:: MATRIX15 semantic presentation

REAL is set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal V103() Element of bool REAL
bool REAL is set
RAT is set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() set is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal V103() set
bool NAT is non empty non trivial non finite V103() set
bool NAT is non empty non trivial non finite V103() set
K96(NAT) is V24() set
INT is set
COMPLEX is set
2 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
[: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:] is Relation-like set
bool [:REAL,REAL:] 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 empty non trivial non finite V103() set
[:[:NAT,NAT:],NAT:] is Relation-like non empty non trivial non finite V103() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite V103() set
K741() is set
3 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() set
Seg 1 is non empty trivial finite 1 -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V37() 1 -element without_zero V103() set
Seg 2 is non empty finite 2 -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V37() without_zero V103() set
Sgm {} is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Element of the carrier of n
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(K * A) * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * (A * B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (K * (A * B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((K * A) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[f,X] is set
{f,X} is non empty finite V37() set
{f} is non empty trivial finite V37() 1 -element set
{{f,X},{f}} is non empty finite V37() without_zero V103() set
Indices (K * (A * B)) is set
dom (K * (A * B)) is finite Element of bool NAT
width (K * (A * B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (K * (A * B))) is finite width (K * (A * B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (K * (A * B)) ) } is set
[:(dom (K * (A * B))),(Seg (width (K * (A * B)))):] is Relation-like finite set
Indices (A * B) is set
dom (A * B) is finite Element of bool NAT
Seg (width (A * B)) is finite width (A * B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (A * B) ) } is set
[:(dom (A * B)),(Seg (width (A * B))):] is Relation-like finite set
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
Seg (len (A * B)) is finite len (A * B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (A * B) ) } is set
dom ((K * A) * B) is finite Element of bool NAT
Indices ((K * A) * B) is set
width ((K * A) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ((K * A) * B)) is finite width ((K * A) * B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((K * A) * B) ) } is set
[:(dom ((K * A) * B)),(Seg (width ((K * A) * B))):] is Relation-like finite set
(K * (A * B)) * (f,X) is Element of the carrier of n
(A * B) * (f,X) is Element of the carrier of n
K * ((A * B) * (f,X)) 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 quasi_total V223( 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
K560( the carrier of n, the multF of n,K,((A * B) * (f,X))) is Element of the carrier of n
Line (A,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
Col (B,X) is Relation-like NAT -defined the carrier of n -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of n
(len B) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len B } is set
(Line (A,f)) "*" (Col (B,X)) is Element of the carrier of n
mlt ((Line (A,f)),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line (A,f)),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line (A,f)),(Col (B,X)))) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the addF of n $$ (mlt ((Line (A,f)),(Col (B,X)))) is Element of the carrier of n
K * ((Line (A,f)) "*" (Col (B,X))) is Element of the carrier of n
K560( the carrier of n, the multF of n,K,((Line (A,f)) "*" (Col (B,X)))) is Element of the carrier of n
K * (mlt ((Line (A,f)),(Col (B,X)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,(mlt ((Line (A,f)),(Col (B,X)))),(K multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (K * (mlt ((Line (A,f)),(Col (B,X))))) is Element of the carrier of n
the addF of n $$ (K * (mlt ((Line (A,f)),(Col (B,X))))) is Element of the carrier of n
K * (Line (A,f)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Line (A,f)),(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
mlt ((K * (Line (A,f))),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(K * (Line (A,f))),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((K * (Line (A,f))),(Col (B,X)))) is Element of the carrier of n
the addF of n $$ (mlt ((K * (Line (A,f))),(Col (B,X)))) is Element of the carrier of n
Line ((K * A),f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (K * A) -element FinSequence-like FinSubsequence-like Element of (width (K * A)) -tuples_on the carrier of n
(width (K * A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (K * A) } is set
(Line ((K * A),f)) "*" (Col (B,X)) is Element of the carrier of n
mlt ((Line ((K * A),f)),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line ((K * A),f)),(Col (B,X))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line ((K * A),f)),(Col (B,X)))) is Element of the carrier of n
the addF of n $$ (mlt ((Line ((K * A),f)),(Col (B,X)))) is Element of the carrier of n
((K * A) * B) * (f,X) is Element of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
K is Element of the carrier of n
A is Element of the carrier of n
K * A 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 quasi_total V223( 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
K560( the carrier of n, the multF of n,K,A) is Element of the carrier of n
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(1_ n) * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * (A * B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(K * A) * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[X,BX] is set
{X,BX} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,BX},{X}} is non empty finite V37() without_zero V103() set
Indices B is set
dom B is finite Element of bool NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
B * (X,BX) is Element of the carrier of n
(1_ n) * (B * (X,BX)) is Element of the carrier of n
K560( the carrier of n, the multF of n,(1_ n),(B * (X,BX))) is Element of the carrier of n
((1_ n) * B) * (X,BX) is Element of the carrier of n
len (K * (A * B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((K * A) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices (A * B) is set
dom (A * B) is finite Element of bool NAT
width (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (A * B)) is finite width (A * B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (A * B) ) } is set
[:(dom (A * B)),(Seg (width (A * B))):] is Relation-like finite set
Indices (K * (A * B)) is set
dom (K * (A * B)) is finite Element of bool NAT
width (K * (A * B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (K * (A * B))) is finite width (K * (A * B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (K * (A * B)) ) } is set
[:(dom (K * (A * B))),(Seg (width (K * (A * B)))):] is Relation-like finite set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[X,BX] is set
{X,BX} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,BX},{X}} is non empty finite V37() without_zero V103() set
(K * (A * B)) * (X,BX) is Element of the carrier of n
(A * B) * (X,BX) is Element of the carrier of n
K * ((A * B) * (X,BX)) is Element of the carrier of n
K560( the carrier of n, the multF of n,K,((A * B) * (X,BX))) is Element of the carrier of n
B * (X,BX) is Element of the carrier of n
A * (B * (X,BX)) is Element of the carrier of n
K560( the carrier of n, the multF of n,A,(B * (X,BX))) is Element of the carrier of n
K * (A * (B * (X,BX))) is Element of the carrier of n
K560( the carrier of n, the multF of n,K,(A * (B * (X,BX)))) is Element of the carrier of n
(K * A) * (B * (X,BX)) is Element of the carrier of n
K560( the carrier of n, the multF of n,(K * A),(B * (X,BX))) is Element of the carrier of n
((K * A) * B) * (X,BX) is Element of the carrier of n
width ((K * A) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((1_ n) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((1_ n) * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
- K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
Indices (- K) is set
dom (- K) is finite Element of bool NAT
width (- K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (- K)) is finite width (- K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (- K) ) } is set
[:(dom (- K)),(Seg (width (- K))):] is Relation-like finite set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
len (- K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (- K)) is finite len (- K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (- K) ) } is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
0. K is zero Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non empty non trivial non finite V103() set
A is Element of the carrier of K
(power K) . (A,n) is Element of the carrier of K
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
(power K) . (A,B) is Element of the carrier of K
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(power K) . (A,BA) is Element of the carrier of K
((power K) . (A,B)) * A is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total quasi_total V223( 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
K560( the carrier of K, the multF of K,((power K) . (A,B)),A) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(power K) . (A,B) is Element of the carrier of K
n is non empty addLoopStr
the carrier of n is non empty set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^ B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
A ^ BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(K ^ B) + (A ^ BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like 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
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(K ^ B),(A ^ BA)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B + BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,B,BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(K + A) ^ (B + BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(len B) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len B } is set
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
(len K) + (len B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of n
X ^ x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(len K) + (len B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
f is Relation-like NAT -defined the carrier of n -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of n
BX ^ f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
X + BX is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,X,BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
x + f is Relation-like NAT -defined the carrier of n -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(X + BX) ^ (x + f) is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((len K) + (len B)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = (len K) + (len B) } is set
MV is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
lA is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
c₁₃ is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg ((len K) + (len B)) is finite (len K) + (len B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (len K) + (len B) ) } is set
x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
dom x is finite (len K) + (len B) -element Element of bool NAT
dom K is finite Element of bool NAT
rng K is finite set
rng A is finite set
dom (X + BX) is finite len K -element Element of bool NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
K . y is set
dom X is finite len K -element Element of bool NAT
dom BX is finite len K -element Element of bool NAT
A . y is set
x . y is set
x is Element of the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
x . y is set
j is Element of the carrier of n
x + x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(x + x) . y is set
x + j is Element of the carrier of n
K560( the carrier of n, the addF of n,x,j) is Element of the carrier of n
(X + BX) . y is set
y is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
y . y is set
dom B is finite Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(len K) + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(len K) + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B . x is set
rng B is finite set
rng BA is finite set
dom x is finite len B -element Element of bool NAT
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
dom f is finite len B -element Element of bool NAT
BA . x is set
x . y is set
j is Element of the carrier of n
dom (x + f) is finite len B -element Element of bool NAT
len (X + BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
x . y is set
x is Element of the carrier of n
x + x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(x + x) . y is set
j + x is Element of the carrier of n
K560( the carrier of n, the addF of n,j,x) is Element of the carrier of n
(x + f) . x is set
y is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
y . y is set
dom K is finite Element of bool NAT
dom B is finite Element of bool NAT
x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
x + x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(x + x) . y is set
y is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
y . y is set
x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
x + x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(x + x) . y is set
y is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len B) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len B)) -tuples_on the carrier of n
y . y is set
n is non empty multMagma
the carrier of n is non empty set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K ^ A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B is Element of the carrier of n
B * (K ^ A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: 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 quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[:[: the carrier of n, the carrier of n:], 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (B,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,(K ^ A),(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B * K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K501( the carrier of n, the carrier of n,K,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B * A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K501( the carrier of n, the carrier of n,A,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * K) ^ (B * A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len A } is set
(len K) + (len A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((len K) + (len A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = (len K) + (len A) } is set
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
AB ^ x is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(len K) + (len A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * AB is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,AB,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B * x is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,x,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * AB) ^ (B * x) is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg ((len K) + (len A)) is finite (len K) + (len A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (len K) + (len A) ) } is set
f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
dom f is finite (len K) + (len A) -element Element of bool NAT
dom K is finite Element of bool NAT
rng K is finite set
K . BX is set
dom AB is finite len K -element Element of bool NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
dom (B * AB) is finite len K -element Element of bool NAT
f . BX is set
MV is Element of the carrier of n
B * f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,f,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * f) . BX is set
B * MV is Element of the carrier of n
K560( the carrier of n, the multF of n,B,MV) is Element of the carrier of n
(B * AB) . BX is set
X is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
X . BX is set
dom A is finite Element of bool NAT
rng A is finite set
dom (B * x) is finite len A -element Element of bool NAT
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
len (B * AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(len K) + MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(len K) + MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A . MV is set
dom x is finite len A -element Element of bool NAT
f . BX is set
lA is Element of the carrier of n
B * f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,f,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * f) . BX is set
B * lA is Element of the carrier of n
K560( the carrier of n, the multF of n,B,lA) is Element of the carrier of n
(B * x) . MV is set
X is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
X . BX is set
dom K is finite Element of bool NAT
dom A is finite Element of bool NAT
B * f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,f,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * f) . BX is set
X is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
X . BX is set
B * f is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,f,(B multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(B * f) . BX is set
X is Relation-like NAT -defined the carrier of n -valued Function-like finite (len K) + (len A) -element FinSequence-like FinSubsequence-like Element of ((len K) + (len A)) -tuples_on the carrier of n
X . BX is set
n is Relation-like Function-like set
dom n is set
K is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng K is finite set
A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng A is finite set
B is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K (#) n is Relation-like NAT -defined Function-like finite set
BA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A (#) n is Relation-like NAT -defined Function-like finite set
K ^ A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K ^ A) (#) n is Relation-like NAT -defined Function-like finite set
B ^ BA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (K ^ A) is finite Element of bool NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) + (len A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg ((len K) + (len A)) is finite (len K) + (len A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (len K) + (len A) ) } is set
rng (K ^ A) is finite set
(rng K) \/ (rng A) is finite set
dom ((K ^ A) (#) n) is finite set
dom B is finite Element of bool NAT
dom K is finite Element of bool NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom BA is finite Element of bool NAT
dom A is finite Element of bool NAT
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (B ^ BA) is finite Element of bool NAT
AB is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(B ^ BA) . x is set
B . x is set
K . x is set
n . (K . x) is set
(K ^ A) . x is set
n . ((K ^ A) . x) is set
((K ^ A) (#) n) . x is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(len K) + f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(B ^ BA) . x is set
BA . f is set
A . f is set
n . (A . f) is set
(K ^ A) . x is set
n . ((K ^ A) . x) is set
((K ^ A) (#) n) . x is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(B ^ BA) . x is set
((K ^ A) (#) n) . x is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(B ^ BA) . x is set
((K ^ A) (#) n) . x is set
(B ^ BA) . AB is set
((K ^ A) (#) n) . AB is set
n is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng n is finite V195() V196() V197() V200() set
dom n is finite Element of bool NAT
Sgm (rng n) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng A is finite V195() V196() V197() V200() set
