seg (len (Len V)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
J is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) (n,K) Element of bool [: the carrier of K, the carrier of K:]
(n,K,J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . b1), len (V . b1), the carrier of n
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . b1), len (V . b1), the carrier of n
V . b1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) . b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Len V) /. b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
b1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) | (b1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (b1 -' 1) is finite b1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= b1 -' 1 ) } is set
(Len V) | (Seg (b1 -' 1)) is Relation-like Seg (b1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len V) | (b1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg T is finite T -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= T ) } is set
(Sum ((Len V) | (b1 -' 1))) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. ((Sum ((Len V) | (b1 -' 1))) + 1) is Element of the carrier of K
(Len V) | b1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg b1 is finite b1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= b1 ) } is set
(Len V) | (Seg b1) is Relation-like Seg b1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len V) | b1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Sum ((Len V) | (len (Len V))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M is finite Element of bool NAT
Seg (Sum ((Len V) | b1)) is finite Sum ((Len V) | b1) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum ((Len V) | b1) ) } is set
Seg (Sum (Len V)) is finite Sum (Len V) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Len V) ) } is set
b1 - 1 is V46() ext-real V165() set
(b1 -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
<*T*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is set
{1,T} is non empty finite V36() set
{{1,T},{1}} is non empty finite V36() set
{[1,T]} is Relation-like non empty trivial finite 1 -element set
((Len V) | (b1 -' 1)) ^ <*T*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sum ((Len V) | (b1 -' 1))) + T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
J |^ c₁₁ is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(J |^ c₁₁) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
(Sum ((Len V) | (b1 -' 1))) + c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum ((Len V) | (b1 -' 1))) + c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (((Sum ((Len V) | (b1 -' 1))) + c₁₁) + 1) is Element of the carrier of K
c₁₁ + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
J |^ (c₁₁ + 1) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(J |^ (c₁₁ + 1)) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
(Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)) + 1) is Element of the carrier of K
(n,(0. n),b1,V) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
V | (Seg b1) is Relation-like Seg b1 -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,(0. n),b1,V) is Relation-like NAT -defined NAT -valued Function-like finite len (n,(0. n),b1,V) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,(0. n),b1,V)) -tuples_on NAT
len (n,(0. n),b1,V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,(0. n),b1,V)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
min ((Len V),((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M /. ((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)) is Element of the carrier of K
J . (M /. ((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1))) is Element of the carrier of K
(0. n) * (M /. ((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1))) is Element of the carrier of K
((0. n) * (M /. ((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)))) + (M /. (((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)) + 1)) is Element of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(0. K) + (M /. (((Sum ((Len V) | (b1 -' 1))) + (c₁₁ + 1)) + 1)) is Element of the carrier of K
dom (J |^ c₁₁) is non empty set
J |^ 1 is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(J |^ 1) * (J |^ c₁₁) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
((J |^ 1) * (J |^ c₁₁)) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
(J |^ 1) . ((J |^ c₁₁) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1))) is Element of the carrier of K
J |^ {} is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(J |^ {}) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
(Sum ((Len V) | (b1 -' 1))) + {} is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum ((Len V) | (b1 -' 1))) + {}) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (((Sum ((Len V) | (b1 -' 1))) + {}) + 1) is Element of the carrier of K
rng M is finite set
bool the carrier of K is set
1 + (n,K,J) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(n,K,J) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Sum ((Len V) | (b1 -' 1))) + ((n,K,J) + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Sum ((Len V) | (b1 -' 1))) + (n,K,J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((Sum ((Len V) | (b1 -' 1))) + (n,K,J)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (((Sum ((Len V) | (b1 -' 1))) + (n,K,J)) + 1) is Element of the carrier of K
M . (((Sum ((Len V) | (b1 -' 1))) + (n,K,J)) + 1) is set
[#] K is non empty non proper Element of bool the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{(0. K)} is non empty trivial finite 1 -element Element of bool the carrier of K
(Omega). K is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace 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 total V18([: the carrier of K, the carrier of K:], the carrier of K) 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 set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the lmult of K is Relation-like [: the carrier of n, the carrier of K:] -defined the carrier of K -valued Function-like total V18([: the carrier of n, the carrier of K:], the carrier of K) 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 set
[:[: the carrier of n, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of n, the carrier of K:], the carrier of K:] is set
VectSpStr(# the carrier of K, the addF of K, the ZeroF of K, the lmult of K #) is non empty strict VectSpStr over n
(0). K is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of K
dim K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
J |^ (n,K,J) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
(J |^ (n,K,J)) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
(ZeroMap (K,K)) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
the carrier of K --> (0. K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like constant non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
{(0. K)} is non empty trivial finite 1 -element set
[: the carrier of K,{(0. K)}:] is Relation-like set
( the carrier of K --> (0. K)) . (M /. ((Sum ((Len V) | (b1 -' 1))) + 1)) is Element of the carrier of K
J |^ T is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) Element of bool [: the carrier of K, the carrier of K:]
ZeroMap (K,K) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . c₁₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . c₁₁), len (V . c₁₁), the carrier of n
V . c₁₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . IM is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . IM), len (V . IM), the carrier of n
V . IM is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (V . IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() VectSpStr over n
the carrier of K is non empty set
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() VectSpStr over n
the carrier of M is non empty set
[: the carrier of K, the carrier of M:] is Relation-like set
bool [: the carrier of K, the carrier of M:] is set
V is Relation-like the carrier of K -defined the carrier of M -valued Function-like non empty total V18( the carrier of K, the carrier of M) additive homogeneous Element of bool [: the carrier of K, the carrier of M:]
B is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() Subspace of K
the carrier of B is non empty set
V | B is Relation-like the carrier of B -defined the carrier of M -valued Function-like non empty total V18( the carrier of B, the carrier of M) additive homogeneous Element of bool [: the carrier of B, the carrier of M:]
[: the carrier of B, the carrier of M:] is Relation-like set
bool [: the carrier of B, the carrier of M:] is set
V | the carrier of B is Relation-like the carrier of B -defined the carrier of K -defined the carrier of M -valued set
M9 is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() Subspace of M
the carrier of M9 is non empty set
[: the carrier of B, the carrier of M9:] is Relation-like set
bool [: the carrier of B, the carrier of M9:] is set
T is Relation-like the carrier of B -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of B, the carrier of M9) Element of bool [: the carrier of B, the carrier of M9:]
the carrier of n is non empty non trivial set
J is Element of the carrier of n
b1 is Element of the carrier of B
J * b1 is Element of the carrier of B
T . (J * b1) is Element of the carrier of M9
(V | B) . b1 is Element of the carrier of M
J * ((V | B) . b1) is Element of the carrier of M
T . b1 is Element of the carrier of M9
J * (T . b1) is Element of the carrier of M9
J is Element of the carrier of B
b1 is Element of the carrier of B
J + b1 is Element of the carrier of B
T . (J + b1) is Element of the carrier of M9
(V | B) . J is Element of the carrier of M
(V | B) . b1 is Element of the carrier of M
((V | B) . J) + ((V | B) . b1) is Element of the carrier of M
T . J is Element of the carrier of M9
T . b1 is Element of the carrier of M9
(T . J) + (T . b1) is Element of the carrier of M9
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
[: the carrier of K, the carrier of M:] is Relation-like set
bool [: the carrier of K, the carrier of M:] is set
V is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is Element of the carrier of n
b1 is Relation-like the carrier of K -defined the carrier of M -valued Function-like non empty total V18( the carrier of K, the carrier of M) additive homogeneous Element of bool [: the carrier of K, the carrier of M:]
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom V & b1 . (V /. b₁) = M9 * (B /. b₁) ) } is set
card { b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom V & b1 . (V /. b₁) = M9 * (B /. b₁) ) } is V25() V26() V27() cardinal set
AutMt (b1,V,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len B, the carrier of n
J is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,M9)
P is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,M9)
block_diagonal (P,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of n
Len P is Relation-like NAT -defined NAT -valued Function-like finite len P -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len P) -tuples_on NAT
len P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len V) is finite len V -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len V ) } is set
V /. (len V) is Element of the carrier of K
b1 . (V /. (len V)) is Element of the carrier of M
B /. (len V) is Element of the carrier of M
M9 * (B /. (len V)) is Element of the carrier of M
(len V) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
V /. (len V) is Element of the carrier of K
b1 . (V /. (len V)) is Element of the carrier of M
B /. (len V) is Element of the carrier of M
M9 * (B /. (len V)) is Element of the carrier of M
B /. ((len V) + 1) is Element of the carrier of M
(M9 * (B /. (len V))) + (B /. ((len V) + 1)) is Element of the carrier of M
V /. (len V) is Element of the carrier of K
b1 . (V /. (len V)) is Element of the carrier of M
B /. (len V) is Element of the carrier of M
M9 * (B /. (len V)) is Element of the carrier of M
(len V) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
B /. ((len V) + 1) is Element of the carrier of M
(M9 * (B /. (len V))) + (B /. ((len V) + 1)) is Element of the carrier of M
len (AutMt (b1,V,B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
V is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of V is non empty set
B is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of B is non empty set
[: the carrier of V, the carrier of B:] is Relation-like set
bool [: the carrier of V, the carrier of B:] is set
M9 is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V
len M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom M9 is finite Element of bool NAT
T is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B
len T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J is Element of the carrier of n
dom T is finite Element of bool NAT
P is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) additive homogeneous Element of bool [: the carrier of V, the carrier of B:]
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom M9 & P . (M9 /. b₁) = J * (T /. b₁) ) } is set
card { b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom M9 & P . (M9 /. b₁) = J * (T /. b₁) ) } is V25() V26() V27() cardinal set
AutMt (P,M9,T) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M9, len T, the carrier of n
IM is finite set
KER is set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 /. FK is Element of the carrier of V
P . (M9 /. FK) is Element of the carrier of B
T /. FK is Element of the carrier of B
J * (T /. FK) is Element of the carrier of B
Seg (len M9) is finite len M9 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len M9 ) } is set
(len M9) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M9 /. (len M9) is Element of the carrier of V
P . (M9 /. (len M9)) is Element of the carrier of B
T /. (len M9) is Element of the carrier of B
J * (T /. (len M9)) is Element of the carrier of B
(n,J,(len M9)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len M9, len M9, the carrier of n
<*(n,J,(len M9))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
[1,(n,J,(len M9))] is set
{1,(n,J,(len M9))} is non empty finite V36() set
{{1,(n,J,(len M9))},{1}} is non empty finite V36() set
{[1,(n,J,(len M9))]} is Relation-like non empty trivial finite 1 -element set
Jk is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
dom Jk is finite Element of bool NAT
{1} is non empty trivial finite V36() 1 -element Element of bool NAT
Bker is set
Jk . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . 1), len (Jk . 1), the carrier of n
Jk . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Jk . Bker is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . Bker), len (Jk . Bker), the carrier of n
Jk . Bker is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . Bker) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Bker is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
block_diagonal (Bker,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len Bker), Sum (Len Bker), the carrier of n
Len Bker is Relation-like NAT -defined NAT -valued Function-like finite len Bker -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len Bker) -tuples_on NAT
len Bker is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len Bker) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len Bker) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M9, len T, the carrier of n
Mx2Tran (Ji,M9,T) is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) additive homogeneous Element of bool [: the carrier of V, the carrier of B:]
BB is set
(Mx2Tran (Ji,M9,T)) * M9 is Relation-like NAT -defined the carrier of B -valued Function-like finite Element of bool [:NAT, the carrier of B:]
[:NAT, the carrier of B:] is Relation-like non trivial non finite set
bool [:NAT, the carrier of B:] is non trivial non finite set
((Mx2Tran (Ji,M9,T)) * M9) . BB is set
M9 . BB is set
(Mx2Tran (Ji,M9,T)) . (M9 . BB) is set
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 /. JJ is Element of the carrier of V
M9 . JJ is set
FI is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) Element of bool [: the carrier of V, the carrier of B:]
FI * M9 is Relation-like NAT -defined the carrier of B -valued Function-like finite Element of bool [:NAT, the carrier of B:]
(FI * M9) . BB is set
P . (M9 . BB) is set
{(len M9)} is non empty trivial finite V36() 1 -element Element of bool NAT
JJ is set
{JJ} is non empty trivial finite 1 -element set
P . (M9 /. JJ) is Element of the carrier of B
T /. JJ is Element of the carrier of B
J * (T /. JJ) is Element of the carrier of B
JJ + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
T /. (JJ + 1) is Element of the carrier of B
(J * (T /. JJ)) + (T /. (JJ + 1)) is Element of the carrier of B
rng M9 is finite set
[#] V is non empty non proper Element of bool the carrier of V
bool the carrier of V is set
dom (Mx2Tran (Ji,M9,T)) is non empty set
dom ((Mx2Tran (Ji,M9,T)) * M9) is finite set
dom P is non empty set
dom (FI * M9) is finite set
card IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
{(len M9)} is non empty trivial finite V36() 1 -element Element of bool NAT
FK is set
Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 /. Jk is Element of the carrier of V
P . (M9 /. Jk) is Element of the carrier of B