dom A is finite Element of bool NAT
Sgm (rng A) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg B is finite B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
A | n is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
A | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined NAT -valued RAT -valued Function-like finite FinSubsequence-like V185() V186() V187() V188() set
A . (n + 1) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
<*(A . (n + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A . (n + 1))] is set
{1,(A . (n + 1))} is non empty finite V37() set
{{1,(A . (n + 1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(A . (n + 1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(A | n) ^ <*(A . (n + 1))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
rng (A | n) is finite V195() V196() V197() V200() set
dom (A | n) is finite Element of bool NAT
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(A | n) . AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(A | n) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A . AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
len (A | n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (n + 1) is non empty finite n + 1 -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{(A . (n + 1))} is non empty trivial finite V37() 1 -element Element of bool NAT
f is set
(A | n) . f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A . X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(A | n) . X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Sgm (rng (A | n)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
rng <*(A . (n + 1))*> is non empty trivial finite 1 -element set
(rng (A | n)) \/ (rng <*(A . (n + 1))*>) is non empty finite set
Sgm {(A . (n + 1))} is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
(A | n) ^ (Sgm {(A . (n + 1))}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
n is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
rng n is finite V195() V196() V197() V200() set
dom n is finite Element of bool NAT
Sgm (rng n) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
len n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng A is finite V195() V196() V197() V200() set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg B is finite B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
dom A is finite Element of bool NAT
Sgm (rng A) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
n is non empty right_complementable V95() Abelian add-associative right_zeroed addLoopStr
the carrier of n is non empty set
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom K is finite Element of bool NAT
Sum K is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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 addF of n $$ K is Element of the carrier of n
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
Seg A is finite A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
K | A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K | (Seg A) is Relation-like NAT -defined Seg A -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like set
AB is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K | A) ^ AB is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (K | A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (K | A) is finite Element of bool NAT
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom x is finite Element of bool NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A + f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A + BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len x) is finite len x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
A + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
X + A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x . X is set
K . (A + BX) is set
Sum x is Element of the carrier of n
the addF of n $$ x is Element of the carrier of n
x . f is set
K . B is set
K /. B is Element of the carrier of n
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(K | A) . X is set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K . BX is set
Sum (K | A) is Element of the carrier of n
the addF of n $$ (K | A) is Element of the carrier of n
(K | A) . A is set
K . A is set
K /. A is Element of the carrier of n
(K /. A) + (K /. B) is Element of the carrier of n
K560( the carrier of n, the addF of n,(K /. A),(K /. B)) is Element of the carrier of n
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K /. A is Element of the carrier of n
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K /. B is Element of the carrier of n
(K /. A) + (K /. B) is Element of the carrier of n
K560( the carrier of n, the addF of n,(K /. A),(K /. B)) is Element of the carrier of n
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K . BA is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A + K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (A + K) is finite A + K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A + K ) } is set
Seg A is finite A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
(Seg (A + K)) \ (Seg A) is finite without_zero Element of bool NAT
Sgm ((Seg (A + K)) \ (Seg A)) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (A + K)) \ (Seg A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (A + K)) \ (Seg A))) -tuples_on NAT
card ((Seg (A + K)) \ (Seg A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card ((Seg (A + K)) \ (Seg A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (A + K)) \ (Seg A)) } is set
(Sgm ((Seg (A + K)) \ (Seg A))) . n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A + n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
idseq K is Relation-like NAT -defined Function-like finite K -element FinSequence-like FinSubsequence-like set
id (Seg K) is Relation-like Seg K -defined Seg K -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg K),(Seg K):]
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is finite V37() set
dom (idseq K) is finite K -element Element of bool NAT
len (idseq K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (idseq K)) is finite len (idseq K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (idseq K) ) } is set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B Shift (idseq K) is Relation-like NAT -defined Function-like FinSubsequence-like set
dom (B Shift (idseq K)) is set
{ (B + b₁) where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : b₁ in dom (idseq K) } is set
AB is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x - A is V105() ext-real complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A + f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
x + A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty set
n * is functional non empty FinSequence-membered FinSequenceSet of n
K is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is finite without_zero Element of bool NAT
B is finite without_zero Element of bool NAT
[:A,B:] is Relation-like finite set
BA is finite without_zero Element of bool NAT
AB is finite without_zero Element of bool NAT
[:BA,AB:] is Relation-like finite set
card A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:A,B:] /\ [:BA,AB:] is Relation-like finite set
Sgm A is Relation-like NAT -defined NAT -valued Function-like finite card A -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card A) -tuples_on NAT
(card A) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card A } is set
(Sgm A) " is Relation-like Function-like set
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite card B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card B) -tuples_on NAT
(card B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card B } is set
(Sgm B) " is Relation-like Function-like set
Sgm BA is Relation-like NAT -defined NAT -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BA) -tuples_on NAT
(card BA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BA } is set
(Sgm BA) " is Relation-like Function-like set
Sgm AB is Relation-like NAT -defined NAT -valued Function-like finite card AB -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card AB) -tuples_on NAT
(card AB) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card AB } is set
(Sgm AB) " is Relation-like Function-like set
[:A,B:] \/ [:BA,AB:] is Relation-like finite set
c₁₃ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card B,n
x is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BA, card AB,n
rng (Sgm B) is finite V195() V196() V197() V200() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
dom (Sgm A) is finite card A -element Element of bool NAT
dom (Sgm B) is finite card B -element Element of bool NAT
dom (Sgm BA) is finite card BA -element Element of bool NAT
dom (Sgm AB) is finite card AB -element Element of bool NAT
rng (Sgm A) is finite V195() V196() V197() V200() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
Seg (card B) is finite card B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card B ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
rng (Sgm AB) is finite V195() V196() V197() V200() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
rng (Sgm BA) is finite V195() V196() V197() V200() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
[:(Seg (len K)),(Seg (width K)):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,y] is set
{x,y} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,y},{x}} is non empty finite V37() without_zero V103() set
y is set
(Sgm AB) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is set
(Sgm B) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is set
(Sgm BA) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is set
(Sgm A) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ * (L,mN) is Element of n
mSN is Element of n
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm BA) ") . x is set
((Sgm AB) ") . y is set
((Sgm B) ") . y is set
((Sgm A) ") . x is set
x * (mSN,j) is Element of n
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (mSN,j) is Element of n
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[mSN,j] is set
{mSN,j} is non empty finite V37() set
{mSN} is non empty trivial finite V37() 1 -element set
{{mSN,j},{mSN}} is non empty finite V37() without_zero V103() set
z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (z,c₂₇) is Element of n
c₂₈ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₂₉ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . c₂₈ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . c₂₉ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x * (c₂₈,c₂₉) is Element of n
K * (mSN,j) is Element of n
y is set
(Sgm B) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is set
(Sgm A) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ * (j,x) is Element of n
L is Element of n
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (mN,mSN) is Element of n
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[mN,mSN] is set
{mN,mSN} is non empty finite V37() set
{mN} is non empty trivial finite V37() 1 -element set
{{mN,mSN},{mN}} is non empty finite V37() without_zero V103() set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (mSN,j) is Element of n
z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x * (z,c₂₇) is Element of n
K * (mN,mSN) is Element of n
y is set
(Sgm AB) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is set
(Sgm BA) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x * (j,x) is Element of n
L is Element of n
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x * (mN,mSN) is Element of n
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[mN,mSN] is set
{mN,mSN} is non empty finite V37() set
{mN} is non empty trivial finite V37() 1 -element set
{{mN,mSN},{mN}} is non empty finite V37() without_zero V103() set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (mSN,j) is Element of n
z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . c₂₇ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x * (z,c₂₇) is Element of n
K * (mN,mSN) is Element of n
K * (x,y) is Element of n
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ * (j,x) is Element of n
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm AB) . mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x * (L,mN) is Element of n
K * (y,x) is Element of n
x is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K,n
Indices x is set
dom x is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width x) is finite width x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
[:(dom x),(Seg (width x)):] is Relation-like finite set
Segm (x,A,B) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card B,n
Segm (x,(Sgm A),(Sgm B)) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card B,n
Segm (x,BA,AB) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BA, card AB,n
Segm (x,(Sgm BA),(Sgm AB)) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BA, card AB,n
(Indices x) \ ([:A,B:] \/ [:BA,AB:]) is Element of bool (Indices x)
bool (Indices x) is set
Seg (card A) is finite card A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card A ) } is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg y is finite y -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= y ) } is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
Indices (Segm (x,A,B)) is set
dom (Segm (x,A,B)) is finite Element of bool NAT
width (Segm (x,A,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (x,A,B))) is finite width (Segm (x,A,B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (x,A,B)) ) } is set
[:(dom (Segm (x,A,B))),(Seg (width (Segm (x,A,B)))):] is Relation-like finite set
[:(Seg (card A)),(Seg (card B)):] is Relation-like finite set
(Sgm B) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm A) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm A) . y),((Sgm B) . x)] is set
{((Sgm A) . y),((Sgm B) . x)} is non empty finite V37() set
{((Sgm A) . y)} is non empty trivial finite V37() 1 -element set
{{((Sgm A) . y),((Sgm B) . x)},{((Sgm A) . y)}} is non empty finite V37() without_zero V103() set
c₁₃ * (y,x) is Element of n
x * (((Sgm A) . y),((Sgm B) . x)) is Element of n
(Segm (x,A,B)) * (y,x) is Element of n
Seg (card AB) is finite card AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card AB ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
Seg (card BA) is finite card BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card BA ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,j] is set
{x,j} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,j},{x}} is non empty finite V37() without_zero V103() set
Indices (Segm (x,BA,AB)) is set
dom (Segm (x,BA,AB)) is finite Element of bool NAT
width (Segm (x,BA,AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (x,BA,AB))) is finite width (Segm (x,BA,AB)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (x,BA,AB)) ) } is set
[:(dom (Segm (x,BA,AB))),(Seg (width (Segm (x,BA,AB)))):] is Relation-like finite set
[:(Seg (card BA)),(Seg (card AB)):] is Relation-like finite set
(Sgm AB) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm BA) . x),((Sgm AB) . j)] is set
{((Sgm BA) . x),((Sgm AB) . j)} is non empty finite V37() set
{((Sgm BA) . x)} is non empty trivial finite V37() 1 -element set
{{((Sgm BA) . x),((Sgm AB) . j)},{((Sgm BA) . x)}} is non empty finite V37() without_zero V103() set
x * (x,j) is Element of n
x * (((Sgm BA) . x),((Sgm AB) . j)) is Element of n
(Segm (x,BA,AB)) * (x,j) is Element of n
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,j] is set
{x,j} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,j},{x}} is non empty finite V37() without_zero V103() set
x * (x,j) is Element of n
K * (x,j) is Element of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is finite without_zero Element of bool NAT
BA is finite without_zero Element of bool NAT
[:A,BA:] is Relation-like finite set
(dom K) \ A is finite Element of bool NAT
B is finite without_zero Element of bool NAT
(Seg (width K)) \ B is finite without_zero Element of bool NAT
Sgm BA is Relation-like NAT -defined NAT -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BA) -tuples_on NAT
card BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card BA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BA } is set
(card BA) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like Element of (card BA) -tuples_on the carrier of n
(card BA) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = card BA } is set
Seg (card BA) is finite card BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card BA ) } is set
(Seg (card BA)) --> (0. n) is Relation-like Seg (card BA) -defined Seg (card BA) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card BA)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (card BA)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (card BA)),{(0. n)}:] is finite V37() set
Segm (K,A,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card B, the carrier of n
card A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm A is Relation-like NAT -defined NAT -valued Function-like finite card A -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card A) -tuples_on NAT
(card A) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card A } is set
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite card B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card B) -tuples_on NAT
(card B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card B } is set
Segm (K,(Sgm A),(Sgm B)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card B, the carrier of n
the_rank_of (Segm (K,A,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (K,AB) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
(Line (K,AB)) * (Sgm BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * (AB,x) is Element of the carrier of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
[:A,B:] is Relation-like finite set
Indices (Segm (K,A,B)) is set
dom (Segm (K,A,B)) is finite Element of bool NAT
width (Segm (K,A,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (K,A,B))) is finite width (Segm (K,A,B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (K,A,B)) ) } is set
[:(dom (Segm (K,A,B))),(Seg (width (Segm (K,A,B)))):] is Relation-like finite set
MV is finite without_zero Element of bool NAT
lA is finite without_zero Element of bool NAT
[:MV,lA:] is Relation-like finite set
card MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
EqSegm ((Segm (K,A,B)),MV,lA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card MV, card MV, the carrier of n
Det (EqSegm ((Segm (K,A,B)),MV,lA)) is Element of the carrier of n
Permutations (card MV) is set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
FinOmega (Permutations (card MV)) is Element of K96((Permutations (card MV)))
K96((Permutations (card MV))) is V24() set
Path_product (EqSegm ((Segm (K,A,B)),MV,lA)) is Relation-like Permutations (card MV) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations (card MV)), the carrier of n:]
[:(Permutations (card MV)), the carrier of n:] is Relation-like set
bool [:(Permutations (card MV)), the carrier of n:] is set
K103((Permutations (card MV)), the carrier of n, the addF of n,(FinOmega (Permutations (card MV))),(Path_product (EqSegm ((Segm (K,A,B)),MV,lA)))) is Element of the carrier of n
(Sgm A) .: MV is finite V195() V196() V197() V200() set
(Sgm B) .: lA is finite V195() V196() V197() V200() set
Segm ((Segm (K,A,B)),MV,lA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card MV, card lA, the carrier of n
Sgm MV is Relation-like NAT -defined NAT -valued Function-like finite card MV -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card MV) -tuples_on NAT
(card MV) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card MV } is set
Sgm lA is Relation-like NAT -defined NAT -valued Function-like finite card lA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card lA) -tuples_on NAT
(card lA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card lA } is set