T /. Jk is Element of the carrier of B
J * (T /. Jk) is Element of the carrier of B
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M9 | FK is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Seg FK is finite FK -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= FK ) } is set
M9 | (Seg FK) is Relation-like Seg FK -defined NAT -defined the carrier of V -valued finite FinSubsequence-like set
len (M9 | FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len M9) -' FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,J,((len M9) -' FK)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of (len M9) -' FK,(len M9) -' FK, the carrier of n
<*(n,J,((len M9) -' FK))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
[1,(n,J,((len M9) -' FK))] is set
{1,(n,J,((len M9) -' FK))} is non empty finite V36() set
{{1,(n,J,((len M9) -' FK))},{1}} is non empty finite V36() set
{[1,(n,J,((len M9) -' FK))]} is Relation-like non empty trivial finite 1 -element set
M9 /^ FK is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
len (M9 /^ FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len M9) - FK is V46() ext-real V165() set
T /^ FK is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len (T /^ FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T | FK is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
T | (Seg FK) is Relation-like Seg FK -defined NAT -defined the carrier of B -valued finite FinSubsequence-like set
len (T | FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
bool the carrier of B is set
rng (T | FK) is finite set
rng (T /^ FK) is finite set
bool the carrier of V is set
rng (M9 | FK) is finite set
rng (M9 /^ FK) is finite set
L1 is linearly-independent Element of bool the carrier of V
Lin L1 is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of V
JJ is linearly-independent Element of bool the carrier of B
Lin JJ is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
i is linearly-independent Element of bool the carrier of V
Lin i is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of V
JJ is linearly-independent Element of bool the carrier of B
Lin JJ is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
P .: L1 is Element of bool the carrier of B
P | (Lin L1) is Relation-like the carrier of (Lin L1) -defined the carrier of B -valued Function-like non empty total V18( the carrier of (Lin L1), the carrier of B) additive homogeneous Element of bool [: the carrier of (Lin L1), the carrier of B:]
the carrier of (Lin L1) is non empty set
[: the carrier of (Lin L1), the carrier of B:] is Relation-like set
bool [: the carrier of (Lin L1), the carrier of B:] is set
P | the carrier of (Lin L1) is Relation-like the carrier of (Lin L1) -defined the carrier of V -defined the carrier of B -valued set
rng (P | (Lin L1)) is non empty set
P .: the carrier of (Lin L1) is set
the carrier of (Lin JJ) is non empty set
K1 is Relation-like NAT -defined the carrier of (Lin L1) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin L1
dom K1 is finite Element of bool NAT
i1 is Relation-like NAT -defined the carrier of (Lin JJ) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin JJ
dom i1 is finite Element of bool NAT
the carrier of (Lin i) is non empty set
the carrier of (Lin JJ) is non empty set
M is Relation-like NAT -defined the carrier of (Lin JJ) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin JJ
i1 ^ M is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K is Relation-like NAT -defined the carrier of (Lin i) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin i
K1 ^ K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K1 /. M9 is Element of the carrier of (Lin L1)
(P | (Lin L1)) . (K1 /. M9) is Element of the carrier of B
i1 /. M9 is Element of the carrier of (Lin JJ)
J * (i1 /. M9) is Element of the carrier of (Lin JJ)
M9 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
i1 /. (M9 + 1) is Element of the carrier of (Lin JJ)
(J * (i1 /. M9)) + (i1 /. (M9 + 1)) is Element of the carrier of (Lin JJ)
K1 . M9 is set
P . (K1 /. M9) is set
T /. M9 is Element of the carrier of B
T . M9 is set
i1 . M9 is set
M9 /. M9 is Element of the carrier of V
M9 . M9 is set
P . (M9 /. M9) is Element of the carrier of B
J * (T /. M9) is Element of the carrier of B
P . (M9 /. M9) is Element of the carrier of B
J * (T /. M9) is Element of the carrier of B
T /. (M9 + 1) is Element of the carrier of B
(J * (T /. M9)) + (T /. (M9 + 1)) is Element of the carrier of B
rng T is finite set
0. B is V65(B) Element of the carrier of B
the ZeroF of B is Element of the carrier of B
(J * (T /. M9)) + (0. B) is Element of the carrier of B
fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 /. fnS is Element of the carrier of V
P . (M9 /. fnS) is Element of the carrier of B
T /. fnS is Element of the carrier of B
J * (T /. fnS) is Element of the carrier of B
ff is Basis of B
T . (M9 + 1) is set
i1 . (M9 + 1) is set
T . (M9 + 1) is set
P . (M9 /. M9) is Element of the carrier of B
J * (T /. M9) is Element of the carrier of B
T /. (M9 + 1) is Element of the carrier of B
(J * (T /. M9)) + (T /. (M9 + 1)) is Element of the carrier of B
M9 is set
dom P is non empty set
fnM is set
P . fnM is set
ff is set
K1 . ff is set
fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 /. fnS is Element of the carrier of (Lin L1)
K1 . fnS is set
(P | (Lin L1)) . (K1 /. fnS) is Element of the carrier of B
i1 /. fnS is Element of the carrier of (Lin JJ)
J * (i1 /. fnS) is Element of the carrier of (Lin JJ)
fnS + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
i1 /. (fnS + 1) is Element of the carrier of (Lin JJ)
(J * (i1 /. fnS)) + (i1 /. (fnS + 1)) is Element of the carrier of (Lin JJ)
Lin (P .: L1) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
the carrier of (Lin (P .: L1)) is non empty set
[: the carrier of (Lin L1), the carrier of (Lin JJ):] is Relation-like set
bool [: the carrier of (Lin L1), the carrier of (Lin JJ):] is set
M9 is Relation-like the carrier of (Lin L1) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin L1), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin L1), the carrier of (Lin JJ):]
fnM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K1 /. fnM is Element of the carrier of (Lin L1)
M9 . (K1 /. fnM) is Element of the carrier of (Lin JJ)
i1 /. fnM is Element of the carrier of (Lin JJ)
J * (i1 /. fnM) is Element of the carrier of (Lin JJ)
fnM + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
i1 /. (fnM + 1) is Element of the carrier of (Lin JJ)
(J * (i1 /. fnM)) + (i1 /. (fnM + 1)) is Element of the carrier of (Lin JJ)
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom K1 & M9 . (K1 /. b₁) = J * (i1 /. b₁) ) } is set
{(len M9)} is non empty trivial finite V36() 1 -element Element of bool NAT
IM \ {(len M9)} is finite Element of bool IM
bool IM is finite V36() set
ff is set
fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 /. fnS is Element of the carrier of V
P . (M9 /. fnS) is Element of the carrier of B
T /. fnS is Element of the carrier of B
J * (T /. fnS) is Element of the carrier of B
K1 /. fnS is Element of the carrier of (Lin L1)
K1 . fnS is set
M9 . fnS is set
M9 . (K1 /. fnS) is Element of the carrier of (Lin JJ)
P . (K1 /. fnS) is set
i1 . fnS is set
T . fnS is set
i1 /. fnS is Element of the carrier of (Lin JJ)
J * (i1 /. fnS) is Element of the carrier of (Lin JJ)
ff is set
fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 /. fnS is Element of the carrier of (Lin L1)
M9 . (K1 /. fnS) is Element of the carrier of (Lin JJ)
i1 /. fnS is Element of the carrier of (Lin JJ)
J * (i1 /. fnS) is Element of the carrier of (Lin JJ)
K1 . fnS is set
M9 . fnS is set
P . (K1 /. fnS) is set
M9 /. fnS is Element of the carrier of V
i1 . fnS is set
T . fnS is set
T /. fnS is Element of the carrier of B
P . (M9 /. fnS) is Element of the carrier of B
J * (T /. fnS) is Element of the carrier of B
card { b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom K1 & M9 . (K1 /. b₁) = J * (i1 /. b₁) ) } is V25() V26() V27() cardinal set
AutMt (M9,K1,i1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len K1, len i1, the carrier of n
len K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
ff is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
block_diagonal (ff,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len ff), Sum (Len ff), the carrier of n
Len ff is Relation-like NAT -defined NAT -valued Function-like finite len ff -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len ff) -tuples_on NAT
len ff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len ff) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len ff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
(n,J,ff,FI) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
Rff is set
dom (n,J,ff,FI) is finite Element of bool NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom ff is finite Element of bool NAT
ff . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (ff . i), len (ff . i), the carrier of n
ff . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (ff . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,J,ff,FI) . Rff is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,J,ff,FI) . Rff), len ((n,J,ff,FI) . Rff), the carrier of n
(n,J,ff,FI) . Rff is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,J,ff,FI) . Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom FI is finite Element of bool NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
len (n,J,((len M9) -' FK)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
F9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len ff) + F9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
F9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len ff) + F9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,J,ff,FI) . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,J,ff,FI) . i), len ((n,J,ff,FI) . i), the carrier of n
(n,J,ff,FI) . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,J,ff,FI) . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (FI . 1), len (FI . 1), the carrier of n
FI . 1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (FI . 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,J,ff,FI) . Rff is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,J,ff,FI) . Rff), len ((n,J,ff,FI) . Rff), the carrier of n
(n,J,ff,FI) . Rff is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,J,ff,FI) . Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom ff is finite Element of bool NAT
dom FI is finite Element of bool NAT
len K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Rff is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len K, len M, the carrier of n
width Rff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,J)
block_diagonal (i,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len i), Sum (Len i), the carrier of n
Len i is Relation-like NAT -defined NAT -valued Function-like finite len i -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len i) -tuples_on NAT
len i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len i) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len K1, len i1, the carrier of n
width B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*B,Rff*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding FinSequence of ( the carrier of n *) *
<*B*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,B] is set
{1,B} is non empty finite V36() set
{{1,B},{1}} is non empty finite V36() set
{[1,B]} is Relation-like non empty trivial finite 1 -element set
<*Rff*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,Rff] is set
{1,Rff} is non empty finite V36() set
{{1,Rff},{1}} is non empty finite V36() set
{[1,Rff]} is Relation-like non empty trivial finite 1 -element set
<*B*> ^ <*Rff*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
Len <*B,Rff*> is Relation-like NAT -defined NAT -valued Function-like finite len <*B,Rff*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*B,Rff*>) -tuples_on NAT
len <*B,Rff*> is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len <*B,Rff*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*B,Rff*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len Rff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len B) + (len Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width <*B,Rff*> is Relation-like NAT -defined NAT -valued Function-like finite len <*B,Rff*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*B,Rff*>) -tuples_on NAT
Sum (Width <*B,Rff*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(width B) + (width Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
block_diagonal (<*B,Rff*>,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len <*B,Rff*>), Sum (Width <*B,Rff*>), the carrier of n
BB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len M9, len T, the carrier of n
Mx2Tran (BB,M9,T) is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) additive homogeneous Element of bool [: the carrier of V, the carrier of B:]
dom K is finite Element of bool NAT
dom M is finite Element of bool NAT
x is set
(Mx2Tran (BB,M9,T)) * M9 is Relation-like NAT -defined the carrier of B -valued Function-like finite Element of bool [:NAT, the carrier of B:]
[:NAT, the carrier of B:] is Relation-like non trivial non finite set
bool [:NAT, the carrier of B:] is non trivial non finite set
((Mx2Tran (BB,M9,T)) * M9) . x is set
I is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 . I is set
(Mx2Tran (BB,M9,T)) . (M9 . I) is set
M9 /. I is Element of the carrier of V
F9 is Relation-like the carrier of V -defined the carrier of B -valued Function-like non empty total V18( the carrier of V, the carrier of B) Element of bool [: the carrier of V, the carrier of B:]
F9 * M9 is Relation-like NAT -defined the carrier of B -valued Function-like finite Element of bool [:NAT, the carrier of B:]
(F9 * M9) . x is set
P . (M9 . I) is set
K1 /. I is Element of the carrier of (Lin L1)
K1 . I is set
M9 . (K1 /. I) is Element of the carrier of (Lin JJ)
P . (K1 /. I) is set
Mx2Tran (B,K1,i1) is Relation-like the carrier of (Lin L1) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin L1), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin L1), the carrier of (Lin JJ):]
(Mx2Tran (B,K1,i1)) . (K1 /. I) is Element of the carrier of (Lin JJ)
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len K1) + j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len K1) + j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . I is set
M . j is set
T /. I is Element of the carrier of B
J * (T /. I) is Element of the carrier of B
M /. j is Element of the carrier of (Lin JJ)
J * (M /. j) is Element of the carrier of (Lin JJ)
Mx2Tran (Rff,K,M) is Relation-like the carrier of (Lin i) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin i), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin i), the carrier of (Lin JJ):]
[: the carrier of (Lin i), the carrier of (Lin JJ):] is Relation-like set
bool [: the carrier of (Lin i), the carrier of (Lin JJ):] is set
K /. j is Element of the carrier of (Lin i)
(Mx2Tran (Rff,K,M)) . (K /. j) is Element of the carrier of (Lin JJ)
T /. I is Element of the carrier of B
J * (T /. I) is Element of the carrier of B
I + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
T /. (I + 1) is Element of the carrier of B
(J * (T /. I)) + (T /. (I + 1)) is Element of the carrier of B
P . (M9 /. I) is Element of the carrier of B
(len K1) + {} is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M /. (j + 1) is Element of the carrier of (Lin JJ)
M . (j + 1) is set
(len K1) + (j + 1) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
T . ((len K1) + (j + 1)) is set
T . I is set
M . j is set
M /. j is Element of the carrier of (Lin JJ)
J * (M /. j) is Element of the carrier of (Lin JJ)
(J * (M /. j)) + (M /. (j + 1)) is Element of the carrier of (Lin JJ)
Mx2Tran (Rff,K,M) is Relation-like the carrier of (Lin i) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin i), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin i), the carrier of (Lin JJ):]
[: the carrier of (Lin i), the carrier of (Lin JJ):] is Relation-like set
bool [: the carrier of (Lin i), the carrier of (Lin JJ):] is set
K /. j is Element of the carrier of (Lin i)
(Mx2Tran (Rff,K,M)) . (K /. j) is Element of the carrier of (Lin JJ)
Mx2Tran (Rff,K,M) is Relation-like the carrier of (Lin i) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin i), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin i), the carrier of (Lin JJ):]
[: the carrier of (Lin i), the carrier of (Lin JJ):] is Relation-like set
bool [: the carrier of (Lin i), the carrier of (Lin JJ):] is set
K /. j is Element of the carrier of (Lin i)
(Mx2Tran (Rff,K,M)) . (K /. j) is Element of the carrier of (Lin JJ)
Mx2Tran (Rff,K,M) is Relation-like the carrier of (Lin i) -defined the carrier of (Lin JJ) -valued Function-like non empty total V18( the carrier of (Lin i), the carrier of (Lin JJ)) additive homogeneous Element of bool [: the carrier of (Lin i), the carrier of (Lin JJ):]
[: the carrier of (Lin i), the carrier of (Lin JJ):] is Relation-like set
bool [: the carrier of (Lin i), the carrier of (Lin JJ):] is set
K /. j is Element of the carrier of (Lin i)
(Mx2Tran (Rff,K,M)) . (K /. j) is Element of the carrier of (Lin JJ)