Segm ((Segm (K,A,B)),(Sgm MV),(Sgm lA)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card MV, card lA, the carrier of n
c₁₃ is finite without_zero Element of bool NAT
x is finite without_zero Element of bool NAT
card c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (K,c₁₃,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card c₁₃, card x, the carrier of n
Sgm c₁₃ is Relation-like NAT -defined NAT -valued Function-like finite card c₁₃ -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card c₁₃) -tuples_on NAT
(card c₁₃) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card c₁₃ } is set
Sgm x is Relation-like NAT -defined NAT -valued Function-like finite card x -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card x) -tuples_on NAT
(card x) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x } is set
Segm (K,(Sgm c₁₃),(Sgm x)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card c₁₃, card x, the carrier of n
X is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{X} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
x \/ {X} is non empty finite without_zero V103() Element of bool NAT
f is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{f} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
c₁₃ \/ {f} is non empty finite without_zero V103() Element of bool NAT
EqSegm (K,(c₁₃ \/ {f}),(x \/ {X})) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (c₁₃ \/ {f}), card (c₁₃ \/ {f}), the carrier of n
card (c₁₃ \/ {f}) is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
Segm (K,(c₁₃ \/ {f}),(x \/ {X})) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (c₁₃ \/ {f}), card (x \/ {X}), the carrier of n
card (x \/ {X}) is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
Sgm (c₁₃ \/ {f}) is Relation-like NAT -defined NAT -valued Function-like finite card (c₁₃ \/ {f}) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (c₁₃ \/ {f})) -tuples_on NAT
(card (c₁₃ \/ {f})) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (c₁₃ \/ {f}) } is set
Sgm (x \/ {X}) is Relation-like NAT -defined NAT -valued Function-like finite card (x \/ {X}) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (x \/ {X})) -tuples_on NAT
(card (x \/ {X})) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (x \/ {X}) } is set
Segm (K,(Sgm (c₁₃ \/ {f})),(Sgm (x \/ {X}))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (c₁₃ \/ {f}), card (x \/ {X}), the carrier of n
[AB,x] is set
{AB,x} is non empty finite V37() set
{AB} is non empty trivial finite V37() 1 -element set
{{AB,x},{AB}} is non empty finite V37() without_zero V103() set
dom (Sgm BA) is finite card BA -element Element of bool NAT
rng (Sgm (x \/ {X})) is finite V195() V196() V197() V200() set
[:(c₁₃ \/ {f}),(x \/ {X}):] is Relation-like finite set
dom (Sgm (c₁₃ \/ {f})) is finite card (c₁₃ \/ {f}) -element Element of bool NAT
Seg (card (c₁₃ \/ {f})) is non empty finite card (c₁₃ \/ {f}) -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (c₁₃ \/ {f}) ) } is set
{AB} is non empty trivial finite V37() 1 -element Element of bool NAT
(card c₁₃) + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
(card (c₁₃ \/ {f})) -' 1 is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card x) + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{x} is non empty trivial finite V37() 1 -element Element of bool NAT
dom (Sgm (x \/ {X})) is finite card (x \/ {X}) -element Element of bool NAT
j is set
(Sgm (x \/ {X})) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
rng (Sgm (c₁₃ \/ {f})) is finite V195() V196() V197() V200() set
x is set
(Sgm (c₁₃ \/ {f})) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
- (1_ n) is Element of the carrier of n
power n is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is Relation-like non empty non trivial non finite V103() set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial non finite V103() set
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
L + mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(power n) . ((- (1_ n)),(L + mN)) is Element of the carrier of n
LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) is finite Element of bool NAT
len (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L))) is finite len (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) ) } is set
Seg (card (x \/ {X})) is non empty finite card (x \/ {X}) -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (x \/ {X}) ) } is set
Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of (card (c₁₃ \/ {f})) -' 1,(card (c₁₃ \/ {f})) -' 1, the carrier of n
(c₁₃ \/ {f}) \ {AB} is finite without_zero Element of bool NAT
(x \/ {X}) \ {x} is finite without_zero Element of bool NAT
EqSegm (K,((c₁₃ \/ {f}) \ {AB}),((x \/ {X}) \ {x})) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((c₁₃ \/ {f}) \ {AB}), card ((c₁₃ \/ {f}) \ {AB}), the carrier of n
card ((c₁₃ \/ {f}) \ {AB}) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
EqSegm (K,c₁₃,((x \/ {X}) \ {x})) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card c₁₃, card c₁₃, the carrier of n
EqSegm (K,c₁₃,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card c₁₃, card c₁₃, the carrier of n
Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)) is Element of the carrier of n
Permutations ((card (c₁₃ \/ {f})) -' 1) is set
FinOmega (Permutations ((card (c₁₃ \/ {f})) -' 1)) is Element of K96((Permutations ((card (c₁₃ \/ {f})) -' 1)))
K96((Permutations ((card (c₁₃ \/ {f})) -' 1))) is V24() set
Path_product (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)) is Relation-like Permutations ((card (c₁₃ \/ {f})) -' 1) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n:]
[:(Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n:] is Relation-like set
bool [:(Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n:] is set
K103((Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n, the addF of n,(FinOmega (Permutations ((card (c₁₃ \/ {f})) -' 1))),(Path_product (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)))) is Element of the carrier of n
((power n) . ((- (1_ n)),(L + mN))) * (Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN))) 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 quasi_total V223( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
K560( the carrier of n, the multF of n,((power n) . ((- (1_ n)),(L + mN))),(Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)))) is Element of the carrier of n
Indices (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) is set
dom (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) is finite Element of bool NAT
width (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X})))) is finite width (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) ) } is set
[:(dom (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X})))),(Seg (width (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))))):] is Relation-like finite set
[:(Seg (card (c₁₃ \/ {f}))),(Seg (card (c₁₃ \/ {f}))):] is Relation-like finite set
[L,mN] is Element of [:NAT,NAT:]
{L,mN} is non empty finite V37() set
{L} is non empty trivial finite V37() 1 -element set
{{L,mN},{L}} is non empty finite V37() without_zero V103() set
rng (Sgm BA) is finite V195() V196() V197() V200() set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm (x \/ {X})) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is set
(Sgm BA) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[L,mSN] is set
{L,mSN} is non empty finite V37() set
{{L,mSN},{L}} is non empty finite V37() without_zero V103() set
(EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) * (L,mSN) is Element of the carrier of n
K * (AB,((Sgm (x \/ {X})) . mSN)) is Element of the carrier of n
z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm BA) . z is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (K,AB)) . ((Sgm BA) . z) is set
((card BA) |-> (0. n)) . z is set
(LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) . mSN is set
Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN) is Element of the carrier of n
L + mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(power n) . ((- (1_ n)),(L + mSN)) is Element of the carrier of n
Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN) is Element of the carrier of n
Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of (card (c₁₃ \/ {f})) -' 1,(card (c₁₃ \/ {f})) -' 1, the carrier of n
Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN)) is Element of the carrier of n
Path_product (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN)) is Relation-like Permutations ((card (c₁₃ \/ {f})) -' 1) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n:]
K103((Permutations ((card (c₁₃ \/ {f})) -' 1)), the carrier of n, the addF of n,(FinOmega (Permutations ((card (c₁₃ \/ {f})) -' 1))),(Path_product (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN)))) is Element of the carrier of n
((power n) . ((- (1_ n)),(L + mSN))) * (Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN)) is Element of the carrier of n
K560( the carrier of n, the multF of n,((power n) . ((- (1_ n)),(L + mSN))),(Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN))) is Element of the carrier of n
(0. n) * (Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN)) is Element of the carrier of n
K560( the carrier of n, the multF of n,(0. n),(Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mSN))) is Element of the carrier of n
(LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) . mN is set
Sum (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) is Element of the carrier of n
the addF of n $$ (LaplaceExpL ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L)) is Element of the carrier of n
Det (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) is Element of the carrier of n
Permutations (card (c₁₃ \/ {f})) is set
FinOmega (Permutations (card (c₁₃ \/ {f}))) is Element of K96((Permutations (card (c₁₃ \/ {f}))))
K96((Permutations (card (c₁₃ \/ {f})))) is V24() set
Path_product (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))) is Relation-like Permutations (card (c₁₃ \/ {f})) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations (card (c₁₃ \/ {f}))), the carrier of n:]
[:(Permutations (card (c₁₃ \/ {f}))), the carrier of n:] is Relation-like set
bool [:(Permutations (card (c₁₃ \/ {f}))), the carrier of n:] is set
K103((Permutations (card (c₁₃ \/ {f}))), the carrier of n, the addF of n,(FinOmega (Permutations (card (c₁₃ \/ {f})))),(Path_product (EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))))) is Element of the carrier of n
(Segm (K,(c₁₃ \/ {f}),(x \/ {X}))) * (L,mN) is Element of the carrier of n
Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN) is Element of the carrier of n
L + mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(power n) . ((- (1_ n)),(L + mN)) is Element of the carrier of n
Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN) is Element of the carrier of n
((power n) . ((- (1_ n)),(L + mN))) * (Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)) is Element of the carrier of n
K560( the carrier of n, the multF of n,((power n) . ((- (1_ n)),(L + mN))),(Minor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN))) is Element of the carrier of n
((Segm (K,(c₁₃ \/ {f}),(x \/ {X}))) * (L,mN)) * (Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)) is Element of the carrier of n
K560( the carrier of n, the multF of n,((Segm (K,(c₁₃ \/ {f}),(x \/ {X}))) * (L,mN)),(Cofactor ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN))) is Element of the carrier of n
(K * (AB,x)) * (((power n) . ((- (1_ n)),(L + mN))) * (Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN)))) is Element of the carrier of n
K560( the carrier of n, the multF of n,(K * (AB,x)),(((power n) . ((- (1_ n)),(L + mN))) * (Det (Delete ((EqSegm (K,(c₁₃ \/ {f}),(x \/ {X}))),L,mN))))) is Element of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is finite without_zero Element of bool NAT
(dom K) \ A is finite Element of bool NAT
Segm (K,A,(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width K)), the carrier of n
card A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm A is Relation-like NAT -defined NAT -valued Function-like finite card A -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card A) -tuples_on NAT
(card A) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card A } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm A),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width K)), the carrier of n
the_rank_of (Segm (K,A,(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
Segm (AB,A,(Seg (width AB))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width AB)), the carrier of n
card (Seg (width AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width AB)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width AB)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width AB))) -tuples_on NAT
(card (Seg (width AB))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width AB)) } is set
Segm (AB,(Sgm A),(Sgm (Seg (width AB)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width AB)), the carrier of n
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
lines AB is finite Element of bool the carrier of ((width K) -VectSp_over n)
the_rank_of AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is finite Element of bool the carrier of ((width K) -VectSp_over n)
card f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
lines (Segm (AB,A,(Seg (width AB)))) is finite Element of bool the carrier of ((card (Seg (width AB))) -VectSp_over n)
(card (Seg (width AB))) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((card (Seg (width AB))) -VectSp_over n) is non empty set
bool the carrier of ((card (Seg (width AB))) -VectSp_over n) is set
X is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (AB,BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
(width AB) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width AB } is set
0. ((width K) -VectSp_over n) is Relation-like Function-like zero Element of the carrier of ((width K) -VectSp_over n)
rng (Sgm A) is finite V195() V196() V197() V200() set
dom (Sgm A) is finite card A -element Element of bool NAT
MV is set
(Sgm A) . MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (card A) is finite card A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card A ) } is set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((Segm (AB,A,(Seg (width AB)))),lA) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (Segm (AB,A,(Seg (width AB)))) -element FinSequence-like FinSubsequence-like Element of (width (Segm (AB,A,(Seg (width AB))))) -tuples_on the carrier of n
width (Segm (AB,A,(Seg (width AB)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (Segm (AB,A,(Seg (width AB))))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (Segm (AB,A,(Seg (width AB)))) } is set
X is finite Element of bool the carrier of ((width K) -VectSp_over n)
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
card X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is finite without_zero Element of bool NAT
(Seg (width K)) \ A is finite without_zero Element of bool NAT
Segm (K,(Seg (len K)),A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card A, the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm A is Relation-like NAT -defined NAT -valued Function-like finite card A -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card A) -tuples_on NAT
(card A) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card A } is set
Segm (K,(Sgm (Seg (len K))),(Sgm A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card A, the carrier of n
the_rank_of (Segm (K,(Seg (len K)),A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom K is finite Element of bool NAT
Indices K is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[B,BA] is set
{B,BA} is non empty finite V37() set
{B} is non empty trivial finite V37() 1 -element set
{{B,BA},{B}} is non empty finite V37() without_zero V103() set
K * (B,BA) is Element of the carrier of n
((len K) |-> (0. n)) . B is set
Col (K,BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(Col (K,BA)) . B is set
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
[:(dom K),A:] is Relation-like finite set
Indices K is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
width (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (K @) is finite Element of bool NAT
Seg (len (K @)) is finite len (K @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (K @) ) } is set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(dom (K @)) \ A is finite Element of bool NAT
Line ((K @),BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (K @) -element FinSequence-like FinSubsequence-like Element of (width (K @)) -tuples_on the carrier of n
(width (K @)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (K @) } is set
Col (K,BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(width (K @)) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (K @) -element FinSequence-like FinSubsequence-like Element of (width (K @)) -tuples_on the carrier of n
Seg (width (K @)) is finite width (K @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (K @) ) } is set
(Seg (width (K @))) --> (0. n) is Relation-like Seg (width (K @)) -defined Seg (width (K @)) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width (K @))),{(0. n)}:]
[:(Seg (width (K @))),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width (K @))),{(0. n)}:] is finite V37() set
the_rank_of (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm ((K @),A,(Seg (len K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (len K)), the carrier of n
Segm ((K @),(Sgm A),(Sgm (Seg (len K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (len K)), the carrier of n
the_rank_of (Segm ((K @),A,(Seg (len K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Segm (K,(Seg (len K)),A)) @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of ((Segm (K,(Seg (len K)),A)) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is non empty right_complementable V95() vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over n
the carrier of K is non empty set
bool the carrier of K is set
A is finite Element of bool the carrier of K
Lin A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of K
B is Element of the carrier of K
BA is Element of the carrier of K
{B} is non empty trivial finite 1 -element Element of bool the carrier of K
A \ {B} is finite Element of bool the carrier of K
AB is Element of the carrier of n
AB * BA is Element of the carrier of K
B + (AB * BA) 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 quasi_total V223( the carrier of K) V224( 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
K560( the carrier of K, the addF of K,B,(AB * BA)) is Element of the carrier of K
{(B + (AB * BA))} is non empty trivial finite 1 -element Element of bool the carrier of K
(A \ {B}) \/ {(B + (AB * BA))} is non empty finite Element of bool the carrier of K
Lin ((A \ {B}) \/ {(B + (AB * BA))}) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of K
the carrier of (Lin A) is non empty set
X is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
- (1_ n) is Element of the carrier of n
K is non empty right_complementable V95() vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over n
the carrier of K is non empty set
bool the carrier of K is set
0. K is zero Element of the carrier of K
A is finite Element of bool the carrier of K
Lin A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of K
B is Element of the carrier of K
BA is Element of the carrier of K
{B} is non empty trivial finite 1 -element Element of bool the carrier of K
A \ {B} is finite Element of bool the carrier of K
AB is Element of the carrier of n
AB * BA is Element of the carrier of K
B + (AB * BA) 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 quasi_total V223( the carrier of K) V224( 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
K560( the carrier of K, the addF of K,B,(AB * BA)) is Element of the carrier of K
{(B + (AB * BA))} is non empty trivial finite 1 -element Element of bool the carrier of K
(A \ {B}) \/ {(B + (AB * BA))} is non empty finite Element of bool the carrier of K
Lin ((A \ {B}) \/ {(B + (AB * BA))}) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of K
the carrier of (Lin ((A \ {B}) \/ {(B + (AB * BA))})) is non empty set
X is set
- AB is Element of the carrier of n
(- AB) * BA is Element of the carrier of K
(B + (AB * BA)) + ((- AB) * BA) is Element of the carrier of K
K560( the carrier of K, the addF of K,(B + (AB * BA)),((- AB) * BA)) is Element of the carrier of K
(B + (AB * BA)) - (AB * BA) is Element of the carrier of K
- (AB * BA) is Element of the carrier of K
(B + (AB * BA)) + (- (AB * BA)) is Element of the carrier of K
K560( the carrier of K, the addF of K,(B + (AB * BA)),(- (AB * BA))) is Element of the carrier of K
(AB * BA) - (AB * BA) is Element of the carrier of K
(AB * BA) + (- (AB * BA)) is Element of the carrier of K
K560( the carrier of K, the addF of K,(AB * BA),(- (AB * BA))) is Element of the carrier of K
B + ((AB * BA) - (AB * BA)) is Element of the carrier of K
K560( the carrier of K, the addF of K,B,((AB * BA) - (AB * BA))) is Element of the carrier of K
B + (0. K) is Element of the carrier of K
K560( the carrier of K, the addF of K,B,(0. K)) is Element of the carrier of K
0. n is zero Element of the carrier of n
(- (1_ n)) - AB is Element of the carrier of n
- AB is Element of the carrier of n
(- (1_ n)) + (- AB) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
K560( the carrier of n, the addF of n,(- (1_ n)),(- AB)) is Element of the carrier of n
- ((- (1_ n)) - AB) is Element of the carrier of n
AB + (1_ n) is Element of the carrier of n
K560( the carrier of n, the addF of n,AB,(1_ n)) is Element of the carrier of n
(1_ n) * B is Element of the carrier of K
AB * B is Element of the carrier of K
((1_ n) * B) + (AB * B) is Element of the carrier of K
K560( the carrier of K, the addF of K,((1_ n) * B),(AB * B)) is Element of the carrier of K
(1. n) + AB is Element of the carrier of n
K560( the carrier of n, the addF of n,(1. n),AB) is Element of the carrier of n
((1. n) + AB) * B is Element of the carrier of K
((1. n) + AB) " is Element of the carrier of n
(((1. n) + AB) ") * (B + (AB * BA)) is Element of the carrier of K
the carrier of (Lin A) is non empty set
n is non empty set
n * is functional non empty FinSequence-membered FinSequenceSet of n
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A,n
AB is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,B,n
BA ^^ AB is Relation-like Function-like FinSequence-yielding set
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA ^^ AB is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
dom AB is finite Element of bool NAT
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len AB) is finite len AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len AB ) } is set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
dom BA is finite Element of bool NAT
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len BA) is finite len BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BA ) } is set
dom (BA ^^ AB) is finite Element of bool NAT
(Seg K) /\ (Seg K) is finite Element of bool NAT
len (BA ^^ AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg x is finite x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
BA . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (BA,x) is Relation-like NAT -defined n -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on n
(width BA) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width BA } is set
AB . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (AB,x) is Relation-like NAT -defined n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on n
(width AB) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width AB } is set
(BA ^^ AB) . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Line (BA,x)) ^ (Line (AB,x)) is Relation-like NAT -defined n -valued Function-like finite (width BA) + (width AB) -element FinSequence-like FinSubsequence-like FinSequence of n
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Line (BA,x)) ^ (Line (AB,x))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng (BA ^^ AB) is finite set
width (BA ^^ AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
BA is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,B
AB is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,B
(B,n,K,A,BA,AB) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width BA) + (width AB),B
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (B,n,K,A,BA,AB) is finite Element of bool NAT
Seg (len (B,n,K,A,BA,AB)) is finite len (B,n,K,A,BA,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (B,n,K,A,BA,AB) ) } is set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line ((B,n,K,A,BA,AB),f) is Relation-like NAT -defined B -valued Function-like finite width (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (width (B,n,K,A,BA,AB)) -tuples_on B
width (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width (B,n,K,A,BA,AB) } is set
Line (BA,f) is Relation-like NAT -defined B -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on B
(width BA) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width BA } is set
Line (AB,f) is Relation-like NAT -defined B -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on B
(width AB) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width AB } is set
(Line (BA,f)) ^ (Line (AB,f)) is Relation-like NAT -defined B -valued Function-like finite (width BA) + (width AB) -element FinSequence-like FinSubsequence-like FinSequence of B
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA . f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
AB . f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(B,n,K,A,BA,AB) . f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
BA is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,B
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
AB is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,B
(B,n,K,A,BA,AB) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width BA) + (width AB),B
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Col ((B,n,K,A,BA,AB),x) is Relation-like NAT -defined B -valued Function-like finite len (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (len (B,n,K,A,BA,AB)) -tuples_on B
len (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len (B,n,K,A,BA,AB) } is set
Col (BA,x) is Relation-like NAT -defined B -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on B
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len BA) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len BA } is set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
width (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (B,n,K,A,BA,AB)) is finite width (B,n,K,A,BA,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (B,n,K,A,BA,AB) ) } is set
dom BA is finite Element of bool NAT
Line (BA,X) is Relation-like NAT -defined B -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on B
(width BA) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width BA } is set
dom (Line (BA,X)) is finite width BA -element Element of bool NAT
dom (B,n,K,A,BA,AB) is finite Element of bool NAT
(Col ((B,n,K,A,BA,AB),x)) . X is set
(B,n,K,A,BA,AB) * (X,x) is Element of B
Line ((B,n,K,A,BA,AB),X) is Relation-like NAT -defined B -valued Function-like finite width (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (width (B,n,K,A,BA,AB)) -tuples_on B
(width (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width (B,n,K,A,BA,AB) } is set
(Line ((B,n,K,A,BA,AB),X)) . x is set
Line (AB,X) is Relation-like NAT -defined B -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on B
(width AB) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width AB } is set
(Line (BA,X)) ^ (Line (AB,X)) is Relation-like NAT -defined B -valued Function-like finite (width BA) + (width AB) -element FinSequence-like FinSubsequence-like FinSequence of B
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((Line (BA,X)) ^ (Line (AB,X))) . x is set
(Line (BA,X)) . x is set
BA * (X,x) is Element of B
(Col (BA,x)) . X is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
BA is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,B
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,B
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
(B,n,K,A,BA,AB) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width BA) + (width AB),B
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width BA) + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col ((B,n,K,A,BA,AB),((width BA) + x)) is Relation-like NAT -defined B -valued Function-like finite len (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (len (B,n,K,A,BA,AB)) -tuples_on B
len (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len (B,n,K,A,BA,AB) } is set
Col (AB,x) is Relation-like NAT -defined B -valued Function-like finite len AB -element FinSequence-like FinSubsequence-like Element of (len AB) -tuples_on B
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len AB) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len AB } is set
dom AB is finite Element of bool NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
width (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (B,n,K,A,BA,AB)) is finite width (B,n,K,A,BA,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (B,n,K,A,BA,AB) ) } is set
Line (AB,X) is Relation-like NAT -defined B -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on B
(width AB) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width AB } is set
dom (Line (AB,X)) is finite width AB -element Element of bool NAT
Line (BA,X) is Relation-like NAT -defined B -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on B
(width BA) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width BA } is set
len (Line (BA,X)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (B,n,K,A,BA,AB) is finite Element of bool NAT
(Col ((B,n,K,A,BA,AB),((width BA) + x))) . X is set
(B,n,K,A,BA,AB) * (X,((width BA) + x)) is Element of B
Line ((B,n,K,A,BA,AB),X) is Relation-like NAT -defined B -valued Function-like finite width (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (width (B,n,K,A,BA,AB)) -tuples_on B
(width (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = width (B,n,K,A,BA,AB) } is set
(Line ((B,n,K,A,BA,AB),X)) . ((width BA) + x) is set
(Line (BA,X)) ^ (Line (AB,X)) is Relation-like NAT -defined B -valued Function-like finite (width BA) + (width AB) -element FinSequence-like FinSubsequence-like FinSequence of B
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((Line (BA,X)) ^ (Line (AB,X))) . ((width BA) + x) is set
(Line (AB,X)) . x is set
AB * (X,x) is Element of B
(Col (AB,x)) . X is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is non empty set
BA * is functional non empty FinSequence-membered FinSequenceSet of BA
AB is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,BA
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,BA
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) + (width x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(BA,n,K,A,AB,x) is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width AB) + (width x),BA
f is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of BA
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of BA
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f ^ X is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of BA
ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)) is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width AB) + (width x),BA
ReplaceLine (AB,B,f) is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,BA
ReplaceLine (x,B,X) is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,BA
(BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is Relation-like NAT -defined BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width (ReplaceLine (AB,B,f))) + (width (ReplaceLine (x,B,X))),BA
width (ReplaceLine (AB,B,f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (x,B,X)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine (AB,B,f))) + (width (ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width AB } is set
(width x) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width x } is set
((width AB) + (width x)) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = (width AB) + (width x) } is set
len (f ^ X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
width (BA,n,K,A,AB,x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((ReplaceLine (x,B,X)),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine (x,B,X)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (x,B,X))) -tuples_on BA
(width (ReplaceLine (x,B,X))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine (x,B,X)) } is set
Line ((ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA
width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) } is set
Line ((ReplaceLine (AB,B,f)),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine (AB,B,f)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (AB,B,f))) -tuples_on BA
(width (ReplaceLine (AB,B,f))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine (AB,B,f)) } is set
Line ((BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) -element FinSequence-like FinSubsequence-like Element of (width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA
width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) } is set
Line ((ReplaceLine (x,B,X)),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine (x,B,X)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (x,B,X))) -tuples_on BA
(width (ReplaceLine (x,B,X))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine (x,B,X)) } is set
Line (x,x) is Relation-like NAT -defined BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on BA
Line ((ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA
width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) } is set
Line ((BA,n,K,A,AB,x),x) is Relation-like NAT -defined BA -valued Function-like finite width (BA,n,K,A,AB,x) -element FinSequence-like FinSubsequence-like Element of (width (BA,n,K,A,AB,x)) -tuples_on BA
(width (BA,n,K,A,AB,x)) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (BA,n,K,A,AB,x) } is set
Line ((ReplaceLine (AB,B,f)),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine (AB,B,f)) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (AB,B,f))) -tuples_on BA
(width (ReplaceLine (AB,B,f))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine (AB,B,f)) } is set
Line (AB,x) is Relation-like NAT -defined BA -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on BA
(Line ((ReplaceLine (AB,B,f)),x)) ^ (Line ((ReplaceLine (x,B,X)),x)) is Relation-like NAT -defined BA -valued Function-like finite (width (ReplaceLine (AB,B,f))) + (width (ReplaceLine (x,B,X))) -element FinSequence-like FinSubsequence-like FinSequence of BA
(width (ReplaceLine (AB,B,f))) + (width (ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) -element FinSequence-like FinSubsequence-like Element of (width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA
width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) } is set
Line ((ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA
width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) } is set
Line ((BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) -element FinSequence-like FinSubsequence-like Element of (width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA
width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) } is set
Line ((ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA
width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) } is set
Line ((BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))),x) is Relation-like NAT -defined BA -valued Function-like finite width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) -element FinSequence-like FinSubsequence-like Element of (width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA
width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X)))) -tuples_on BA is functional non empty FinSequence-membered FinSequenceSet of BA
{ b₁ where b₁ is Relation-like NAT -defined BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of BA * : len b₁ = width (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) } is set
(ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (ReplaceLine ((BA,n,K,A,AB,x),B,(f ^ X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (BA,n,K,A,(ReplaceLine (AB,B,f)),(ReplaceLine (x,B,X))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
BA is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K,B
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
AB is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A,B
(B,n,K,A,BA,AB) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,(width BA) + (width AB),B
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA))) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card (Seg (width BA)),B
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Sgm (Seg (width BA)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width BA)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width BA))) -tuples_on NAT
(card (Seg (width BA))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width BA)) } is set
Segm ((B,n,K,A,BA,AB),(Sgm (Seg n)),(Sgm (Seg (width BA)))) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card (Seg (width BA)),B
Seg ((width BA) + (width AB)) is finite (width BA) + (width AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (width BA) + (width AB) ) } is set
(Seg ((width BA) + (width AB))) \ (Seg (width BA)) is finite without_zero Element of bool NAT
Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg ((width BA) + (width AB))) \ (Seg (width BA))),B
card ((Seg ((width BA) + (width AB))) \ (Seg (width BA))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg ((width BA) + (width AB))) \ (Seg (width BA))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg ((width BA) + (width AB))) \ (Seg (width BA))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) -tuples_on NAT
(card ((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg ((width BA) + (width AB))) \ (Seg (width BA))) } is set
Segm ((B,n,K,A,BA,AB),(Sgm (Seg n)),(Sgm ((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg ((width BA) + (width AB))) \ (Seg (width BA))),B
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (B,n,K,A,BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BX,MV] is set
{BX,MV} is non empty finite V37() set
{BX} is non empty trivial finite V37() 1 -element set
{{BX,MV},{BX}} is non empty finite V37() without_zero V103() set
f is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, width BA,B
Indices f is set
dom f is finite Element of bool NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width f) is finite width f -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width f ) } is set
[:(dom f),(Seg (width f)):] is Relation-like finite set
dom BA is finite Element of bool NAT
[:(Seg n),(Seg (width BA)):] is Relation-like finite set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
idseq (width BA) is Relation-like NAT -defined Function-like finite width BA -element FinSequence-like FinSubsequence-like set
id (Seg (width BA)) is Relation-like Seg (width BA) -defined Seg (width BA) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width BA)),(Seg (width BA)):]
[:(Seg (width BA)),(Seg (width BA)):] is Relation-like finite set
bool [:(Seg (width BA)),(Seg (width BA)):] is finite V37() set
(idseq (width BA)) . c₁₃ is set
(Sgm (Seg (width BA))) . c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
dom (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) is finite Element of bool NAT
len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA))))) is finite len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) ) } is set
(Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) * (BX,MV) is Element of B
Col ((Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))),c₁₃) is Relation-like NAT -defined B -valued Function-like finite len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) -element FinSequence-like FinSubsequence-like Element of (len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA))))) -tuples_on B
(len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA))))) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len (Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))) } is set
(Col ((Segm ((B,n,K,A,BA,AB),(Seg n),(Seg (width BA)))),c₁₃)) . lA is set
Col ((B,n,K,A,BA,AB),((Sgm (Seg (width BA))) . c₁₃)) is Relation-like NAT -defined B -valued Function-like finite len (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (len (B,n,K,A,BA,AB)) -tuples_on B
(len (B,n,K,A,BA,AB)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len (B,n,K,A,BA,AB) } is set
(Col ((B,n,K,A,BA,AB),((Sgm (Seg (width BA))) . c₁₃))) . lA is set
Col (BA,c₁₃) is Relation-like NAT -defined B -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on B
(len BA) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len BA } is set
(Col (BA,c₁₃)) . lA is set
BA * (BX,MV) is Element of B
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg ((width BA) + (width AB))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((width BA) + (width AB)) - (width BA) is V105() ext-real complex set
dom AB is finite Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,x] is set
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
c₁₃ is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, width AB,B
Indices c₁₃ is set
dom c₁₃ is finite Element of bool NAT
width c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width c₁₃) is finite width c₁₃ -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width c₁₃ ) } is set
[:(dom c₁₃),(Seg (width c₁₃)):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
[:(Seg n),(Seg (width AB)):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) is finite Element of bool NAT
len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA)))))) is finite len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) ) } is set
(Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) * (x,x) is Element of B
Col ((Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))),y) is Relation-like NAT -defined B -valued Function-like finite len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) -element FinSequence-like FinSubsequence-like Element of (len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA)))))) -tuples_on B
(len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA)))))) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len (Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))) } is set
(Col ((Segm ((B,n,K,A,BA,AB),(Seg n),((Seg ((width BA) + (width AB))) \ (Seg (width BA))))),y)) . y is set
(Sgm ((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Col ((B,n,K,A,BA,AB),((Sgm ((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) . y)) is Relation-like NAT -defined B -valued Function-like finite len (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (len (B,n,K,A,BA,AB)) -tuples_on B
(Col ((B,n,K,A,BA,AB),((Sgm ((Seg ((width BA) + (width AB))) \ (Seg (width BA)))) . y))) . y is set
(width BA) + y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col ((B,n,K,A,BA,AB),((width BA) + y)) is Relation-like NAT -defined B -valued Function-like finite len (B,n,K,A,BA,AB) -element FinSequence-like FinSubsequence-like Element of (len (B,n,K,A,BA,AB)) -tuples_on B
(Col ((B,n,K,A,BA,AB),((width BA) + y))) . y is set
Col (AB,y) is Relation-like NAT -defined B -valued Function-like finite len AB -element FinSequence-like FinSubsequence-like Element of (len AB) -tuples_on B
(len AB) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b₁ where b₁ is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b₁ = len AB } is set
(Col (AB,y)) . y is set
c₁₃ * (x,x) is Element of B
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^^ A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K ^^ A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),AB,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width AB) + (width BA), the carrier of n
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) + (width BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Seg (len K)),(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width K)), the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Sgm (Seg (len K))),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width K)), the carrier of n
Indices ( the carrier of n,(len K),(width K),(width A),AB,BA) is set
dom ( the carrier of n,(len K),(width K),(width A),AB,BA) is finite Element of bool NAT
width ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA)) is finite width ( the carrier of n,(len K),(width K),(width A),AB,BA) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,(len K),(width K),(width A),AB,BA) ) } is set
[:(dom ( the carrier of n,(len K),(width K),(width A),AB,BA)),(Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))):] is Relation-like finite set
(width K) + (width A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg ((width K) + (width A)) is finite (width K) + (width A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (width K) + (width A) ) } is set
[:(Seg (len K)),(Seg ((width K) + (width A))):] is Relation-like finite set
[:(Seg (len K)),(Seg (width K)):] is Relation-like finite set
(Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K)) is finite without_zero Element of bool NAT
[:(Seg (len K)),((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))):] is Relation-like finite set
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Seg (len K)),((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))), the carrier of n
card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K)))) -tuples_on NAT
(card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))) } is set
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Sgm (Seg (len K))),(Sgm ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg (width ( the carrier of n,(len K),(width K),(width A),AB,BA))) \ (Seg (width K))), the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^^ A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K ^^ A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),AB,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width AB) + (width BA), the carrier of n
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) + (width BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^^ A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A ^^ K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),AB,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width AB) + (width BA), the carrier of n
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) + (width BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,(len K),(width A),(width K),BA,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width BA) + (width AB), the carrier of n
(width BA) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width A),(width K),BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width K),(width A),AB,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) \ (Seg (width K)) is finite without_zero Element of bool NAT
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Seg (len K)),((Seg (width A)) \ (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg (width A)) \ (Seg (width K))), the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card ((Seg (width A)) \ (Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm ((Seg (width A)) \ (Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width A)) \ (Seg (width K))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (width A)) \ (Seg (width K)))) -tuples_on NAT
(card ((Seg (width A)) \ (Seg (width K)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width A)) \ (Seg (width K))) } is set
Segm (( the carrier of n,(len K),(width K),(width A),AB,BA),(Sgm (Seg (len K))),(Sgm ((Seg (width A)) \ (Seg (width K))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg (width A)) \ (Seg (width K))), the carrier of n
width ( the carrier of n,(len K),(width A),(width K),BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width A),(width K),BA,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (( the carrier of n,(len K),(width A),(width K),BA,AB),(Seg (len K)),(Seg (width A))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width A)), the carrier of n
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (( the carrier of n,(len K),(width A),(width K),BA,AB),(Sgm (Seg (len K))),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width A)), the carrier of n
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. (K,(len A),n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,n, the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
(len A) |-> (n |-> (0. K)) is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len A) -tuples_on (n -tuples_on the carrier of K)
(len A) -tuples_on (n -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
(n -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (n -tuples_on the carrier of K) * : len b₁ = len A } is set
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
(Seg (len A)) --> (n |-> (0. K)) is Relation-like Seg (len A) -defined Seg (len A) -defined n -tuples_on the carrier of K -valued {(n |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len A)),{(n |-> (0. K))}:]
{(n |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len A)),{(n |-> (0. K))}:] is Relation-like finite set
bool [:(Seg (len A)),{(n |-> (0. K))}:] is finite V37() set
the_rank_of A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A ^^ B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
the_rank_of (A ^^ B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A, width A, the carrier of K
AB is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A, width B, the carrier of K
( the carrier of K,(len A),(width A),(width B),x,AB) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,(width x) + (width AB), the carrier of K
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width x) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of K,(len A),(width A),(width B),x,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) + (width B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of K,(len A),(width A),(width B),x,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len ( the carrier of K,(len A),(width A),(width B),x,AB)) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ( the carrier of K,(len A),(width A),(width B),x,AB) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of K,(len A),(width A),(width B),x,AB)) -tuples_on the carrier of K
(len ( the carrier of K,(len A),(width A),(width B),x,AB)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len ( the carrier of K,(len A),(width A),(width B),x,AB) } is set
Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)) is finite len ( the carrier of K,(len A),(width A),(width B),x,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of K,(len A),(width A),(width B),x,AB) ) } is set
(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))) --> (0. K) is Relation-like Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)) -defined Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))),{(0. K)}:]
[:(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))),{(0. K)}:] is Relation-like finite set
bool [:(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))),{(0. K)}:] is finite V37() set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (width ( the carrier of K,(len A),(width A),(width B),x,AB)) is finite width ( the carrier of K,(len A),(width A),(width B),x,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,(len A),(width A),(width B),x,AB) ) } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width ( the carrier of K,(len A),(width A),(width B),x,AB))) \ (Seg (width A)) is finite without_zero Element of bool NAT
BX - (width A) is V105() ext-real complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV + (width A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((len ( the carrier of K,(len A),(width A),(width B),x,AB)) |-> (0. K)) . lA is set
dom B is finite Element of bool NAT
Col (AB,MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite len AB -element FinSequence-like FinSubsequence-like Element of (len AB) -tuples_on the carrier of K
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len AB) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len AB } is set
(Col (AB,MV)) . lA is set
AB * (lA,MV) is Element of the carrier of K
[lA,MV] is set
{lA,MV} is non empty finite V37() set
{lA} is non empty trivial finite V37() 1 -element set
{{lA,MV},{lA}} is non empty finite V37() without_zero V103() set
Indices B is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
Col (B,MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of K
(len B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len B } is set
len (Col (B,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((len ( the carrier of K,(len A),(width A),(width B),x,AB)) |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (( the carrier of K,(len A),(width A),(width B),x,AB),BX) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ( the carrier of K,(len A),(width A),(width B),x,AB) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of K,(len A),(width A),(width B),x,AB)) -tuples_on the carrier of K
Segm (( the carrier of K,(len A),(width A),(width B),x,AB),(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))),(Seg (width A))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))), card (Seg (width A)), the carrier of K
card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)))) -tuples_on NAT
(card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))) } is set
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (( the carrier of K,(len A),(width A),(width B),x,AB),(Sgm (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB)))),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))), card (Seg (width A)), the carrier of K
the_rank_of (Segm (( the carrier of K,(len A),(width A),(width B),x,AB),(Seg (len ( the carrier of K,(len A),(width A),(width B),x,AB))),(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K ^^ A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K ^^ A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) --> (0. n) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. n)}:]
[:(Seg (width A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. n)}:] is finite V37() set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),BA,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width BA) + (width B), the carrier of n
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) + (width B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is finite without_zero Element of bool NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (A,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
width ( the carrier of n,(len K),(width K),(width A),BA,B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) + (width A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ( the carrier of n,(len K),(width K),(width A),BA,B)) is finite width ( the carrier of n,(len K),(width K),(width A),BA,B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,(len K),(width K),(width A),BA,B) ) } is set
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (( the carrier of n,(len K),(width K),(width A),BA,B),(Seg (len K)),(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width K)), the carrier of n
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (( the carrier of n,(len K),(width K),(width A),BA,B),(Sgm (Seg (len K))),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width K)), the carrier of n
(Sgm (Seg (len K))) . f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
idseq (len K) is Relation-like NAT -defined Function-like finite len K -element FinSequence-like FinSubsequence-like set
id (Seg (len K)) is Relation-like Seg (len K) -defined Seg (len K) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len K)),(Seg (len K)):]
[:(Seg (len K)),(Seg (len K)):] is Relation-like finite set
bool [:(Seg (len K)),(Seg (len K)):] is finite V37() set
(idseq (len K)) . f is set
(card (Seg (width K))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like Element of (card (Seg (width K))) -tuples_on the carrier of n
(card (Seg (width K))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = card (Seg (width K)) } is set
Seg (card (Seg (width K))) is finite card (Seg (width K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (width K)) ) } is set
(Seg (card (Seg (width K)))) --> (0. n) is Relation-like Seg (card (Seg (width K))) -defined Seg (card (Seg (width K))) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card (Seg (width K)))),{(0. n)}:]
[:(Seg (card (Seg (width K)))),{(0. n)}:] is Relation-like finite set
bool [:(Seg (card (Seg (width K)))),{(0. n)}:] is finite V37() set
Line (K,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Line (( the carrier of n,(len K),(width K),(width A),BA,B),f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),BA,B) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),BA,B)) -tuples_on the carrier of n
(width ( the carrier of n,(len K),(width K),(width A),BA,B)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,(len K),(width K),(width A),BA,B) } is set
(Line (( the carrier of n,(len K),(width K),(width A),BA,B),f)) * (Sgm (Seg (width K))) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (A,f)) . X is set
((width A) |-> (0. n)) . X is set
Line (BA,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of n
(width BA) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width BA } is set
len (Line (BA,f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (B,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of n
(width B) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width B } is set
len (Line (B,f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X + (width K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,(len K),(width K),(width A),BA,B) * (f,(X + (width K))) is Element of the carrier of n
(Line (( the carrier of n,(len K),(width K),(width A),BA,B),f)) . (X + (width K)) is set
(Line (BA,f)) ^ (Line (B,f)) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width BA) + (width B) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(width BA) + (width B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((Line (BA,f)) ^ (Line (B,f))) . (X + (width K)) is set
(Line (B,f)) . X is set
Indices BA is set
dom BA is finite Element of bool NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
[:(dom BA),(Seg (width BA)):] is Relation-like finite set
the_rank_of BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is finite without_zero Element of bool NAT
MV is finite without_zero Element of bool NAT
[:BX,MV:] is Relation-like finite set
card BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
EqSegm (BA,BX,MV) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BX, card BX, the carrier of n
Det (EqSegm (BA,BX,MV)) is Element of the carrier of n
Permutations (card BX) is set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
FinOmega (Permutations (card BX)) is Element of K96((Permutations (card BX)))
K96((Permutations (card BX))) is V24() set
Path_product (EqSegm (BA,BX,MV)) is Relation-like Permutations (card BX) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations (card BX)), the carrier of n:]
[:(Permutations (card BX)), the carrier of n:] is Relation-like set
bool [:(Permutations (card BX)), the carrier of n:] is set
K103((Permutations (card BX)), the carrier of n, the addF of n,(FinOmega (Permutations (card BX))),(Path_product (EqSegm (BA,BX,MV)))) is Element of the carrier of n