rng M9 is finite set
[#] V is non empty non proper Element of bool the carrier of V
dom (Mx2Tran (BB,M9,T)) is non empty set
dom ((Mx2Tran (BB,M9,T)) * M9) is finite set
dom P is non empty set
dom (F9 * M9) is finite set
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
M is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of M is non empty set
[: the carrier of K, the carrier of M:] is Relation-like set
bool [: the carrier of K, the carrier of M:] is set
V is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of M -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is Element of the carrier of n
T is Relation-like the carrier of K -defined the carrier of M -valued Function-like non empty total V18( the carrier of K, the carrier of M) additive homogeneous Element of bool [: the carrier of K, the carrier of M:]
AutMt (T,V,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len V, len B, the carrier of n
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom V & T . (V /. b₁) = M9 * (B /. b₁) ) } is set
b1 is set
P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V /. P is Element of the carrier of K
T . (V /. P) is Element of the carrier of M
B /. P is Element of the carrier of M
M9 * (B /. P) is Element of the carrier of M
b1 is finite set
card b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of P is non empty set
c₁₁ is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of c₁₁ is non empty set
[: the carrier of P, the carrier of c₁₁:] is Relation-like set
bool [: the carrier of P, the carrier of c₁₁:] is set
IM is Relation-like NAT -defined the carrier of P -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of P
len IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER is Relation-like NAT -defined the carrier of c₁₁ -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of c₁₁
len KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom IM is finite Element of bool NAT
Jk is Relation-like the carrier of P -defined the carrier of c₁₁ -valued Function-like non empty total V18( the carrier of P, the carrier of c₁₁) additive homogeneous Element of bool [: the carrier of P, the carrier of c₁₁:]
FK is Element of the carrier of n
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom IM & Jk . (IM /. b₁) = FK * (KER /. b₁) ) } is set
card { b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( b₁ in dom IM & Jk . (IM /. b₁) = FK * (KER /. b₁) ) } is V25() V26() V27() cardinal set
AutMt (Jk,IM,KER) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len IM, len KER, the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous (n,K) Element of bool [: the carrier of K, the carrier of K:]
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
B is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of B is non empty set
[: the carrier of B, the carrier of B:] is Relation-like set
bool [: the carrier of B, the carrier of B:] is set
T is Relation-like the carrier of B -defined the carrier of B -valued Function-like non empty total V18( the carrier of B, the carrier of B) additive homogeneous (n,B) Element of bool [: the carrier of B, the carrier of B:]
(n,B,T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
the Basis of B is Basis of B
Lin the Basis of B is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
the addF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like total V18([: the carrier of B, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[:[: the carrier of B, the carrier of B:], the carrier of B:] is Relation-like set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the ZeroF of B is Element of the carrier of B
the lmult of B is Relation-like [: the carrier of n, the carrier of B:] -defined the carrier of B -valued Function-like total V18([: the carrier of n, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of n, the carrier of B:], the carrier of B:]
[: the carrier of n, the carrier of B:] is Relation-like set
[:[: the carrier of n, the carrier of B:], the carrier of B:] is Relation-like set
bool [:[: the carrier of n, the carrier of B:], the carrier of B:] is set
VectSpStr(# the carrier of B, the addF of B, the ZeroF of B, the lmult of B #) is non empty strict VectSpStr over n
T |^ 1 is Relation-like the carrier of B -defined the carrier of B -valued Function-like non empty total V18( the carrier of B, the carrier of B) additive homogeneous Element of bool [: the carrier of B, the carrier of B:]
im (T |^ 1) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
the carrier of (im (T |^ 1)) is non empty set
[: the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1)):] is Relation-like set
bool [: the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1)):] is set
T | (im (T |^ 1)) is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of B -valued Function-like non empty total V18( the carrier of (im (T |^ 1)), the carrier of B) additive homogeneous Element of bool [: the carrier of (im (T |^ 1)), the carrier of B:]
[: the carrier of (im (T |^ 1)), the carrier of B:] is Relation-like set
bool [: the carrier of (im (T |^ 1)), the carrier of B:] is set
T | the carrier of (im (T |^ 1)) is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of B -defined the carrier of B -valued set
P is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of (im (T |^ 1)) -valued Function-like non empty total V18( the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1))) additive homogeneous Element of bool [: the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1)):]
c₁₁ is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of (im (T |^ 1)) -valued Function-like non empty total V18( the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1))) additive homogeneous (n, im (T |^ 1)) Element of bool [: the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1)):]
(n,(im (T |^ 1)),c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,(im (T |^ 1)),c₁₁) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER is Relation-like NAT -defined the carrier of (im (T |^ 1)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im (T |^ 1)
AutMt (c₁₁,KER,KER) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len KER, len KER, the carrier of n
len KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
IM is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (IM,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len IM), Sum (Len IM), the carrier of n
Len IM is Relation-like NAT -defined NAT -valued Function-like finite len IM -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len IM) -tuples_on NAT
len IM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len IM) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom IM is finite Element of bool NAT
len (AutMt (c₁₁,KER,KER)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom KER is finite Element of bool NAT
Seg (Sum (Len IM)) is finite Sum (Len IM) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Sum (Len IM) ) } is set
dom (Len IM) is finite len IM -element Element of bool NAT
(Sum (Len IM)) + (len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg ((Sum (Len IM)) + (len IM)) is finite (Sum (Len IM)) + (len IM) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (Sum (Len IM)) + (len IM) ) } is set
FI is set
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of NAT
1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (1 -' 1) is finite 1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= 1 -' 1 ) } is set
(Len IM) | (Seg (1 -' 1)) is Relation-like Seg (1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Bim -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (Bim -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (Bim -' 1) is finite Bim -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Bim -' 1 ) } is set
(Len IM) | (Seg (Bim -' 1)) is Relation-like Seg (Bim -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (Bim -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji -' Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Bim -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (Bim -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (Bim -' 1) is finite Bim -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Bim -' 1 ) } is set
(Len IM) | (Seg (Bim -' 1)) is Relation-like Seg (Bim -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (Bim -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji -' Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | Bim is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg Bim is finite Bim -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Bim ) } is set
(Len IM) | (Seg Bim) is Relation-like Seg Bim -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | Bim) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Sum ((Len IM) | Bim)) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
((Sum ((Len IM) | Bim)) + 1) - 1 is V46() ext-real V165() set
(Ji -' Bim) - 1 is V46() ext-real V165() set
Ji - Bim is V46() ext-real V165() set
Bim + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Ji - Bim) + Bim is V46() ext-real V165() set
(Sum (Len IM)) + Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
1 + Bim is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Ji - Bim) + Bim is V46() ext-real V165() set
Ji -' (Bim + 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji - (Bim + 1) is V46() ext-real V165() set
(Bim + 1) -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | ((Bim + 1) -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg ((Bim + 1) -' 1) is finite (Bim + 1) -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (Bim + 1) -' 1 ) } is set
(Len IM) | (Seg ((Bim + 1) -' 1)) is Relation-like Seg ((Bim + 1) -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | ((Bim + 1) -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (JJ -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (JJ -' 1) is finite JJ -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= JJ -' 1 ) } is set
(Len IM) | (Seg (JJ -' 1)) is Relation-like Seg (JJ -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (JJ -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB -' JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | JJ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg JJ is finite JJ -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= JJ ) } is set
(Len IM) | (Seg JJ) is Relation-like Seg JJ -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (JJ -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (JJ -' 1) is finite JJ -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= JJ -' 1 ) } is set
(Len IM) | (Seg (JJ -' 1)) is Relation-like Seg (JJ -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (JJ -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB -' JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
[:(Seg ((Sum (Len IM)) + (len IM))),NAT:] is Relation-like set
bool [:(Seg ((Sum (Len IM)) + (len IM))),NAT:] is set
FI is Relation-like Seg ((Sum (Len IM)) + (len IM)) -defined NAT -valued Function-like total V18( Seg ((Sum (Len IM)) + (len IM)), NAT ) finite V174() V175() V176() V177() Element of bool [:(Seg ((Sum (Len IM)) + (len IM))),NAT:]
0. B is V65(B) Element of the carrier of B
dom FI is finite set
Mx2Tran ((AutMt (c₁₁,KER,KER)),KER,KER) is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of (im (T |^ 1)) -valued Function-like non empty total V18( the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1))) additive homogeneous Element of bool [: the carrier of (im (T |^ 1)), the carrier of (im (T |^ 1)):]
Ji is set
Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI /. Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Bim -' BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | BB is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg BB is finite BB -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= BB ) } is set
(Len IM) | (Seg BB) is Relation-like Seg BB -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | BB) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (BB -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (BB -' 1) is finite BB -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= BB -' 1 ) } is set
(Len IM) | (Seg (BB -' 1)) is Relation-like Seg (BB -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (BB -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) . BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM . BB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (IM . BB), len (IM . BB), the carrier of n
IM . BB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (IM . BB) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (IM . BB) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (len (Len IM)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (len (Len IM)) is finite len (Len IM) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Len IM) ) } is set
(Len IM) | (Seg (len (Len IM))) is Relation-like Seg (len (Len IM)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (len (Len IM))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB - 1 is V46() ext-real V165() set
(BB -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
<*((Len IM) . BB)*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((Len IM) . BB)] is set
{1,((Len IM) . BB)} is non empty finite V36() set
{{1,((Len IM) . BB)},{1}} is non empty finite V36() set
{[1,((Len IM) . BB)]} is Relation-like non empty trivial finite 1 -element set
((Len IM) | (BB -' 1)) ^ <*((Len IM) . BB)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sum ((Len IM) | (BB -' 1))) + (len (IM . BB)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Bim -' BB) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER /. ((Bim -' BB) + 1) is Element of the carrier of (im (T |^ 1))
JJ is Element of the carrier of B
(T |^ 1) . JJ is Element of the carrier of B
T . JJ is set
KER . ((Bim -' BB) + 1) is set
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ -' L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (L1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (L1 -' 1) is finite L1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= L1 -' 1 ) } is set
(Len IM) | (Seg (L1 -' 1)) is Relation-like Seg (L1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (L1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(JJ -' L1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((JJ -' L1) + 1) is set
KER . (JJ -' L1) is set
min ((Len IM),(JJ -' L1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | L1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg L1 is finite L1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= L1 ) } is set
(Len IM) | (Seg L1) is Relation-like Seg L1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER /. (Bim -' BB) is Element of the carrier of (im (T |^ 1))
JJ is Element of the carrier of (im (T |^ 1))
T . JJ is set
min ((Len IM),(Bim -' BB)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (min ((Len IM),(Bim -' BB))) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (min ((Len IM),(Bim -' BB))) is finite min ((Len IM),(Bim -' BB)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len IM),(Bim -' BB)) ) } is set
(Len IM) | (Seg (min ((Len IM),(Bim -' BB)))) is Relation-like Seg (min ((Len IM),(Bim -' BB))) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (min ((Len IM),(Bim -' BB)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,(0. n),BB,IM) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
IM | (Seg BB) is Relation-like Seg BB -defined NAT -defined ( the carrier of n *) * -valued finite FinSubsequence-like Function-yielding set
Len (n,(0. n),BB,IM) is Relation-like NAT -defined NAT -valued Function-like finite len (n,(0. n),BB,IM) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (n,(0. n),BB,IM)) -tuples_on NAT
len (n,(0. n),BB,IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (n,(0. n),BB,IM)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
T . (KER /. (Bim -' BB)) is set
c₁₁ . (KER /. (Bim -' BB)) is Element of the carrier of (im (T |^ 1))
(0. n) * (KER /. (Bim -' BB)) is Element of the carrier of (im (T |^ 1))
0. (im (T |^ 1)) is V65( im (T |^ 1)) Element of the carrier of (im (T |^ 1))
the ZeroF of (im (T |^ 1)) is Element of the carrier of (im (T |^ 1))
(Bim -' BB) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER /. ((Bim -' BB) + 1) is Element of the carrier of (im (T |^ 1))
((0. n) * (KER /. (Bim -' BB))) + (KER /. ((Bim -' BB) + 1)) is Element of the carrier of (im (T |^ 1))
(0. (im (T |^ 1))) + (KER /. ((Bim -' BB) + 1)) is Element of the carrier of (im (T |^ 1))
KER . ((Bim -' BB) + 1) is set
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ -' L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (L1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (L1 -' 1) is finite L1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= L1 -' 1 ) } is set
(Len IM) | (Seg (L1 -' 1)) is Relation-like Seg (L1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (L1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(JJ -' L1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((JJ -' L1) + 1) is set
KER . (JJ -' L1) is set
min ((Len IM),(JJ -' L1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | L1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg L1 is finite L1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= L1 ) } is set
(Len IM) | (Seg L1) is Relation-like Seg L1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
[:(Seg ((Sum (Len IM)) + (len IM))), the carrier of B:] is Relation-like set
bool [:(Seg ((Sum (Len IM)) + (len IM))), the carrier of B:] is set
Ji is Relation-like Seg ((Sum (Len IM)) + (len IM)) -defined the carrier of B -valued Function-like total V18( Seg ((Sum (Len IM)) + (len IM)), the carrier of B) finite Element of bool [:(Seg ((Sum (Len IM)) + (len IM))), the carrier of B:]
rng Ji is finite set
dom Ji is finite set
Bim is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
bool the carrier of B is set
BB is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
rng BB is finite set
rng KER is finite set
bool the carrier of (im (T |^ 1)) is set
(len IM) + (Sum (Len IM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg ((len IM) + (Sum (Len IM))) is finite (len IM) + (Sum (Len IM)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (len IM) + (Sum (Len IM)) ) } is set
{ b₁ where b₁ is Element of the carrier of B : ex b₂, b₃ being V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set st
( b₂ in Seg ((len IM) + (Sum (Len IM))) & b₃ = FI . b₂ & b₁ = BB . b₂ & not b₂ -' b₃ = Sum ((Len IM) | (b₃ -' 1)) & b₂ -' b₃ = Sum ((Len IM) | b₃) ) } is set
{ b₁ where b₁ is Element of the carrier of B : ex b₂, b₃ being V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set st
( b₂ in Seg ((len IM) + (Sum (Len IM))) & b₃ = FI . b₂ & b₁ = BB . b₂ & not Sum ((Len IM) | b₃) <= b₂ -' b₃ ) } is set
j is set
w is Element of the carrier of B
W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . W is set
(Len IM) | i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg i is finite i -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
(Len IM) | (Seg i) is Relation-like Seg i -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
W -' i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is set
w is Element of the carrier of B
W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . W is set
W -' i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (i -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (i -' 1) is finite i -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i -' 1 ) } is set
(Len IM) | (Seg (i -' 1)) is Relation-like Seg (i -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (i -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg i is finite i -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
(Len IM) | (Seg i) is Relation-like Seg i -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ is Element of bool the carrier of B
W is Relation-like the carrier of B -defined the carrier of n -valued Function-like total V18( the carrier of B, the carrier of n) Linear_Combination of JJ
Sum W is Element of the carrier of B
j is Element of bool the carrier of B
T | j is Relation-like the carrier of B -defined j -defined the carrier of B -defined the carrier of B -valued Function-like Element of bool [: the carrier of B, the carrier of B:]
i is set
dom (T | j) is set
k1 is set
(T | j) . i is set
(T | j) . k1 is set
T . i is set
T . k1 is set
dom T is non empty set
(dom T) /\ j is Element of bool the carrier of B
i1 is Element of the carrier of B
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . K1 is set
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . K1 is set
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 - K is V46() ext-real V165() set
K -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K - 1 is V46() ext-real V165() set
(K -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) . (K -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M is Element of the carrier of B
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
fnM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . M9 is set
(Len IM) | fnM is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg fnM is finite fnM -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM ) } is set
(Len IM) | (Seg fnM) is Relation-like Seg fnM -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | fnM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 -' fnM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
fnM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . M9 is set
(Len IM) | fnM is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg fnM is finite fnM -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM ) } is set
(Len IM) | (Seg fnM) is Relation-like Seg fnM -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | fnM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 -' fnM is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M9 - fnM is V46() ext-real V165() set
fnM -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
fnM - 1 is V46() ext-real V165() set
(fnM -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len IM) . (fnM -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (fnM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (fnM -' 1) is finite fnM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM -' 1 ) } is set
(Len IM) | (Seg (fnM -' 1)) is Relation-like Seg (fnM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (fnM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . M is Element of the carrier of B
(M9 -' fnM) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((M9 -' fnM) + 1) is set
T . i1 is Element of the carrier of B
(K1 -' K) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((K1 -' K) + 1) is set
M9 - K is V46() ext-real V165() set
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (fnM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (fnM -' 1) is finite fnM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM -' 1 ) } is set
(Len IM) | (Seg (fnM -' 1)) is Relation-like Seg (fnM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (fnM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len IM),(M9 -' fnM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . i1 is Element of the carrier of B
(K1 -' K) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((K1 -' K) + 1) is set
T . M is Element of the carrier of B
(M9 -' fnM) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((M9 -' fnM) + 1) is set
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (fnM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (fnM -' 1) is finite fnM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM -' 1 ) } is set
(Len IM) | (Seg (fnM -' 1)) is Relation-like Seg (fnM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (fnM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len IM),(K1 -' K)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . M is Element of the carrier of B
(M9 -' fnM) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((M9 -' fnM) + 1) is set
T . i1 is Element of the carrier of B
(K1 -' K) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((K1 -' K) + 1) is set
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (fnM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (fnM -' 1) is finite fnM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM -' 1 ) } is set
(Len IM) | (Seg (fnM -' 1)) is Relation-like Seg (fnM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (fnM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len IM),(M9 -' fnM)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . M is Element of the carrier of B
(M9 -' fnM) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((M9 -' fnM) + 1) is set
T . i1 is Element of the carrier of B
(K1 -' K) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((K1 -' K) + 1) is set
M9 - K is V46() ext-real V165() set
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (fnM -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (fnM -' 1) is finite fnM -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= fnM -' 1 ) } is set
(Len IM) | (Seg (fnM -' 1)) is Relation-like Seg (fnM -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (fnM -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
w is Element of bool the carrier of B
JJ is linearly-independent Element of bool the carrier of B
i is set
k1 is Element of the carrier of B
i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . i1 is set
i1 -' K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K1 -' 1) is finite K1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 -' 1 ) } is set
(Len IM) | (Seg (K1 -' 1)) is Relation-like Seg (K1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | K1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K1 is finite K1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 ) } is set
(Len IM) | (Seg K1) is Relation-like Seg K1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . i1 is set
i1 -' K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K1 -' 1) is finite K1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 -' 1 ) } is set
(Len IM) | (Seg (K1 -' 1)) is Relation-like Seg (K1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | K1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K1 is finite K1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 ) } is set
(Len IM) | (Seg K1) is Relation-like Seg K1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER . (i1 -' K1) is set
Carrier W is finite Element of bool the carrier of B
w \/ j is Element of bool the carrier of B
i is set
dom BB is finite Element of bool NAT
k1 is Element of the carrier of B
i1 is set
BB . i1 is set
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI /. K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K - 1 is V46() ext-real V165() set
(K -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len IM) . K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
<*((Len IM) . K)*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((Len IM) . K)] is set
{1,((Len IM) . K)} is non empty finite V36() set
{{1,((Len IM) . K)},{1}} is non empty finite V36() set
{[1,((Len IM) . K)]} is Relation-like non empty trivial finite 1 -element set
((Len IM) | (K -' 1)) ^ <*((Len IM) . K)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sum ((Len IM) | (K -' 1))) + ((Len IM) . K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T .: j is Element of bool the carrier of B
i is set
dom T is non empty set
k1 is set
T . k1 is set
i1 is Element of the carrier of B
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . K1 is set
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . K1 is set
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . i1 is Element of the carrier of B
(K1 -' K) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((K1 -' K) + 1) is set
T .: w is Element of bool the carrier of B
{(0. B)} is non empty trivial finite 1 -element Element of bool the carrier of B
i is set
dom T is non empty set
k1 is set
T . k1 is set
i1 is Element of the carrier of B
T . i1 is Element of the carrier of B
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI . K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
BB . K1 is set
K1 -' K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K -' 1) is finite K -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K -' 1 ) } is set
(Len IM) | (Seg (K -' 1)) is Relation-like Seg (K -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | K is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K is finite K -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Len IM) | (Seg K) is Relation-like Seg K -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ is finite Element of bool the carrier of B
card JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 is finite Element of bool the carrier of B
card L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(card JJ) - (card L1) is V46() ext-real V165() set
i is finite Element of bool the carrier of B
card i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 \/ i is finite Element of bool the carrier of B
Lin (L1 \/ i) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
Lin JJ is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
(Omega). (Lin JJ) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of Lin JJ
the carrier of (Lin JJ) is non empty set
the addF of (Lin JJ) is Relation-like [: the carrier of (Lin JJ), the carrier of (Lin JJ):] -defined the carrier of (Lin JJ) -valued Function-like total V18([: the carrier of (Lin JJ), the carrier of (Lin JJ):], the carrier of (Lin JJ)) Element of bool [:[: the carrier of (Lin JJ), the carrier of (Lin JJ):], the carrier of (Lin JJ):]
[: the carrier of (Lin JJ), the carrier of (Lin JJ):] is Relation-like set
[:[: the carrier of (Lin JJ), the carrier of (Lin JJ):], the carrier of (Lin JJ):] is Relation-like set
bool [:[: the carrier of (Lin JJ), the carrier of (Lin JJ):], the carrier of (Lin JJ):] is set
the ZeroF of (Lin JJ) is Element of the carrier of (Lin JJ)
the lmult of (Lin JJ) is Relation-like [: the carrier of n, the carrier of (Lin JJ):] -defined the carrier of (Lin JJ) -valued Function-like total V18([: the carrier of n, the carrier of (Lin JJ):], the carrier of (Lin JJ)) Element of bool [:[: the carrier of n, the carrier of (Lin JJ):], the carrier of (Lin JJ):]
[: the carrier of n, the carrier of (Lin JJ):] is Relation-like set
[:[: the carrier of n, the carrier of (Lin JJ):], the carrier of (Lin JJ):] is Relation-like set
bool [:[: the carrier of n, the carrier of (Lin JJ):], the carrier of (Lin JJ):] is set
VectSpStr(# the carrier of (Lin JJ), the addF of (Lin JJ), the ZeroF of (Lin JJ), the lmult of (Lin JJ) #) is non empty strict VectSpStr over n
(Omega). B is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
dim B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dim (Lin JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
j + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
W is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng W is finite set
L1 \/ (rng W) is finite set
dom W is finite Element of bool NAT
W is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng W is finite set
L1 \/ (rng W) is finite set
dom W is finite Element of bool NAT
W . (j + 1) is set
T . (W . (j + 1)) is set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
W . i is set
T . (W . i) is set
W . (j + 1) is set
T . (W . (j + 1)) is set
W /. (j + 1) is Element of the carrier of B
T .: L1 is finite Element of bool the carrier of B
dom T is non empty set
[#] B is non empty non proper Element of bool the carrier of B
K1 is set
K is set
KER . K is set
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
min ((Len IM),M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (min ((Len IM),M)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (min ((Len IM),M)) is finite min ((Len IM),M) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= min ((Len IM),M) ) } is set
(Len IM) | (Seg (min ((Len IM),M))) is Relation-like Seg (min ((Len IM),M)) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (min ((Len IM),M))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(min ((Len IM),M)) + M is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((min ((Len IM),M)) + M) - 1 is V46() ext-real V165() set
((len IM) + (Sum (Len IM))) - 1 is V46() ext-real V165() set
M -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M - 1 is V46() ext-real V165() set
1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
2 - 1 is V46() ext-real V165() set
(min ((Len IM),M)) + (M -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((min ((Len IM),M)) + (M -' 1)) -' (min ((Len IM),M)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((min ((Len IM),M)) + (M -' 1)) - (min ((Len IM),M)) is V46() ext-real V165() set
((len IM) + (Sum (Len IM))) - {} is V46() ext-real non negative V165() set
FI . ((min ((Len IM),M)) + (M -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI /. ((min ((Len IM),M)) + (M -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB . ((min ((Len IM),M)) + (M -' 1)) is set
(M -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(min ((Len IM),M)) -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | ((min ((Len IM),M)) -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg ((min ((Len IM),M)) -' 1) is finite (min ((Len IM),M)) -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= (min ((Len IM),M)) -' 1 ) } is set
(Len IM) | (Seg ((min ((Len IM),M)) -' 1)) is Relation-like Seg ((min ((Len IM),M)) -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | ((min ((Len IM),M)) -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
fnS - 1 is V46() ext-real V165() set
Rff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Rff + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len IM) | Rff is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg Rff is finite Rff -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= Rff ) } is set
(Len IM) | (Seg Rff) is Relation-like Seg Rff -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((min ((Len IM),M)) + (M -' 1)) -' fnS is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
fnS -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . (BB . ((min ((Len IM),M)) + (M -' 1))) is set
(((min ((Len IM),M)) + (M -' 1)) -' (min ((Len IM),M))) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((((min ((Len IM),M)) + (M -' 1)) -' (min ((Len IM),M))) + 1) is set
im T is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