(Sgm (Seg (len K))) .: BX is finite V195() V196() V197() V200() set
(Sgm (Seg (width K))) .: MV is finite V195() V196() V197() V200() set
Segm (K,BX,MV) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BX, card MV, the carrier of n
Sgm BX is Relation-like NAT -defined NAT -valued Function-like finite card BX -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BX) -tuples_on NAT
(card BX) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BX } is set
Sgm MV is Relation-like NAT -defined NAT -valued Function-like finite card MV -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card MV) -tuples_on NAT
(card MV) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card MV } is set
Segm (K,(Sgm BX),(Sgm MV)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BX, card MV, the carrier of n
lA is finite without_zero Element of bool NAT
c₁₃ is finite without_zero Element of bool NAT
card lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (( the carrier of n,(len K),(width K),(width A),BA,B),lA,c₁₃) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card lA, card c₁₃, the carrier of n
Sgm lA is Relation-like NAT -defined NAT -valued Function-like finite card lA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card lA) -tuples_on NAT
(card lA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card lA } is set
Sgm c₁₃ is Relation-like NAT -defined NAT -valued Function-like finite card c₁₃ -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card c₁₃) -tuples_on NAT
(card c₁₃) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card c₁₃ } is set
Segm (( the carrier of n,(len K),(width K),(width A),BA,B),(Sgm lA),(Sgm c₁₃)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card lA, card c₁₃, the carrier of n
EqSegm (K,BX,MV) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BX, card BX, the carrier of n
dom ( the carrier of n,(len K),(width K),(width A),BA,B) is finite Element of bool NAT
len ( the carrier of n,(len K),(width K),(width A),BA,B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len ( the carrier of n,(len K),(width K),(width A),BA,B)) is finite len ( the carrier of n,(len K),(width K),(width A),BA,B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of n,(len K),(width K),(width A),BA,B) ) } is set
[:lA,(Seg (width K)):] is Relation-like finite set
Indices ( the carrier of n,(len K),(width K),(width A),BA,B) is set
[:(dom ( the carrier of n,(len K),(width K),(width A),BA,B)),(Seg (width ( the carrier of n,(len K),(width K),(width A),BA,B))):] is Relation-like finite set
1 + (width K) is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
(Seg (width ( the carrier of n,(len K),(width K),(width A),BA,B))) \ c₁₃ is finite without_zero Element of bool NAT
(0. n) * (Line (K,f)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(0. n) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is 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 quasi_total V223( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] ((0. n),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,(Line (K,f)),((0. n) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng (Sgm BX) is finite V195() V196() V197() V200() set
dom (Sgm BX) is finite card BX -element Element of bool NAT
x is set
(Sgm BX) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
rng (Sgm MV) is finite V195() V196() V197() V200() set
Seg (card BX) is finite card BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card BX ) } is set
dom (Line (K,f)) is finite width K -element Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((Segm (K,BX,MV)),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (Segm (K,BX,MV)) -element FinSequence-like FinSubsequence-like Element of (width (Segm (K,BX,MV))) -tuples_on the carrier of n
width (Segm (K,BX,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (Segm (K,BX,MV))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (Segm (K,BX,MV)) } is set
(Line (K,f)) * (Sgm MV) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
(0. n) * (Line ((Segm (K,BX,MV)),x)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (Segm (K,BX,MV)) -element FinSequence-like FinSubsequence-like Element of (width (Segm (K,BX,MV))) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Line ((Segm (K,BX,MV)),x)),((0. n) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Det (EqSegm (K,BX,MV)) is Element of the carrier of n
Path_product (EqSegm (K,BX,MV)) is Relation-like Permutations (card BX) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations (card BX)), the carrier of n:]
K103((Permutations (card BX)), the carrier of n, the addF of n,(FinOmega (Permutations (card BX))),(Path_product (EqSegm (K,BX,MV)))) is Element of the carrier of n
(0. n) * (Det (EqSegm (K,BX,MV))) is Element of the carrier of n
K560( the carrier of n, the multF of n,(0. n),(Det (EqSegm (K,BX,MV)))) is Element of the carrier of n
Line ((EqSegm (K,BX,MV)),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (EqSegm (K,BX,MV)) -element FinSequence-like FinSubsequence-like Element of (width (EqSegm (K,BX,MV))) -tuples_on the carrier of n
width (EqSegm (K,BX,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (EqSegm (K,BX,MV))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (EqSegm (K,BX,MV)) } is set
ReplaceLine ((EqSegm (K,BX,MV)),x,(Line ((EqSegm (K,BX,MV)),x))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card BX, card BX, the carrier of n
Det (ReplaceLine ((EqSegm (K,BX,MV)),x,(Line ((EqSegm (K,BX,MV)),x)))) is Element of the carrier of n
Path_product (ReplaceLine ((EqSegm (K,BX,MV)),x,(Line ((EqSegm (K,BX,MV)),x)))) is Relation-like Permutations (card BX) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations (card BX)), the carrier of n:]
K103((Permutations (card BX)), the carrier of n, the addF of n,(FinOmega (Permutations (card BX))),(Path_product (ReplaceLine ((EqSegm (K,BX,MV)),x,(Line ((EqSegm (K,BX,MV)),x)))))) is Element of the carrier of n
(dom ( the carrier of n,(len K),(width K),(width A),BA,B)) \ lA is finite Element of bool NAT
the_rank_of (Segm (( the carrier of n,(len K),(width K),(width A),BA,B),lA,c₁₃)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty set
K is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of n
<*K*> is Relation-like NAT -defined n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
n * is functional non empty FinSequence-membered FinSequenceSet of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*K*> @ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
len <*K*> is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
width <*K*> is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (<*K*> @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (<*K*> @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty set
n * is functional non empty FinSequence-membered FinSequenceSet of n
K is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of n
(n,K) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*K*> is Relation-like NAT -defined n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
A is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
Line (A,1) is Relation-like NAT -defined n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on n
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width A } is set
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((n,K),1) is Relation-like NAT -defined n -valued Function-like finite width (n,K) -element FinSequence-like FinSubsequence-like Element of (width (n,K)) -tuples_on n
width (n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (n,K)) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width (n,K) } is set
(n,K) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, width A,n
B . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is non empty set
n * is functional non empty FinSequence-membered FinSequenceSet of n
K is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,K) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1,n
<*K*> is Relation-like NAT -defined n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*K*> @ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
A is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (A,1) is Relation-like NAT -defined n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on n
(len A) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = len A } is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,K) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
len (n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (n,K) is finite Element of bool NAT
Line ((n,K),1) is Relation-like NAT -defined n -valued Function-like finite width (n,K) -element FinSequence-like FinSubsequence-like Element of (width (n,K)) -tuples_on n
width (n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (n,K)) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width (n,K) } is set
A @ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
len (A @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((A @),1) is Relation-like NAT -defined n -valued Function-like finite width (A @) -element FinSequence-like FinSubsequence-like Element of (width (A @)) -tuples_on n
width (A @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (A @)) -tuples_on n is functional non empty FinSequence-membered FinSequenceSet of n
{ b₁ where b₁ is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like Element of n * : len b₁ = width (A @) } is set
(n,K) is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K,n
(n,K) @ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
(A @) @ is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,K) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K, 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 constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K, the carrier of n
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
( the carrier of n,K) + ( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(K + A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K + A), the carrier of n
len (K + A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(K + A)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K + A), the carrier of n
[1,(K + A)] is set
{1,(K + A)} is non empty finite V37() set
{{1,(K + A)},{1}} is non empty finite V37() without_zero V103() set
{[1,(K + A)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len ( the carrier of n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
AB + x is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,AB,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(AB + x)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (AB + x), the carrier of n
len (AB + x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(AB + x)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (AB + x), the carrier of n
[1,(AB + x)] is set
{1,(AB + x)} is non empty finite V37() set
{{1,(AB + x)},{1}} is non empty finite V37() without_zero V103() set
{[1,(AB + x)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len ( the carrier of n,(AB + x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,(AB + x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (( the carrier of n,K) + ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (( the carrier of n,K) + ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
Indices (( the carrier of n,K) + ( the carrier of n,A)) is set
dom (( the carrier of n,K) + ( the carrier of n,A)) is finite Element of bool NAT
Seg (width (( the carrier of n,K) + ( the carrier of n,A))) is finite width (( the carrier of n,K) + ( the carrier of n,A)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (( the carrier of n,K) + ( the carrier of n,A)) ) } is set
[:(dom (( the carrier of n,K) + ( the carrier of n,A))),(Seg (width (( the carrier of n,K) + ( the carrier of n,A)))):] is Relation-like finite set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BX,MV] is set
{BX,MV} is non empty finite V37() set
{BX} is non empty trivial finite V37() 1 -element set
{{BX,MV},{BX}} is non empty finite V37() without_zero V103() set
(( the carrier of n,K) + ( the carrier of n,A)) * (BX,MV) is Element of the carrier of n
( the carrier of n,(AB + x)) * (BX,MV) is Element of the carrier of n
dom ( the carrier of n,K) is finite Element of bool NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
[1,(len K)] is Element of [:NAT,NAT:]
{1,(len K)} is non empty finite V37() set
{{1,(len K)},{1}} is non empty finite V37() without_zero V103() set
Indices ( the carrier of n,K) is set
Seg (width ( the carrier of n,K)) is finite width ( the carrier of n,K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,K) ) } is set
[:(dom ( the carrier of n,K)),(Seg (width ( the carrier of n,K))):] is Relation-like finite set
Line ((( the carrier of n,K) + ( the carrier of n,A)),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (( the carrier of n,K) + ( the carrier of n,A)) -element FinSequence-like FinSubsequence-like Element of (width (( the carrier of n,K) + ( the carrier of n,A))) -tuples_on the carrier of n
(width (( the carrier of n,K) + ( the carrier of n,A))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (( the carrier of n,K) + ( the carrier of n,A)) } is set
Line (( the carrier of n,K),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,K) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,K)) -tuples_on the carrier of n
(width ( the carrier of n,K)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,K) } is set
Line (( the carrier of n,A),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,A)) -tuples_on the carrier of n
(width ( the carrier of n,A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,A) } is set
(Line (( the carrier of n,K),1)) + (Line (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (( the carrier of n,K),1)),(Line (( the carrier of n,A),1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K + (Line (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,K,(Line (( the carrier of n,A),1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,K) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, 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 constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len K, the carrier of n
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() without_zero V103() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*K*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,K) + ( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(K + A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (K + A),1, the carrier of n
len (K + A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(K + A)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K + A), the carrier of n
[1,(K + A)] is set
{1,(K + A)} is non empty finite V37() set
{{1,(K + A)},{1}} is non empty finite V37() without_zero V103() set
{[1,(K + A)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(K + A)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ( the carrier of n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
AB + x is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,AB,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(AB + x)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (AB + x),1, the carrier of n
len (AB + x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(AB + x)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (AB + x), the carrier of n
[1,(AB + x)] is set
{1,(AB + x)} is non empty finite V37() set
{{1,(AB + x)},{1}} is non empty finite V37() without_zero V103() set
{[1,(AB + x)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(AB + x)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ( the carrier of n,(AB + x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (( the carrier of n,K) + ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (( the carrier of n,K) + ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom ( the carrier of n,K) is finite Element of bool NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
[(len K),1] is Element of [:NAT,NAT:]
{(len K),1} is non empty finite V37() set
{(len K)} is non empty trivial finite V37() 1 -element set
{{(len K),1},{(len K)}} is non empty finite V37() without_zero V103() set
Indices ( the carrier of n,K) is set
Seg (width ( the carrier of n,K)) is finite width ( the carrier of n,K) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,K) ) } is set
[:(dom ( the carrier of n,K)),(Seg (width ( the carrier of n,K))):] is Relation-like finite set
Col ((( the carrier of n,K) + ( the carrier of n,A)),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len (( the carrier of n,K) + ( the carrier of n,A)) -element FinSequence-like FinSubsequence-like Element of (len (( the carrier of n,K) + ( the carrier of n,A))) -tuples_on the carrier of n
(len (( the carrier of n,K) + ( the carrier of n,A))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len (( the carrier of n,K) + ( the carrier of n,A)) } is set
Col (( the carrier of n,K),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len ( the carrier of n,K) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of n,K)) -tuples_on the carrier of n
(len ( the carrier of n,K)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len ( the carrier of n,K) } is set
Col (( the carrier of n,A),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of n,A)) -tuples_on the carrier of n
(len ( the carrier of n,A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len ( the carrier of n,A) } is set
(Col (( the carrier of n,K),1)) + (Col (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Col (( the carrier of n,K),1)),(Col (( the carrier of n,A),1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K + (Col (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,K,(Col (( the carrier of n,A),1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is Element of the carrier of n
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K * ( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
K * A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,A,(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
( the carrier of n,(K * A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K * A), the carrier of n
len (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(K * A)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K * A), the carrier of n
[1,(K * A)] is set
{1,(K * A)} is non empty finite V37() set
{{1,(K * A)},{1}} is non empty finite V37() without_zero V103() set
{[1,(K * A)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len (K * ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((K * ( the carrier of n,A)),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (K * ( the carrier of n,A)) -element FinSequence-like FinSubsequence-like Element of (width (K * ( the carrier of n,A))) -tuples_on the carrier of n
width (K * ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (K * ( the carrier of n,A))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (K * ( the carrier of n,A)) } is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (( the carrier of n,A),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,A)) -tuples_on the carrier of n
(width ( the carrier of n,A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,A) } is set
K * (Line (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Line (( the carrier of n,A),1)),(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
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is Element of the carrier of n
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * ( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
K * A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (K,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,A,(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
( the carrier of n,(K * A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (K * A),1, the carrier of n
len (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(K * A)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (K * A), the carrier of n
[1,(K * A)] is set