T . (W /. (j + 1)) is Element of the carrier of B
k1 is Basis of im (T |^ 1)
Lin k1 is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of im (T |^ 1)
K1 is Relation-like the carrier of (im (T |^ 1)) -defined the carrier of n -valued Function-like total V18( the carrier of (im (T |^ 1)), the carrier of n) Linear_Combination of k1
Sum K1 is Element of the carrier of (im (T |^ 1))
Carrier K1 is finite Element of bool the carrier of (im (T |^ 1))
K is Relation-like the carrier of B -defined the carrier of n -valued Function-like total V18( the carrier of B, the carrier of n) Linear_Combination of B
Carrier K is finite Element of bool the carrier of B
Sum K is Element of the carrier of B
M is Relation-like the carrier of B -defined the carrier of n -valued Function-like total V18( the carrier of B, the carrier of n) Linear_Combination of L1
Sum M is Element of the carrier of B
T . (Sum M) is Element of the carrier of B
i is finite Element of bool the carrier of B
{(W /. (j + 1))} is non empty trivial finite 1 -element Element of bool the carrier of B
i \ {(W /. (j + 1))} is finite Element of bool the carrier of B
Carrier M is finite Element of bool the carrier of B
- M is Relation-like the carrier of B -defined the carrier of n -valued Function-like total V18( the carrier of B, the carrier of n) Linear_Combination of B
Carrier (- M) is finite Element of bool the carrier of B
M9 is Relation-like the carrier of B -defined the carrier of n -valued Function-like total V18( the carrier of B, the carrier of n) Linear_Combination of i \ {(W /. (j + 1))}
Sum M9 is Element of the carrier of B
(W /. (j + 1)) + (Sum M9) is Element of the carrier of B
((W /. (j + 1)) + (Sum M9)) - (W /. (j + 1)) is Element of the carrier of B
- (W /. (j + 1)) is Element of the carrier of B
((W /. (j + 1)) + (Sum M9)) + (- (W /. (j + 1))) is Element of the carrier of B
(W /. (j + 1)) - (W /. (j + 1)) is Element of the carrier of B
(W /. (j + 1)) + (- (W /. (j + 1))) is Element of the carrier of B
(Sum M9) + ((W /. (j + 1)) - (W /. (j + 1))) is Element of the carrier of B
(Sum M9) + (0. B) is Element of the carrier of B
- (Sum M) is Element of the carrier of B
W +* ((j + 1),((W /. (j + 1)) + (Sum M9))) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
ff is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len ff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng ff is finite set
L1 \/ (rng ff) is finite set
dom ff is finite Element of bool NAT
(j + 1) .--> ((W /. (j + 1)) + (Sum M9)) is Relation-like NAT -defined {(j + 1)} -defined the carrier of B -valued Function-like one-to-one finite set
{(j + 1)} is non empty trivial finite V36() 1 -element set
{(j + 1)} --> ((W /. (j + 1)) + (Sum M9)) is Relation-like {(j + 1)} -defined the carrier of B -valued {((W /. (j + 1)) + (Sum M9))} -valued Function-like constant non empty total V18({(j + 1)},{((W /. (j + 1)) + (Sum M9))}) finite Element of bool [:{(j + 1)},{((W /. (j + 1)) + (Sum M9))}:]
{((W /. (j + 1)) + (Sum M9))} is non empty trivial finite 1 -element set
[:{(j + 1)},{((W /. (j + 1)) + (Sum M9))}:] is Relation-like finite set
bool [:{(j + 1)},{((W /. (j + 1)) + (Sum M9))}:] is finite V36() set
card i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(card i) - 1 is V46() ext-real V165() set
{((W /. (j + 1)) + (Sum M9))} is non empty trivial finite 1 -element Element of bool the carrier of B
(i \ {(W /. (j + 1))}) \/ {((W /. (j + 1)) + (Sum M9))} is non empty finite Element of bool the carrier of B
Rff is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Rff + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
card (i \ {(W /. (j + 1))}) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
card ((i \ {(W /. (j + 1))}) \/ {((W /. (j + 1)) + (Sum M9))}) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
{((W /. (j + 1)) + (Sum M9))} is non empty trivial finite 1 -element Element of bool the carrier of B
rng ((j + 1) .--> ((W /. (j + 1)) + (Sum M9))) is finite set
W +* ((j + 1) .--> ((W /. (j + 1)) + (Sum M9))) is Relation-like Function-like finite set
(rng W) \ {(W /. (j + 1))} is finite Element of bool (rng W)
bool (rng W) is finite V36() set
((rng W) \ {(W /. (j + 1))}) \/ {((W /. (j + 1)) + (Sum M9))} is non empty finite set
Rff is set
(i \ {(W /. (j + 1))}) \/ {((W /. (j + 1)) + (Sum M9))} is non empty finite Element of bool the carrier of B
L1 \ {(W /. (j + 1))} is finite Element of bool the carrier of B
(L1 \ {(W /. (j + 1))}) \/ ((rng W) \ {(W /. (j + 1))}) is finite set
((L1 \ {(W /. (j + 1))}) \/ ((rng W) \ {(W /. (j + 1))})) \/ {((W /. (j + 1)) + (Sum M9))} is non empty finite set
L1 \/ ((rng W) \ {(W /. (j + 1))}) is finite set
(L1 \/ ((rng W) \ {(W /. (j + 1))})) \/ {((W /. (j + 1)) + (Sum M9))} is non empty finite set
card i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
card (L1 \/ (rng ff)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Rff is finite Element of bool the carrier of B
Lin Rff is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
dim (Lin Rff) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Omega). (Lin Rff) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of Lin Rff
the carrier of (Lin Rff) is non empty set
the addF of (Lin Rff) is Relation-like [: the carrier of (Lin Rff), the carrier of (Lin Rff):] -defined the carrier of (Lin Rff) -valued Function-like total V18([: the carrier of (Lin Rff), the carrier of (Lin Rff):], the carrier of (Lin Rff)) Element of bool [:[: the carrier of (Lin Rff), the carrier of (Lin Rff):], the carrier of (Lin Rff):]
[: the carrier of (Lin Rff), the carrier of (Lin Rff):] is Relation-like set
[:[: the carrier of (Lin Rff), the carrier of (Lin Rff):], the carrier of (Lin Rff):] is Relation-like set
bool [:[: the carrier of (Lin Rff), the carrier of (Lin Rff):], the carrier of (Lin Rff):] is set
the ZeroF of (Lin Rff) is Element of the carrier of (Lin Rff)
the lmult of (Lin Rff) is Relation-like [: the carrier of n, the carrier of (Lin Rff):] -defined the carrier of (Lin Rff) -valued Function-like total V18([: the carrier of n, the carrier of (Lin Rff):], the carrier of (Lin Rff)) Element of bool [:[: the carrier of n, the carrier of (Lin Rff):], the carrier of (Lin Rff):]
[: the carrier of n, the carrier of (Lin Rff):] is Relation-like set
[:[: the carrier of n, the carrier of (Lin Rff):], the carrier of (Lin Rff):] is Relation-like set
bool [:[: the carrier of n, the carrier of (Lin Rff):], the carrier of (Lin Rff):] is set
VectSpStr(# the carrier of (Lin Rff), the addF of (Lin Rff), the ZeroF of (Lin Rff), the lmult of (Lin Rff) #) is non empty strict VectSpStr over n
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
ff . i is set
T . (ff . i) is set
W . i is set
(W /. (j + 1)) - (Sum M) is Element of the carrier of B
- (Sum M) is Element of the carrier of B
(W /. (j + 1)) + (- (Sum M)) is Element of the carrier of B
T . ((W /. (j + 1)) - (Sum M)) is Element of the carrier of B
(T . (W /. (j + 1))) - (T . (Sum M)) is Element of the carrier of B
- (T . (Sum M)) is Element of the carrier of B
(T . (W /. (j + 1))) + (- (T . (Sum M))) is Element of the carrier of B
W . (j + 1) is set
T . (W . (j + 1)) is set
card (L1 \/ i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(card L1) + (card i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 /\ i is finite Element of bool the carrier of B
card (L1 /\ i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
((card L1) + (card i)) - (card (L1 /\ i)) is V46() ext-real V165() set
(card JJ) - (card (L1 /\ i)) is V46() ext-real V165() set
(card (L1 \/ i)) + (card (L1 /\ i)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dim (Lin (L1 \/ i)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Seg (card i) is finite card i -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= card i ) } is set
card (Seg (card i)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is Relation-like Function-like set
dom j is set
rng j is set
w is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
W is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng W is finite set
L1 \/ (rng W) is finite set
dom W is finite Element of bool NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
W . i is set
T . (W . i) is set
j is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng j is finite set
L1 \/ (rng j) is finite set
dom j is finite Element of bool NAT
BB ^ j is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
rng (BB ^ j) is finite set
(rng BB) \/ (rng j) is finite set
w is set
dom BB is finite Element of bool NAT
W is set
BB . w is set
BB . W is set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI /. i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI . i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
k1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI /. k1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI . k1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i -' i1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (i1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (i1 -' 1) is finite i1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i1 -' 1 ) } is set
(Len IM) | (Seg (i1 -' 1)) is Relation-like Seg (i1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (i1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . (BB . w) is set
(i -' i1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((i -' i1) + 1) is set
i - i1 is V46() ext-real V165() set
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (K1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (K1 -' 1) is finite K1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 -' 1 ) } is set
(Len IM) | (Seg (K1 -' 1)) is Relation-like Seg (K1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (K1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
k1 -' K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
k1 - K1 is V46() ext-real V165() set
KER . (i -' i1) is set
min ((Len IM),(i -' i1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | i1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg i1 is finite i1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i1 ) } is set
(Len IM) | (Seg i1) is Relation-like Seg i1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | i1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Len IM) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i1 - 1 is V46() ext-real V165() set
KER . (k1 -' K1) is set
min ((Len IM),(k1 -' K1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | K1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K1 is finite K1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 ) } is set
(Len IM) | (Seg K1) is Relation-like Seg K1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
T . (BB . W) is set
(k1 -' K1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((k1 -' K1) + 1) is set
K1 - 1 is V46() ext-real V165() set
(Len IM) . (K1 -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Len IM) . (i1 -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Len IM) . (i1 -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
0. (im (T |^ 1)) is V65( im (T |^ 1)) Element of the carrier of (im (T |^ 1))
the ZeroF of (im (T |^ 1)) is Element of the carrier of (im (T |^ 1))
(Len IM) . (K1 -' 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
0. (im (T |^ 1)) is V65( im (T |^ 1)) Element of the carrier of (im (T |^ 1))
the ZeroF of (im (T |^ 1)) is Element of the carrier of (im (T |^ 1))
k1 - i1 is V46() ext-real V165() set
w is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B
dom w is finite Element of bool NAT
W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
w /. W is Element of the carrier of B
T . (w /. W) is Element of the carrier of B
(0. n) * (w /. W) is Element of the carrier of B
W + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
w /. (W + 1) is Element of the carrier of B
((0. n) * (w /. W)) + (w /. (W + 1)) is Element of the carrier of B
w . W is set
FI /. W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI . W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
W -' i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg i is finite i -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
(Len IM) | (Seg i) is Relation-like Seg i -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (i -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (i -' 1) is finite i -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= i -' 1 ) } is set
(Len IM) | (Seg (i -' 1)) is Relation-like Seg (i -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (i -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i - 1 is V46() ext-real V165() set
(i -' 1) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len IM) . i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
<*((Len IM) . i)*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((Len IM) . i)] is set
{1,((Len IM) . i)} is non empty finite V36() set
{{1,((Len IM) . i)},{1}} is non empty finite V36() set
{[1,((Len IM) . i)]} is Relation-like non empty trivial finite 1 -element set
((Len IM) | (i -' 1)) ^ <*((Len IM) . i)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sum ((Len IM) | (i -' 1))) + ((Len IM) . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB . W is set
T . (BB . W) is set
BB . W is set
T . (BB . W) is set
(W -' i) + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
KER . ((W -' i) + 1) is set
W - i is V46() ext-real V165() set
1 + {} is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(W - i) + 1 is V46() ext-real V165() set
((W - i) + 1) + i is V46() ext-real V165() set
FI /. (W + 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI . (W + 1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
1 + i is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(W + 1) -' i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(W + 1) - i is V46() ext-real V165() set
k1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
k1 -' 1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len IM) | (k1 -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg (k1 -' 1) is finite k1 -' 1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= k1 -' 1 ) } is set
(Len IM) | (Seg (k1 -' 1)) is Relation-like Seg (k1 -' 1) -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | (k1 -' 1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(W + 1) -' k1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
k1 - 1 is V46() ext-real V165() set
(W + 1) - k1 is V46() ext-real V165() set
K1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
K1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(Len IM) | K1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V174() V175() V176() V177() FinSequence of NAT
Seg K1 is finite K1 -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= K1 ) } is set
(Len IM) | (Seg K1) is Relation-like Seg K1 -defined NAT -defined NAT -valued RAT -valued finite FinSubsequence-like V174() V175() V176() V177() set
Sum ((Len IM) | K1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB . (W + 1) is set
w . (W + 1) is set
(0. B) + (w /. (W + 1)) is Element of the carrier of B
len BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len BB) + i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len BB) + i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j . i is set
T . (j . i) is set
len BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
AutMt (T,w,w) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len w, len w, the carrier of n
len w is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
W is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (W,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len W), Sum (Len W), the carrier of n
Len W is Relation-like NAT -defined NAT -valued Function-like finite len W -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len W) -tuples_on NAT
len W is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len W) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len W) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom (Len W) is finite len W -element Element of bool NAT
dom W is finite Element of bool NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
W . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (W . i), len (W . i), the carrier of n
W . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (W . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len W) . i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
B is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of B is non empty set
[: the carrier of B, the carrier of B:] is Relation-like set
bool [: the carrier of B, the carrier of B:] is set
the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B
T is Relation-like the carrier of B -defined the carrier of B -valued Function-like non empty total V18( the carrier of B, the carrier of B) additive homogeneous (n,B) Element of bool [: the carrier of B, the carrier of B:]
(n,B,T) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[#] B is non empty non proper Element of bool the carrier of B
bool the carrier of B is set
0. B is V65(B) Element of the carrier of B
the ZeroF of B is Element of the carrier of B
{(0. B)} is non empty trivial finite 1 -element Element of bool the carrier of B
(0). B is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
the carrier of ((0). B) is non empty set
(Omega). B is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of B
the addF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like total V18([: the carrier of B, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[:[: the carrier of B, the carrier of B:], the carrier of B:] is Relation-like set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the lmult of B is Relation-like [: the carrier of n, the carrier of B:] -defined the carrier of B -valued Function-like total V18([: the carrier of n, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of n, the carrier of B:], the carrier of B:]
[: the carrier of n, the carrier of B:] is Relation-like set
[:[: the carrier of n, the carrier of B:], the carrier of B:] is Relation-like set
bool [:[: the carrier of n, the carrier of B:], the carrier of B:] is set
VectSpStr(# the carrier of B, the addF of B, the ZeroF of B, the lmult of B #) is non empty strict VectSpStr over n
dim B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
len the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
AutMt (T, the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B, the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B, len the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B, the carrier of n
len (AutMt (T, the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B, the Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of B)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (V,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len V), Sum (Len V), the carrier of n
Len V is Relation-like NAT -defined NAT -valued Function-like finite len V -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len V) -tuples_on NAT
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len V) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(Len V) . J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,K,M) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V is non empty right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of V is non empty set
[: the carrier of V, the carrier of V:] is Relation-like set
bool [: the carrier of V, the carrier of V:] is set
B is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous (n,V) Element of bool [: the carrier of V, the carrier of V:]
(n,V,B) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
B is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of K
AutMt (M,B,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len B, len B, the carrier of n
len B is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
V is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n, 0. n)
block_diagonal (V,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len V), Sum (Len V), the carrier of n
Len V is Relation-like NAT -defined NAT -valued Function-like finite len V -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len V) -tuples_on NAT
len V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len V) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len V) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom V is finite Element of bool NAT
M9 is set
dom (Len V) is finite len V -element Element of bool NAT
T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(Len V) . T is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (V . M9), len (V . M9), the carrier of n
V . M9 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (V . M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
0. K is V65(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
M is Element of the carrier of K
- M is Element of the carrier of K
V is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() VectSpStr over K
the carrier of V is non empty set
[: the carrier of V, the carrier of V:] is Relation-like set
bool [: the carrier of V, the carrier of V:] is set
id V is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
id the carrier of V is Relation-like the carrier of V -defined the carrier of V -valued non empty total V18( the carrier of V, the carrier of V) Element of bool [: the carrier of V, the carrier of V:]
(- M) * (id V) is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
B is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
B + ((- M) * (id V)) is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
(B + ((- M) * (id V))) |^ n is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
ker ((B + ((- M) * (id V))) |^ n) is non empty right_complementable strict V145(K) V146(K) V147(K) V148(K) V152() V153() V154() Subspace of V
M9 is non empty right_complementable V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional Subspace of V
the carrier of M9 is non empty set
[: the carrier of M9, the carrier of M9:] is Relation-like set
bool [: the carrier of M9, the carrier of M9:] is set
B | M9 is Relation-like the carrier of M9 -defined the carrier of V -valued Function-like non empty total V18( the carrier of M9, the carrier of V) additive homogeneous Element of bool [: the carrier of M9, the carrier of V:]
[: the carrier of M9, the carrier of V:] is Relation-like set
bool [: the carrier of M9, the carrier of V:] is set
B | the carrier of M9 is Relation-like the carrier of M9 -defined the carrier of V -defined the carrier of V -valued set
T is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
(B + ((- M) * (id V))) | M9 is Relation-like the carrier of M9 -defined the carrier of V -valued Function-like non empty total V18( the carrier of M9, the carrier of V) additive homogeneous Element of bool [: the carrier of M9, the carrier of V:]
(B + ((- M) * (id V))) | the carrier of M9 is Relation-like the carrier of M9 -defined the carrier of V -defined the carrier of V -valued set
P is set
b1 is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous (K,M9) Element of bool [: the carrier of M9, the carrier of M9:]
dom b1 is non empty set
c₁₁ is Element of the carrier of M9
IM is Element of the carrier of V
(id V) . IM is Element of the carrier of V
(- M) * ((id V) . IM) is Element of the carrier of V
id M9 is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
id the carrier of M9 is Relation-like the carrier of M9 -defined the carrier of M9 -valued non empty total V18( the carrier of M9, the carrier of M9) Element of bool [: the carrier of M9, the carrier of M9:]
(id M9) . c₁₁ is Element of the carrier of M9
(- M) * ((id M9) . c₁₁) is Element of the carrier of M9
B . c₁₁ is set
T . c₁₁ is Element of the carrier of M9
b1 . P is set
(B + ((- M) * (id V))) . IM is Element of the carrier of V
B . IM is Element of the carrier of V
((- M) * (id V)) . IM is Element of the carrier of V
(B . IM) + (((- M) * (id V)) . IM) is Element of the carrier of V
(B . IM) + ((- M) * ((id V) . IM)) is Element of the carrier of V
(T . c₁₁) + ((- M) * ((id M9) . c₁₁)) is Element of the carrier of M9
(- M) * (id M9) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
((- M) * (id M9)) . c₁₁ is Element of the carrier of M9
(T . c₁₁) + (((- M) * (id M9)) . c₁₁) is Element of the carrier of M9
T + ((- M) * (id M9)) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
(T + ((- M) * (id M9))) . P is set
dom (T + ((- M) * (id M9))) is non empty set
c₁₁ is Relation-like NAT -defined the carrier of M9 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M9
AutMt (b1,c₁₁,c₁₁) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
len c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P is Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) (K, 0. K)
block_diagonal (P,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
Len P is Relation-like NAT -defined NAT -valued Function-like finite len P -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len P) -tuples_on NAT
len P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len P) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
M + (0. K) is Element of the carrier of K
1. (K,(Len P)) is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of K *) *
(len P) |-> M is Relation-like NAT -defined the carrier of K -valued Function-like finite len P -element FinSequence-like FinSubsequence-like Element of (len P) -tuples_on the carrier of K
(len P) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
mlt (((len P) |-> M),(1. (K,(Len P)))) is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of K *) *
P (+) (mlt (((len P) |-> M),(1. (K,(Len P))))) is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of K *) *
KER is set
IM is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) (K,M)
dom IM is finite Element of bool NAT
FK is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
IM . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (IM . FK), len (IM . FK), the carrier of K
IM . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (IM . FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (P . FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
P . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (P . FK), len (P . FK), the carrier of K
(mlt (((len P) |-> M),(1. (K,(Len P))))) . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((mlt (((len P) |-> M),(1. (K,(Len P))))) . FK), len ((mlt (((len P) |-> M),(1. (K,(Len P))))) . FK), the carrier of K
(mlt (((len P) |-> M),(1. (K,(Len P))))) . FK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len ((mlt (((len P) |-> M),(1. (K,(Len P))))) . FK) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(P . FK) + ((mlt (((len P) |-> M),(1. (K,(Len P))))) . FK) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (P . FK), len (P . FK), the carrier of K
dom P is finite Element of bool NAT
IM . KER is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (IM . KER), len (IM . KER), the carrier of K
IM . KER is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
len (IM . KER) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
KER is Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) (K,M)
block_diagonal (KER,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len KER), Sum (Len KER), the carrier of K
Len KER is Relation-like NAT -defined NAT -valued Function-like finite len KER -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len KER) -tuples_on NAT
len KER is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len KER) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len KER) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
AutMt (T,c₁₁,c₁₁) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
AutMt ((id M9),c₁₁,c₁₁) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
Indices (AutMt ((id M9),c₁₁,c₁₁)) is set
dom (AutMt ((id M9),c₁₁,c₁₁)) is finite Element of bool NAT
width (AutMt ((id M9),c₁₁,c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (AutMt ((id M9),c₁₁,c₁₁))) is finite width (AutMt ((id M9),c₁₁,c₁₁)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (AutMt ((id M9),c₁₁,c₁₁)) ) } is set
[:(dom (AutMt ((id M9),c₁₁,c₁₁))),(Seg (width (AutMt ((id M9),c₁₁,c₁₁)))):] is Relation-like finite set
Indices (AutMt (T,c₁₁,c₁₁)) is set
dom (AutMt (T,c₁₁,c₁₁)) is finite Element of bool NAT
width (AutMt (T,c₁₁,c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (AutMt (T,c₁₁,c₁₁))) is finite width (AutMt (T,c₁₁,c₁₁)) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (AutMt (T,c₁₁,c₁₁)) ) } is set
[:(dom (AutMt (T,c₁₁,c₁₁))),(Seg (width (AutMt (T,c₁₁,c₁₁)))):] is Relation-like finite set
Bker is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
FI is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
[Bker,FI] is set
{Bker,FI} is non empty finite V36() set
{Bker} is non empty trivial finite V36() 1 -element set
{{Bker,FI},{Bker}} is non empty finite V36() set
(- M) * (AutMt ((id M9),c₁₁,c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
(AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
Indices ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) is set
dom ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) is finite Element of bool NAT
width ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁))))) is finite width ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) ) } is set
[:(dom ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁))))),(Seg (width ((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))))):] is Relation-like finite set
M * (AutMt ((id M9),c₁₁,c₁₁)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) + (M * (AutMt ((id M9),c₁₁,c₁₁))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
(((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) + (M * (AutMt ((id M9),c₁₁,c₁₁)))) * (Bker,FI) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) * (Bker,FI) is Element of the carrier of K
(M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI) is Element of the carrier of K
(((AutMt (T,c₁₁,c₁₁)) + ((- M) * (AutMt ((id M9),c₁₁,c₁₁)))) * (Bker,FI)) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) is Element of the carrier of K
(AutMt (T,c₁₁,c₁₁)) * (Bker,FI) is Element of the carrier of K
((- M) * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (((- M) * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) is Element of the carrier of K
(((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (((- M) * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI))) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) is Element of the carrier of K
(((- M) * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + ((((- M) * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI))) is Element of the carrier of K
(AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI) is Element of the carrier of K
(- M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI)) is Element of the carrier of K
((- M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI)) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (((- M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) + ((M * (AutMt ((id M9),c₁₁,c₁₁))) * (Bker,FI))) is Element of the carrier of K
M * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI)) is Element of the carrier of K
((- M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) + (M * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (((- M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) + (M * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI)))) is Element of the carrier of K
(- M) + M is Element of the carrier of K
((- M) + M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI)) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (((- M) + M) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) is Element of the carrier of K
(0. K) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI)) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + ((0. K) * ((AutMt ((id M9),c₁₁,c₁₁)) * (Bker,FI))) is Element of the carrier of K
((AutMt (T,c₁₁,c₁₁)) * (Bker,FI)) + (0. K) is Element of the carrier of K
Len (mlt (((len P) |-> M),(1. (K,(Len P))))) is Relation-like NAT -defined NAT -valued Function-like finite len (mlt (((len P) |-> M),(1. (K,(Len P))))) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (mlt (((len P) |-> M),(1. (K,(Len P)))))) -tuples_on NAT
len (mlt (((len P) |-> M),(1. (K,(Len P))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (mlt (((len P) |-> M),(1. (K,(Len P)))))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Len (1. (K,(Len P))) is Relation-like NAT -defined NAT -valued Function-like finite len (1. (K,(Len P))) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (1. (K,(Len P)))) -tuples_on NAT
len (1. (K,(Len P))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len (1. (K,(Len P)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
dom (Len P) is finite len P -element Element of bool NAT
dom (1. (K,(Len P))) is finite Element of bool NAT
len (Len P) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (AutMt (b1,c₁₁,c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Width (mlt (((len P) |-> M),(1. (K,(Len P))))) is Relation-like NAT -defined NAT -valued Function-like finite len (mlt (((len P) |-> M),(1. (K,(Len P))))) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (mlt (((len P) |-> M),(1. (K,(Len P)))))) -tuples_on NAT
Width (1. (K,(Len P))) is Relation-like NAT -defined NAT -valued Function-like finite len (1. (K,(Len P))) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (1. (K,(Len P)))) -tuples_on NAT
Width P is Relation-like NAT -defined NAT -valued Function-like finite len P -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len P) -tuples_on NAT
AutMt (((- M) * (id M9)),c₁₁,c₁₁) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
Mx2Tran ((AutMt (((- M) * (id M9)),c₁₁,c₁₁)),c₁₁,c₁₁) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
Mx2Tran ((AutMt ((id M9),c₁₁,c₁₁)),c₁₁,c₁₁) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
(- M) * (Mx2Tran ((AutMt ((id M9),c₁₁,c₁₁)),c₁₁,c₁₁)) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
Mx2Tran (((- M) * (AutMt ((id M9),c₁₁,c₁₁))),c₁₁,c₁₁) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
(AutMt (b1,c₁₁,c₁₁)) + (M * (AutMt ((id M9),c₁₁,c₁₁))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len c₁₁, len c₁₁, the carrier of K
1. (K,(Sum (Len P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
M * (1. (K,(Sum (Len P)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
(block_diagonal (P,(0. K))) + (M * (1. (K,(Sum (Len P))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