{1,(K * A)} is non empty finite V37() set
{{1,(K * A)},{1}} is non empty finite V37() without_zero V103() set
{[1,(K * A)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(K * A)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (K * ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(K * A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (K * ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col ((K * ( the carrier of n,A)),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len (K * ( the carrier of n,A)) -element FinSequence-like FinSubsequence-like Element of (len (K * ( the carrier of n,A))) -tuples_on the carrier of n
len (K * ( the carrier of n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len (K * ( the carrier of n,A))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len (K * ( the carrier of n,A)) } is set
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (( the carrier of n,A),1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of n,A)) -tuples_on the carrier of n
(len ( the carrier of n,A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len ( the carrier of n,A) } is set
K * (Col (( the carrier of n,A),1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite len ( the carrier of n,A) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of n,A)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Col (( the carrier of n,A),1)),(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
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
( the carrier of K,(n |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (n |-> (0. K)), the carrier of K
len (n |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(n |-> (0. K))*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (n |-> (0. K)), the carrier of K
[1,(n |-> (0. K))] is set
{1,(n |-> (0. K))} is non empty finite V37() set
{{1,(n |-> (0. K))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
0. (K,1,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1,n, the carrier of K
1 |-> (n |-> (0. K)) is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of 1 -tuples_on (n -tuples_on the carrier of K)
1 -tuples_on (n -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
(n -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (n -tuples_on the carrier of K) * : len b₁ = 1 } is set
(Seg 1) --> (n |-> (0. K)) is Relation-like Seg 1 -defined Seg 1 -defined n -tuples_on the carrier of K -valued {(n |-> (0. K))} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg 1),{(n |-> (0. K))}:]
{(n |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg 1),{(n |-> (0. K))}:] is Relation-like finite set
bool [:(Seg 1),{(n |-> (0. K))}:] is finite V37() set
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1,n, the carrier of K
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom A is finite Element of bool NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BA,AB] is set
{BA,AB} is non empty finite V37() set
{BA} is non empty trivial finite V37() 1 -element set
{{BA,AB},{BA}} is non empty finite V37() without_zero V103() set
Indices A is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
Indices (0. (K,1,n)) is set
dom (0. (K,1,n)) is finite Element of bool NAT
width (0. (K,1,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (K,1,n))) is finite width (0. (K,1,n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,1,n)) ) } is set
[:(dom (0. (K,1,n))),(Seg (width (0. (K,1,n)))):] is Relation-like finite set
(0. (K,1,n)) * (BA,AB) is Element of the carrier of K
(n |-> (0. K)) . AB is set
Line (A,BA) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
(width A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width A } is set
(Line (A,BA)) . AB is set
A * (BA,AB) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
( the carrier of K,(n |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (n |-> (0. K)),1, the carrier of K
len (n |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(n |-> (0. K))*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (n |-> (0. K)), the carrier of K
[1,(n |-> (0. K))] is set
{1,(n |-> (0. K))} is non empty finite V37() set
{{1,(n |-> (0. K))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n |-> (0. K))*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
0. (K,n,1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,1, the carrier of K
1 -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = 1 } is set
1 |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of K
(Seg 1) --> (0. K) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of K -valued {(0. K)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. K)}:]
[:(Seg 1),{(0. K)}:] is Relation-like finite set
bool [:(Seg 1),{(0. K)}:] is finite V37() set
n |-> (1 |-> (0. K)) is Relation-like NAT -defined 1 -tuples_on the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (1 -tuples_on the carrier of K)
n -tuples_on (1 -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of K
(1 -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of K) * : len b₁ = n } is set
(Seg n) --> (1 |-> (0. K)) is Relation-like non-empty Seg n -defined Seg n -defined 1 -tuples_on the carrier of K -valued {(1 |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(1 |-> (0. K))}:]
{(1 |-> (0. K))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg n),{(1 |-> (0. K))}:] is Relation-like finite set
bool [:(Seg n),{(1 |-> (0. K))}:] is finite V37() set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BA,AB] is set
{BA,AB} is non empty finite V37() set
{BA} is non empty trivial finite V37() 1 -element set
{{BA,AB},{BA}} is non empty finite V37() without_zero V103() set
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,1, the carrier of K
Indices A is set
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
Indices (0. (K,n,1)) is set
dom (0. (K,n,1)) is finite Element of bool NAT
width (0. (K,n,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (K,n,1))) is finite width (0. (K,n,1)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,n,1)) ) } is set
[:(dom (0. (K,n,1))),(Seg (width (0. (K,n,1)))):] is Relation-like finite set
(0. (K,n,1)) * (BA,AB) is Element of the carrier of K
(n |-> (0. K)) . BA is set
Col (A,AB) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
(len A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len A } is set
(Col (A,AB)) . BA is set
A * (BA,AB) is Element of the carrier of K
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,A) is set
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B is set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
Col (A,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len A } is set
(len A) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
(Seg (len A)) --> (0. K) is Relation-like Seg (len A) -defined Seg (len A) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len A)),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg (len A)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (len A)),{(0. K)}:] is finite V37() set
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
BA is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,B,BA) is set
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width B & width b₁ = width BA & B * b₁ = BA ) } is set
Col (BA,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on the carrier of K
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len BA) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len BA } is set
(len BA) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on the carrier of K
Seg (len BA) is finite len BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BA ) } is set
(Seg (len BA)) --> (0. K) is Relation-like Seg (len BA) -defined Seg (len BA) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len BA)),{(0. K)}:]
[:(Seg (len BA)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (len BA)),{(0. K)}:] is finite V37() set
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Indices BA is set
dom BA is finite Element of bool NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
[:(dom BA),(Seg (width BA)):] is Relation-like finite set
[:(Seg (len BA)),(Seg (width BA)):] is Relation-like finite set
[f,n] is set
{f,n} is non empty finite V37() set
{f} is non empty trivial finite V37() 1 -element set
{{f,n},{f}} is non empty finite V37() without_zero V103() set
BA * (f,n) is Element of the carrier of K
Line (B,f) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width B } is set
Col (x,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of K
(len x) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len x } is set
(Line (B,f)) "*" (Col (x,n)) is Element of the carrier of K
mlt ((Line (B,f)),(Col (x,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 quasi_total V223( 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
K498( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (B,f)),(Col (x,n))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (B,f)),(Col (x,n)))) 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 quasi_total V223( the carrier of K) V224( the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (mlt ((Line (B,f)),(Col (x,n)))) is Element of the carrier of K
(0. K) * (Line (B,f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(0. K) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((0. K),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K501( the carrier of K, the carrier of K,(Line (B,f)),((0. K) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum ((0. K) * (Line (B,f))) is Element of the carrier of K
the addF of K $$ ((0. K) * (Line (B,f))) is Element of the carrier of K
Sum (Line (B,f)) is Element of the carrier of K
the addF of K $$ (Line (B,f)) is Element of the carrier of K
(0. K) * (Sum (Line (B,f))) is Element of the carrier of K
K560( the carrier of K, the multF of K,(0. K),(Sum (Line (B,f)))) is Element of the carrier of K
((len BA) |-> (0. K)) . f is set
(Col (BA,n)) . f is set
len (Col (BA,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((len BA) |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Element of the carrier of n
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,B,BA) is set
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width B & width b₁ = width BA & B * b₁ = BA ) } is set
K * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,B,(K * BA)) is set
width (K * BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width B & width b₁ = width (K * BA) & B * b₁ = K * BA ) } is set
(n,(K * B),(K * BA)) is set
width (K * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (K * B) & width b₁ = width (K * BA) & (K * B) * b₁ = K * BA ) } is set
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * (K * A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
B * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * (B * A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(K * B) * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Element of the carrier of n
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,A,B) is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width A & width b₁ = width B & A * b₁ = B ) } is set
K * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,(K * A),(K * B)) is set
width (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (K * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (K * A) & width b₁ = width (K * B) & (K * A) * b₁ = K * B ) } is set
BA is set
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K " is Element of the carrier of n
(K ") * (K * A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(K ") * 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 quasi_total V223( 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
K560( the carrier of n, the multF of n,(K "),K) is Element of the carrier of n
((K ") * K) * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
(1_ n) * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(K ") * (K * B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
((K ") * K) * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(1_ n) * B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BA is set
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(K * A) * AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is non empty set
n * is functional non empty FinSequence-membered FinSequenceSet of n
K is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
Indices K is set
dom K is finite Element of bool NAT
[:(dom K),(Seg (width K)):] is Relation-like finite set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[B,BA] is set
{B,BA} is non empty finite V37() set
{B} is non empty trivial finite V37() 1 -element set
{{B,BA},{B}} is non empty finite V37() without_zero V103() set
K * (B,BA) is Element of n
A * (B,BA) is Element of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,A,B) is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width A & width b₁ = width B & A * b₁ = B ) } is set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
K + BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,A,AB) is set
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width A & width b₁ = width AB & A * b₁ = AB ) } is set
B + AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,A,(B + AB)) is set
width (B + AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width A & width b₁ = width (B + AB) & A * b₁ = B + AB ) } is set
width (K + BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (K + BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (A * K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A * K) + (A * BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ((A * K) + (A * BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * (K + BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (A * (K + BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width (A * K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A * K) + (A * BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width ((A * K) + (A * BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * (K + BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width (A * (K + BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (A * K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ((A * K) + (A * BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A * (K + BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * (K + BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A * K) + (A * BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * (K + BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A * K) + (A * BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * (K + BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A * K) + (A * BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of BA is non empty non trivial V103() set
the carrier of BA * is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
AB is Element of the carrier of BA
x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
f is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (f,B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
(width f) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width f } is set
AB * (Line (f,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
AB multfield is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
[: the carrier of BA, the carrier of BA:] is Relation-like set
bool [: the carrier of BA, the carrier of BA:] is set
the multF of BA is Relation-like [: the carrier of BA, the carrier of BA:] -defined the carrier of BA -valued Function-like total quasi_total V223( the carrier of BA) Element of bool [:[: the carrier of BA, the carrier of BA:], the carrier of BA:]
[:[: the carrier of BA, the carrier of BA:], the carrier of BA:] is Relation-like set
bool [:[: the carrier of BA, the carrier of BA:], the carrier of BA:] is set
id the carrier of BA is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
the multF of BA [;] (AB,(id the carrier of BA)) is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
K501( the carrier of BA, the carrier of BA,(Line (f,B)),(AB multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
ReplaceLine (f,B,(AB * (Line (f,B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
X is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
(BA,f,X) is set
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA * : ( len b₁ = width f & width b₁ = width X & f * b₁ = X ) } is set
Line (X,B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of BA
(width X) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width X } is set
AB * (Line (X,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(Line (X,B)),(AB multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
ReplaceLine (X,B,(AB * (Line (X,B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
(BA,(ReplaceLine (f,B,(AB * (Line (f,B))))),(ReplaceLine (X,B,(AB * (Line (X,B)))))) is set
width (ReplaceLine (f,B,(AB * (Line (f,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (X,B,(AB * (Line (X,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA * : ( len b₁ = width (ReplaceLine (f,B,(AB * (Line (f,B))))) & width b₁ = width (ReplaceLine (X,B,(AB * (Line (X,B))))) & (ReplaceLine (f,B,(AB * (Line (f,B))))) * b₁ = ReplaceLine (X,B,(AB * (Line (X,B)))) ) } is set
Indices (ReplaceLine (X,B,(AB * (Line (X,B))))) is set
dom (ReplaceLine (X,B,(AB * (Line (X,B))))) is finite Element of bool NAT
Seg (width (ReplaceLine (X,B,(AB * (Line (X,B)))))) is finite width (ReplaceLine (X,B,(AB * (Line (X,B))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (X,B,(AB * (Line (X,B))))) ) } is set
[:(dom (ReplaceLine (X,B,(AB * (Line (X,B)))))),(Seg (width (ReplaceLine (X,B,(AB * (Line (X,B))))))):] is Relation-like finite set
Indices X is set
dom X is finite Element of bool NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
len (AB * (Line (X,B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Line (X,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (AB * (Line (f,B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Line (f,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (ReplaceLine (f,B,(AB * (Line (f,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f * x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
(ReplaceLine (f,B,(AB * (Line (f,B))))) * x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
len ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (ReplaceLine (f,B,(AB * (Line (f,B)))))) is finite len (ReplaceLine (f,B,(AB * (Line (f,B))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (ReplaceLine (f,B,(AB * (Line (f,B))))) ) } is set
Indices ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) is set
dom ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) is finite Element of bool NAT
Seg (width ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x)) is finite width ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) ) } is set
[:(dom ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x)),(Seg (width ((ReplaceLine (f,B,(AB * (Line (f,B))))) * x))):] is Relation-like finite set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,y] is set
{y,y} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,y},{y}} is non empty finite V37() without_zero V103() set
X * (B,y) is Element of the carrier of BA
(Line (X,B)) . y is set
X * (y,y) is Element of the carrier of BA
Line (f,y) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
Col (x,y) is Relation-like NAT -defined the carrier of BA -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of BA
(len x) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = len x } is set
(Line (f,y)) "*" (Col (x,y)) is Element of the carrier of BA
mlt ((Line (f,y)),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
K498( the carrier of BA, the carrier of BA, the carrier of BA, the multF of BA,(Line (f,y)),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
Sum (mlt ((Line (f,y)),(Col (x,y)))) is Element of the carrier of BA