Sum (Len (1. (K,(Len P)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
block_diagonal ((1. (K,(Len P))),(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len (1. (K,(Len P)))), Sum (Len (1. (K,(Len P)))), the carrier of K
M * (block_diagonal ((1. (K,(Len P))),(0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len (1. (K,(Len P)))), Sum (Len (1. (K,(Len P)))), the carrier of K
(block_diagonal (P,(0. K))) + (M * (block_diagonal ((1. (K,(Len P))),(0. K)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
M * (0. K) is Element of the carrier of K
block_diagonal ((mlt (((len P) |-> M),(1. (K,(Len P))))),(M * (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len (mlt (((len P) |-> M),(1. (K,(Len P)))))), Sum (Len (mlt (((len P) |-> M),(1. (K,(Len P)))))), the carrier of K
Sum (Len (mlt (((len P) |-> M),(1. (K,(Len P)))))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(block_diagonal (P,(0. K))) + (block_diagonal ((mlt (((len P) |-> M),(1. (K,(Len P))))),(M * (0. K)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
block_diagonal ((mlt (((len P) |-> M),(1. (K,(Len P))))),(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len (mlt (((len P) |-> M),(1. (K,(Len P)))))), Sum (Len (mlt (((len P) |-> M),(1. (K,(Len P)))))), the carrier of K
(block_diagonal (P,(0. K))) + (block_diagonal ((mlt (((len P) |-> M),(1. (K,(Len P))))),(0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len P), Sum (Len P), the carrier of K
(0. K) + (0. K) is Element of the carrier of K
block_diagonal (KER,((0. K) + (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len KER), Sum (Len KER), the carrier of K
n is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() algebraic-closed doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
( the carrier of n *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n *
0. n is V65(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
K is non empty non trivial right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of K is non empty non trivial set
[: the carrier of K, the carrier of K:] is Relation-like set
bool [: the carrier of K, the carrier of K:] is set
M is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total V18( the carrier of K, the carrier of K) additive homogeneous Element of bool [: the carrier of K, the carrier of K:]
V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
M9 is non empty non trivial right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
dim M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
the carrier of M9 is non empty non trivial set
[: the carrier of M9, the carrier of M9:] is Relation-like set
bool [: the carrier of M9, the carrier of M9:] is set
T is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
T is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
0. M9 is V65(M9) Element of the carrier of M9
the ZeroF of M9 is Element of the carrier of M9
J is Element of the carrier of M9
T . J is Element of the carrier of M9
b1 is Element of the carrier of n
b1 * J is Element of the carrier of M9
id M9 is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
id the carrier of M9 is Relation-like the carrier of M9 -defined the carrier of M9 -valued non empty total V18( the carrier of M9, the carrier of M9) Element of bool [: the carrier of M9, the carrier of M9:]
- b1 is Element of the carrier of n
(- b1) * (id M9) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
T + ((- b1) * (id M9)) is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
ker (T + ((- b1) * (id M9))) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
dim (ker (T + ((- b1) * (id M9)))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
UnionKers (T + ((- b1) * (id M9))) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
c₁₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(T + ((- b1) * (id M9))) |^ c₁₁ is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
ker ((T + ((- b1) * (id M9))) |^ c₁₁) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
im ((T + ((- b1) * (id M9))) |^ c₁₁) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
dim (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dim (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(dim (ker ((T + ((- b1) * (id M9))) |^ c₁₁))) + (dim (im ((T + ((- b1) * (id M9))) |^ c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) is non empty set
[: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)):] is Relation-like set
bool [: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)):] is set
T | (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) is Relation-like the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9) additive homogeneous Element of bool [: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:]
[: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:] is Relation-like set
bool [: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:] is set
T | the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) is Relation-like the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of M9 -defined the carrier of M9 -valued set
FK is Relation-like the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like non empty total V18( the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁))) additive homogeneous Element of bool [: the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)):]
Bker is Relation-like NAT -defined the carrier of (ker ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of ker ((T + ((- b1) * (id M9))) |^ c₁₁)
AutMt (FK,Bker,Bker) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len Bker, len Bker, the carrier of n
len Bker is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Jk is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) (n,b1)
block_diagonal (Jk,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len Jk), Sum (Len Jk), the carrier of n
Len Jk is Relation-like NAT -defined NAT -valued Function-like finite len Jk -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len Jk) -tuples_on NAT
len Jk is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len Jk) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len Jk) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(T + ((- b1) * (id M9))) |^ 1 is Relation-like the carrier of M9 -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of M9, the carrier of M9) additive homogeneous Element of bool [: the carrier of M9, the carrier of M9:]
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of NAT
len (block_diagonal (Jk,(0. n))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is non empty set
[: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)):] is Relation-like set
bool [: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)):] is set
T | (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is Relation-like the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of M9 -valued Function-like non empty total V18( the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9) additive homogeneous Element of bool [: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:]
[: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:] is Relation-like set
bool [: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of M9:] is set
T | the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is Relation-like the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of M9 -defined the carrier of M9 -valued set
(ker ((T + ((- b1) * (id M9))) |^ c₁₁)) /\ (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
(0). M9 is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
(ker ((T + ((- b1) * (id M9))) |^ c₁₁)) + (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
(Omega). M9 is non empty right_complementable strict V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional Subspace of M9
the addF of M9 is Relation-like [: the carrier of M9, the carrier of M9:] -defined the carrier of M9 -valued Function-like total V18([: the carrier of M9, the carrier of M9:], the carrier of M9) Element of bool [:[: the carrier of M9, the carrier of M9:], the carrier of M9:]
[:[: the carrier of M9, the carrier of M9:], the carrier of M9:] is Relation-like set
bool [:[: the carrier of M9, the carrier of M9:], the carrier of M9:] is set
the lmult of M9 is Relation-like [: the carrier of n, the carrier of M9:] -defined the carrier of M9 -valued Function-like total V18([: the carrier of n, the carrier of M9:], the carrier of M9) Element of bool [:[: the carrier of n, the carrier of M9:], the carrier of M9:]
[: the carrier of n, the carrier of M9:] is Relation-like set
[:[: the carrier of n, the carrier of M9:], the carrier of M9:] is Relation-like set
bool [:[: the carrier of n, the carrier of M9:], the carrier of M9:] is set
VectSpStr(# the carrier of M9, the addF of M9, the ZeroF of M9, the lmult of M9 #) is non empty strict VectSpStr over n
the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁) is Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)
len the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
FI is Relation-like the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like non empty total V18( the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))) additive homogeneous Element of bool [: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)):]
AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), len the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the carrier of n
len (AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁))) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Bker ^ the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
the carrier of ((ker ((T + ((- b1) * (id M9))) |^ c₁₁)) + (im ((T + ((- b1) * (id M9))) |^ c₁₁))) is non empty set
dom Jk is finite Element of bool NAT
Bim is Relation-like NAT -defined the carrier of M9 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M9
AutMt (T,Bim,Bim) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len Bim, len Bim, the carrier of n
len Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
<*(AutMt (FK,Bker,Bker))*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(AutMt (FK,Bker,Bker))] is set
{1,(AutMt (FK,Bker,Bker))} is non empty finite V36() set
{{1,(AutMt (FK,Bker,Bker))},{1}} is non empty finite V36() set
{[1,(AutMt (FK,Bker,Bker))]} is Relation-like non empty trivial finite 1 -element set
<*(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))] is set
{1,(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))} is non empty finite V36() set
{{1,(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))},{1}} is non empty finite V36() set
{[1,(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))]} is Relation-like non empty trivial finite 1 -element set
<*(AutMt (FK,Bker,Bker))*> ^ <*(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
block_diagonal (<*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>), Sum (Len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>), the carrier of n
Len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> is Relation-like NAT -defined NAT -valued Function-like finite len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>) -tuples_on NAT
len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*> is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*(AutMt (FK,Bker,Bker)),(AutMt (FI, the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁), the Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)))*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*(AutMt (FK,Bker,Bker))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
block_diagonal (<*(AutMt (FK,Bker,Bker))*>,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len <*(AutMt (FK,Bker,Bker))*>), Sum (Len <*(AutMt (FK,Bker,Bker))*>), the carrier of n
Len <*(AutMt (FK,Bker,Bker))*> is Relation-like NAT -defined NAT -valued Function-like finite len <*(AutMt (FK,Bker,Bker))*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*(AutMt (FK,Bker,Bker))*>) -tuples_on NAT
len <*(AutMt (FK,Bker,Bker))*> is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len <*(AutMt (FK,Bker,Bker))*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*(AutMt (FK,Bker,Bker))*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
BB is Element of the carrier of n
0. (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is V65( im ((T + ((- b1) * (id M9))) |^ c₁₁)) Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
the ZeroF of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
JJ is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
FI . JJ is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
BB * JJ is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Jk . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . JJ), len (Jk . JJ), the carrier of n
Jk . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Jk . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,BB,(len (Jk . JJ))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (Jk . JJ), len (Jk . JJ), the carrier of n
Seg (len Jk) is finite len Jk -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len Jk ) } is set
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,b1,JJ) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of JJ,JJ, the carrier of n
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Jk . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . JJ), len (Jk . JJ), the carrier of n
Jk . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Jk . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,BB,(len (Jk . JJ))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (Jk . JJ), len (Jk . JJ), the carrier of n
Seg (len (Jk . JJ)) is finite len (Jk . JJ) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Jk . JJ) ) } is set
[(len (Jk . JJ)),(len (Jk . JJ))] is Element of [:NAT,NAT:]
{(len (Jk . JJ)),(len (Jk . JJ))} is non empty finite V36() set
{(len (Jk . JJ))} is non empty trivial finite V36() 1 -element set
{{(len (Jk . JJ)),(len (Jk . JJ))},{(len (Jk . JJ))}} is non empty finite V36() set
[:(Seg (len (Jk . JJ))),(Seg (len (Jk . JJ))):] is Relation-like finite set
Indices (Jk . JJ) is set
dom (Jk . JJ) is finite Element of bool NAT
width (Jk . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (Jk . JJ)) is finite width (Jk . JJ) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (Jk . JJ) ) } is set
[:(dom (Jk . JJ)),(Seg (width (Jk . JJ))):] is Relation-like finite set
(Jk . JJ) * ((len (Jk . JJ)),(len (Jk . JJ))) is Element of the carrier of n
JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,b1,JJ) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of JJ,JJ, the carrier of n
FI is Relation-like the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like non empty total V18( the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))) additive homogeneous Element of bool [: the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)), the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)):]
Bim is Relation-like NAT -defined the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of im ((T + ((- b1) * (id M9))) |^ c₁₁)
AutMt (FI,Bim,Bim) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len Bim, len Bim, the carrier of n
len Bim is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
block_diagonal (Ji,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len Ji), Sum (Len Ji), the carrier of n
Len Ji is Relation-like NAT -defined NAT -valued Function-like finite len Ji -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len Ji) -tuples_on NAT
len Ji is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len Ji) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len Ji) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom Ji is finite Element of bool NAT
Bker ^ Bim is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
the carrier of ((ker ((T + ((- b1) * (id M9))) |^ c₁₁)) + (im ((T + ((- b1) * (id M9))) |^ c₁₁))) is non empty set
(n,Jk,Ji) is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
JJ is set
dom (n,Jk,Ji) is finite Element of bool NAT
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom Jk is finite Element of bool NAT
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,Jk,Ji) . L1), len ((n,Jk,Ji) . L1), the carrier of n
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,Jk,Ji) . L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Jk . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . L1), len (Jk . L1), the carrier of n
Jk . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len Jk) + i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len Jk) + i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,Jk,Ji) . L1), len ((n,Jk,Ji) . L1), the carrier of n
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,Jk,Ji) . L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Ji . i), len (Ji . i), the carrier of n
Ji . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Ji . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
dom Jk is finite Element of bool NAT
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,Jk,Ji) . L1), len ((n,Jk,Ji) . L1), the carrier of n
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,Jk,Ji) . L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,Jk,Ji) . L1), len ((n,Jk,Ji) . L1), the carrier of n
(n,Jk,Ji) . L1 is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,Jk,Ji) . L1) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,Jk,Ji) . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len ((n,Jk,Ji) . JJ), len ((n,Jk,Ji) . JJ), the carrier of n
(n,Jk,Ji) . JJ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len ((n,Jk,Ji) . JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ is Relation-like non-empty NAT -defined ( the carrier of n *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (n) FinSequence of ( the carrier of n *) *
block_diagonal (JJ,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len JJ), Sum (Len JJ), the carrier of n
Len JJ is Relation-like NAT -defined NAT -valued Function-like finite len JJ -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len JJ) -tuples_on NAT
len JJ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len JJ) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len JJ) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom JJ is finite Element of bool NAT
BB is Relation-like NAT -defined the carrier of M9 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of M9
AutMt (T,BB,BB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len BB, len BB, the carrier of n
len BB is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
<*(block_diagonal (Jk,(0. n)))*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(block_diagonal (Jk,(0. n)))] is set
{1,(block_diagonal (Jk,(0. n)))} is non empty finite V36() set
{{1,(block_diagonal (Jk,(0. n)))},{1}} is non empty finite V36() set
{[1,(block_diagonal (Jk,(0. n)))]} is Relation-like non empty trivial finite 1 -element set
<*(block_diagonal (Ji,(0. n)))*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(block_diagonal (Ji,(0. n)))] is set
{1,(block_diagonal (Ji,(0. n)))} is non empty finite V36() set
{{1,(block_diagonal (Ji,(0. n)))},{1}} is non empty finite V36() set
{[1,(block_diagonal (Ji,(0. n)))]} is Relation-like non empty trivial finite 1 -element set
<*(block_diagonal (Jk,(0. n)))*> ^ <*(block_diagonal (Ji,(0. n)))*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
block_diagonal (<*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>,(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>), Sum (Len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>), the carrier of n
Len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*> is Relation-like NAT -defined NAT -valued Function-like finite len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*> -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>) -tuples_on NAT
len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*> is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*(block_diagonal (Jk,(0. n))),(block_diagonal (Ji,(0. n)))*>) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
<*(block_diagonal (Jk,(0. n)))*> is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
<*(block_diagonal (Jk,(0. n)))*> ^ Ji is Relation-like NAT -defined ( the carrier of n *) * -valued Function-like non empty finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of n *) *
block_diagonal ((<*(block_diagonal (Jk,(0. n)))*> ^ Ji),(0. n)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji)), Sum (Len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji)), the carrier of n
Len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji) is Relation-like NAT -defined NAT -valued Function-like finite len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji) -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji)) -tuples_on NAT
len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji) is non empty V25() V26() V27() V31() finite cardinal V46() ext-real positive non negative V165() Element of NAT
(len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len (<*(block_diagonal (Jk,(0. n)))*> ^ Ji)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
L1 is Element of the carrier of n
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (JJ . i), len (JJ . i), the carrier of n
JJ . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (JJ . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (JJ . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,L1,(len (JJ . i))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (JJ . i), len (JJ . i), the carrier of n
dom Jk is finite Element of bool NAT
Seg (len Jk) is finite len Jk -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len Jk ) } is set
Jk . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . i), len (Jk . i), the carrier of n
Jk . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,b1,j) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of j,j, the carrier of n
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
Ji . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Ji . i), len (Ji . i), the carrier of n
Ji . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Ji . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Ji . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,L1,(len (Ji . i))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (Ji . i), len (Ji . i), the carrier of n
(len Jk) + i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
JJ . j is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (JJ . j), len (JJ . j), the carrier of n
JJ . j is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (JJ . j) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (JJ . j) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,L1,(len (JJ . j))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (JJ . j), len (JJ . j), the carrier of n
i is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
JJ . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (JJ . i), len (JJ . i), the carrier of n
JJ . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (JJ . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (JJ . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(n,L1,(len (JJ . i))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of len (JJ . i), len (JJ . i), the carrier of n
dom Jk is finite Element of bool NAT
Jk . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Jk . i), len (Jk . i), the carrier of n
Jk . i is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Jk . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (Jk . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (len (Jk . i)) is finite len (Jk . i) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= len (Jk . i) ) } is set
[(len (Jk . i)),(len (Jk . i))] is Element of [:NAT,NAT:]
{(len (Jk . i)),(len (Jk . i))} is non empty finite V36() set
{(len (Jk . i))} is non empty trivial finite V36() 1 -element set
{{(len (Jk . i)),(len (Jk . i))},{(len (Jk . i))}} is non empty finite V36() set
[:(Seg (len (Jk . i))),(Seg (len (Jk . i))):] is Relation-like finite set
Indices (Jk . i) is set
dom (Jk . i) is finite Element of bool NAT
width (Jk . i) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Seg (width (Jk . i)) is finite width (Jk . i) -element V43() Element of bool NAT
{ b₁ where b₁ is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT : ( 1 <= b₁ & b₁ <= width (Jk . i) ) } is set
[:(dom (Jk . i)),(Seg (width (Jk . i))):] is Relation-like finite set
(Jk . i) * ((len (Jk . i)),(len (Jk . i))) is Element of the carrier of n
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(n,b1,j) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() upper_triangular Matrix of j,j, the carrier of n
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len Jk) + j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
(len Jk) + j is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
Ji . j is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len (Ji . j), len (Ji . j), the carrier of n
Ji . j is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of n *
len (Ji . j) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
0. (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is V65( im ((T + ((- b1) * (id M9))) |^ c₁₁)) Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
the ZeroF of (im ((T + ((- b1) * (id M9))) |^ c₁₁)) is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
w is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
FI . w is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
L1 * w is Element of the carrier of (im ((T + ((- b1) * (id M9))) |^ c₁₁))
W is Element of the carrier of M9
L1 * W is Element of the carrier of M9
T . W is Element of the carrier of M9
dom Jk is finite Element of bool NAT
V is non empty non trivial right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
dim V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
the carrier of V is non empty non trivial set
[: the carrier of V, the carrier of V:] is Relation-like set
bool [: the carrier of V, the carrier of V:] is set
B is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
dim K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
V is non empty non trivial right_complementable V145(n) V146(n) V147(n) V148(n) V152() V153() V154() finite-dimensional VectSpStr over n
the carrier of V is non empty non trivial set
[: the carrier of V, the carrier of V:] is Relation-like set
bool [: the carrier of V, the carrier of V:] is set
dim V is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
B is Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous Element of bool [: the carrier of V, the carrier of V:]
n is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital V131() V133() right-distributive left-distributive right_unital well-unital V142() left_unital V152() V153() V154() algebraic-closed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
0. K is V65(K) 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 tabular Function-yielding V201() Matrix of n,n, the carrier of K
B is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) FinSequence of ( the carrier of K *) *
M9 is Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) FinSequence of ( the carrier of K *) *
Len M9 is Relation-like NAT -defined NAT -valued Function-like finite len M9 -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len M9) -tuples_on NAT
len M9 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len M9) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len M9) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
block_diagonal (M9,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len M9), Sum (Len M9), the carrier of K
V is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
V * (block_diagonal (M9,(0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
V ~ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
(V * (block_diagonal (M9,(0. K)))) * (V ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like functional empty V25() V26() V27() V29() V30() V31() finite finite-yielding V36() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V165() V174() V175() V176() V177() Function-yielding V201() FinSequence of NAT
1_ K is Element of the carrier of K
1. K is V65(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
Det V is Element of the carrier of K
T is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
V * T is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
(V * T) * (V ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
n -VectSp_over K is non empty right_complementable strict V145(K) V146(K) V147(K) V148(K) V152() V153() V154() finite-dimensional VectSpStr over K
the carrier of (n -VectSp_over K) is non empty set
the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K is Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K
len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dim (n -VectSp_over K) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() set
M9 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -valued Function-like non empty total V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) additive homogeneous Element of bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):]
[: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):] is Relation-like set
bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):] is set
b1 is Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K
AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len b1, the carrier of K
len b1 is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
J is Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding (K) FinSequence of ( the carrier of K *) *
block_diagonal (J,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of Sum (Len J), Sum (Len J), the carrier of K
Len J is Relation-like NAT -defined NAT -valued Function-like finite len J -element FinSequence-like FinSubsequence-like V174() V175() V176() V177() Element of (len J) -tuples_on NAT
len J is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
(len J) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len J) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
dom J is finite Element of bool NAT
dom (Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is non empty set
[#] (n -VectSp_over K) is non empty non proper Element of bool the carrier of (n -VectSp_over K)
bool the carrier of (n -VectSp_over K) is set
id (n -VectSp_over K) is Relation-like the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -valued Function-like non empty total V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) additive homogeneous Element of bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):]
id the carrier of (n -VectSp_over K) is Relation-like the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -valued non empty total V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) Element of bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):]
AutEqMt ((id (n -VectSp_over K)), the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K,b1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
P * (block_diagonal (J,(0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
P ~ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of n,n, the carrier of K
(P * (block_diagonal (J,(0. K)))) * (P ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
width P is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
len (P ~) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
rng (Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is non empty set
len (AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
width (AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) is V25() V26() V27() V31() finite cardinal V46() ext-real non negative V165() Element of NAT
AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)), the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
(Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) * (id (n -VectSp_over K)) is Relation-like the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -valued the carrier of (n -VectSp_over K) -valued Function-like non empty total V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) additive homogeneous Element of bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):]
AutMt (((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) * (id (n -VectSp_over K))), the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
AutMt ((id (n -VectSp_over K)), the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K,b1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, len b1, the carrier of K
AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
(AutMt ((id (n -VectSp_over K)), the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K,b1)) * (AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
P * (AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
(id (n -VectSp_over K)) * (Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -defined the carrier of (n -VectSp_over K) -valued the carrier of (n -VectSp_over K) -valued Function-like non empty total V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) V18( the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K)) additive homogeneous Element of bool [: the carrier of (n -VectSp_over K), the carrier of (n -VectSp_over K):]
AutMt (((id (n -VectSp_over K)) * (Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K))),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
P * (AutMt (((id (n -VectSp_over K)) * (Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K))),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
AutMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the carrier of K
(AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (AutMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
P * ((AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (AutMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
AutEqMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len b1, the carrier of K
(AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (AutEqMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() Matrix of len b1, len b1, the carrier of K
P * ((AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (AutEqMt ((id (n -VectSp_over K)),b1, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
(AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (P ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *
P * ((AutMt ((Mx2Tran (M9, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K, the Relation-like NAT -defined the carrier of (n -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over K)),b1,b1)) * (P ~)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Function-yielding V201() FinSequence of the carrier of K *