the addF of BA is Relation-like [: the carrier of BA, the carrier of BA:] -defined the carrier of BA -valued Function-like total quasi_total V223( the carrier of BA) V224( the carrier of BA) Element of bool [:[: the carrier of BA, the carrier of BA:], the carrier of BA:]
the addF of BA $$ (mlt ((Line (f,y)),(Col (x,y)))) is Element of the carrier of BA
Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (f,B,(AB * (Line (f,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA
(width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width (ReplaceLine (f,B,(AB * (Line (f,B))))) } is set
((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) * (y,y) is Element of the carrier of BA
(AB * (Line (f,B))) "*" (Col (x,y)) is Element of the carrier of BA
mlt ((AB * (Line (f,B))),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
K498( the carrier of BA, the carrier of BA, the carrier of BA, the multF of BA,(AB * (Line (f,B))),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
Sum (mlt ((AB * (Line (f,B))),(Col (x,y)))) is Element of the carrier of BA
the addF of BA $$ (mlt ((AB * (Line (f,B))),(Col (x,y)))) is Element of the carrier of BA
mlt ((Line (f,B)),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
K498( the carrier of BA, the carrier of BA, the carrier of BA, the multF of BA,(Line (f,B)),(Col (x,y))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
AB * (mlt ((Line (f,B)),(Col (x,y)))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
K501( the carrier of BA, the carrier of BA,(mlt ((Line (f,B)),(Col (x,y)))),(AB multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
Sum (AB * (mlt ((Line (f,B)),(Col (x,y))))) is Element of the carrier of BA
the addF of BA $$ (AB * (mlt ((Line (f,B)),(Col (x,y))))) is Element of the carrier of BA
Sum (mlt ((Line (f,B)),(Col (x,y)))) is Element of the carrier of BA
the addF of BA $$ (mlt ((Line (f,B)),(Col (x,y)))) is Element of the carrier of BA
AB * (Sum (mlt ((Line (f,B)),(Col (x,y))))) is Element of the carrier of BA
K560( the carrier of BA, the multF of BA,AB,(Sum (mlt ((Line (f,B)),(Col (x,y)))))) is Element of the carrier of BA
x is Element of the carrier of BA
AB * x is Element of the carrier of BA
K560( the carrier of BA, the multF of BA,AB,x) is Element of the carrier of BA
(AB * (Line (X,B))) . y is set
(ReplaceLine (X,B,(AB * (Line (X,B))))) * (y,y) is Element of the carrier of BA
Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),y) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (f,B,(AB * (Line (f,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA
(width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width (ReplaceLine (f,B,(AB * (Line (f,B))))) } is set
((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) * (y,y) is Element of the carrier of BA
(ReplaceLine (X,B,(AB * (Line (X,B))))) * (y,y) is Element of the carrier of BA
((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) * (y,y) is Element of the carrier of BA
(ReplaceLine (X,B,(AB * (Line (X,B))))) * (y,y) is Element of the carrier of BA
((ReplaceLine (f,B,(AB * (Line (f,B))))) * x) * (y,y) is Element of the carrier of BA
(ReplaceLine (X,B,(AB * (Line (X,B))))) * (y,y) is Element of the carrier of BA
len (ReplaceLine (X,B,(AB * (Line (X,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of BA is non empty non trivial V103() set
the carrier of BA * is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
0. BA is zero Element of the carrier of BA
AB is Element of the carrier of BA
x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (x,B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of BA
(width x) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width x } is set
AB * (Line (x,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of BA
AB multfield is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
[: the carrier of BA, the carrier of BA:] is Relation-like set
bool [: the carrier of BA, the carrier of BA:] is set
the multF of BA is Relation-like [: the carrier of BA, the carrier of BA:] -defined the carrier of BA -valued Function-like total quasi_total V223( the carrier of BA) Element of bool [:[: the carrier of BA, the carrier of BA:], the carrier of BA:]
[:[: the carrier of BA, the carrier of BA:], the carrier of BA:] is Relation-like set
bool [:[: the carrier of BA, the carrier of BA:], the carrier of BA:] is set
id the carrier of BA is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
the multF of BA [;] (AB,(id the carrier of BA)) is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
K501( the carrier of BA, the carrier of BA,(Line (x,B)),(AB multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
ReplaceLine (x,B,(AB * (Line (x,B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
f is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
(BA,x,f) is set
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA * : ( len b₁ = width x & width b₁ = width f & x * b₁ = f ) } is set
Line (f,B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
(width f) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width f } is set
AB * (Line (f,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(Line (f,B)),(AB multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
ReplaceLine (f,B,(AB * (Line (f,B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
(BA,(ReplaceLine (x,B,(AB * (Line (x,B))))),(ReplaceLine (f,B,(AB * (Line (f,B)))))) is set
width (ReplaceLine (x,B,(AB * (Line (x,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (f,B,(AB * (Line (f,B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA * : ( len b₁ = width (ReplaceLine (x,B,(AB * (Line (x,B))))) & width b₁ = width (ReplaceLine (f,B,(AB * (Line (f,B))))) & (ReplaceLine (x,B,(AB * (Line (x,B))))) * b₁ = ReplaceLine (f,B,(AB * (Line (f,B)))) ) } is set
MV is set
lA is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x * lA is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
MV is set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
Line ((ReplaceLine (x,B,(AB * (Line (x,B))))),B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (x,B,(AB * (Line (x,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (x,B,(AB * (Line (x,B)))))) -tuples_on the carrier of BA
(width (ReplaceLine (x,B,(AB * (Line (x,B)))))) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width (ReplaceLine (x,B,(AB * (Line (x,B))))) } is set
AB " is Element of the carrier of BA
(AB ") * (Line ((ReplaceLine (x,B,(AB * (Line (x,B))))),B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (x,B,(AB * (Line (x,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (x,B,(AB * (Line (x,B)))))) -tuples_on the carrier of BA
(AB ") multfield is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
the multF of BA [;] ((AB "),(id the carrier of BA)) is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
K501( the carrier of BA, the carrier of BA,(Line ((ReplaceLine (x,B,(AB * (Line (x,B))))),B)),((AB ") multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (f,B,(AB * (Line (f,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA
(width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA * : len b₁ = width (ReplaceLine (f,B,(AB * (Line (f,B))))) } is set
(AB ") * (Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width (ReplaceLine (f,B,(AB * (Line (f,B))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,B,(AB * (Line (f,B)))))) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B)),((AB ") multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
ReplaceLine ((ReplaceLine (f,B,(AB * (Line (f,B))))),B,((AB ") * (Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
ReplaceLine ((ReplaceLine (x,B,(AB * (Line (x,B))))),B,((AB ") * (Line ((ReplaceLine (x,B,(AB * (Line (x,B))))),B)))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
len (AB * (Line (x,B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(AB ") * (AB * (Line (x,B))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(AB * (Line (x,B))),((AB ") multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
(AB ") * AB is Element of the carrier of BA
K560( the carrier of BA, the multF of BA,(AB "),AB) is Element of the carrier of BA
((AB ") * AB) * (Line (x,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of BA
((AB ") * AB) multfield is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
the multF of BA [;] (((AB ") * AB),(id the carrier of BA)) is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
K501( the carrier of BA, the carrier of BA,(Line (x,B)),(((AB ") * AB) multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
1_ BA is Element of the carrier of BA
1. BA is non zero Element of the carrier of BA
(1_ BA) * (Line (x,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of BA
(1_ BA) multfield is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
the multF of BA [;] ((1_ BA),(id the carrier of BA)) is Relation-like the carrier of BA -defined the carrier of BA -valued Function-like non empty total quasi_total Element of bool [: the carrier of BA, the carrier of BA:]
K501( the carrier of BA, the carrier of BA,(Line (x,B)),((1_ BA) multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
len (AB * (Line (f,B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(AB ") * (AB * (Line (f,B))) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(AB * (Line (f,B))),((AB ") multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
((AB ") * AB) * (Line (f,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(Line (f,B)),(((AB ") * AB) multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
(1_ BA) * (Line (f,B)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of BA
K501( the carrier of BA, the carrier of BA,(Line (f,B)),((1_ BA) multfield)) is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of BA
len ((AB ") * (Line ((ReplaceLine (f,B,(AB * (Line (f,B))))),B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA *
Replace ((ReplaceLine (f,B,(AB * (Line (f,B))))),B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
c₁₃ is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA *
Replace (f,B,c₁₃) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
Replace ((Replace (f,B,c₁₃)),B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
Replace (f,B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
ReplaceLine (f,B,(Line (f,B))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
len ((AB ") * (Line ((ReplaceLine (x,B,(AB * (Line (x,B))))),B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA *
Replace ((ReplaceLine (x,B,(AB * (Line (x,B))))),B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
lA is Relation-like NAT -defined the carrier of BA -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of BA *
Replace (x,B,lA) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
Replace ((Replace (x,B,lA)),B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
Replace (x,B,x) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of BA *
ReplaceLine (x,B,(Line (x,B))) is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,n, the carrier of BA
x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (x,B,(AB * (Line (x,B))))) * x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg B is finite B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of AB is non empty non trivial V103() set
the carrier of AB * is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
x is Element of the carrier of AB
f is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
X is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (X,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(width X) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width X } is set
Line (X,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
x * (Line (X,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
x multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
[: the carrier of AB, the carrier of AB:] is Relation-like set
bool [: the carrier of AB, the carrier of AB:] is set
the multF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
[:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is Relation-like set
bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is set
id the carrier of AB is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] (x,(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (X,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (X,BA)) + (x * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
the addF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) V224( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),(x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
BX is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,X,BX) is set
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width X & width b₁ = width BX & X * b₁ = BX ) } is set
Line (BX,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of AB
(width BX) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width BX } is set
Line (BX,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of AB
x * (Line (BX,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (BX,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (BX,BA)) + (x * (Line (BX,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (BX,BA)),(x * (Line (BX,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,(ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),(ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A))))))) is set
width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) & width b₁ = width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) & (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * b₁ = ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A))))) ) } is set
MV is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X * MV is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
len ((Line (X,BA)) + (x * (Line (X,A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
len ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom BX is finite Element of bool NAT
Seg (len (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) is finite len (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) ) } is set
Indices ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) is set
dom ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) is finite Element of bool NAT
Seg (width ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV)) is finite width ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) ) } is set
[:(dom ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV)),(Seg (width ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV))):] is Relation-like finite set
Indices BX is set
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
[:(dom BX),(Seg (width BX)):] is Relation-like finite set
Indices (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) is set
dom (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) is finite Element of bool NAT
Seg (width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A))))))) is finite width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) ) } is set
[:(dom (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A))))))),(Seg (width (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))))):] is Relation-like finite set
len ((Line (BX,BA)) + (x * (Line (BX,A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (x * (Line (X,A))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Line (X,BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng (x * (Line (BX,A))) is finite set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[j,x] is set
{j,x} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,x},{j}} is non empty finite V37() without_zero V103() set
BX * (j,x) is Element of the carrier of AB
Line (X,j) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
Col (MV,x) is Relation-like NAT -defined the carrier of AB -valued Function-like finite len MV -element FinSequence-like FinSubsequence-like Element of (len MV) -tuples_on the carrier of AB
(len MV) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = len MV } is set
(Line (X,j)) "*" (Col (MV,x)) is Element of the carrier of AB
mlt ((Line (X,j)),(Col (MV,x))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,(Line (X,j)),(Col (MV,x))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Sum (mlt ((Line (X,j)),(Col (MV,x)))) is Element of the carrier of AB
the addF of AB $$ (mlt ((Line (X,j)),(Col (MV,x)))) is Element of the carrier of AB
len (Col (MV,x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (BX,j) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of AB
(Line (BX,j)) . x is set
BX * (A,x) is Element of the carrier of AB
(Line (BX,A)) . x is set
dom (x * (Line (BX,A))) is finite width BX -element Element of bool NAT
(x * (Line (BX,A))) . x is set
[A,x] is set
{A,x} is non empty finite V37() set
{A} is non empty trivial finite V37() 1 -element set
{{A,x},{A}} is non empty finite V37() without_zero V103() set
(Line (X,A)) "*" (Col (MV,x)) is Element of the carrier of AB
mlt ((Line (X,A)),(Col (MV,x))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,(Line (X,A)),(Col (MV,x))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Sum (mlt ((Line (X,A)),(Col (MV,x)))) is Element of the carrier of AB
the addF of AB $$ (mlt ((Line (X,A)),(Col (MV,x)))) is Element of the carrier of AB
Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),j) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
(width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) } is set
((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) * (j,x) is Element of the carrier of AB
L is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
((Line (X,BA)) + (x * (Line (X,A)))) "*" L is Element of the carrier of AB
mlt (((Line (X,BA)) + (x * (Line (X,A)))),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,((Line (X,BA)) + (x * (Line (X,A)))),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Sum (mlt (((Line (X,BA)) + (x * (Line (X,A)))),L)) is Element of the carrier of AB
the addF of AB $$ (mlt (((Line (X,BA)) + (x * (Line (X,A)))),L)) is Element of the carrier of AB
mlt ((Line (X,BA)),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,(Line (X,BA)),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
mlt ((x * (Line (X,A))),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,(x * (Line (X,A))),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(mlt ((Line (X,BA)),L)) + (mlt ((x * (Line (X,A))),L)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(mlt ((Line (X,BA)),L)),(mlt ((x * (Line (X,A))),L))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Sum ((mlt ((Line (X,BA)),L)) + (mlt ((x * (Line (X,A))),L))) is Element of the carrier of AB
the addF of AB $$ ((mlt ((Line (X,BA)),L)) + (mlt ((x * (Line (X,A))),L))) is Element of the carrier of AB
mlt ((Line (X,A)),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the multF of AB,(Line (X,A)),L) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
x * (mlt ((Line (X,A)),L)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(mlt ((Line (X,A)),L)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(mlt ((Line (X,BA)),L)) + (x * (mlt ((Line (X,A)),L))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(mlt ((Line (X,BA)),L)),(x * (mlt ((Line (X,A)),L)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Sum ((mlt ((Line (X,BA)),L)) + (x * (mlt ((Line (X,A)),L)))) is Element of the carrier of AB
the addF of AB $$ ((mlt ((Line (X,BA)),L)) + (x * (mlt ((Line (X,A)),L)))) is Element of the carrier of AB
Sum (mlt ((Line (X,BA)),L)) is Element of the carrier of AB
the addF of AB $$ (mlt ((Line (X,BA)),L)) is Element of the carrier of AB
Sum (x * (mlt ((Line (X,A)),L))) is Element of the carrier of AB
the addF of AB $$ (x * (mlt ((Line (X,A)),L))) is Element of the carrier of AB
(Sum (mlt ((Line (X,BA)),L))) + (Sum (x * (mlt ((Line (X,A)),L)))) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(Sum (mlt ((Line (X,BA)),L))),(Sum (x * (mlt ((Line (X,A)),L))))) is Element of the carrier of AB
x * (BX * (A,x)) is Element of the carrier of AB
K560( the carrier of AB, the multF of AB,x,(BX * (A,x))) is Element of the carrier of AB
(BX * (j,x)) + (x * (BX * (A,x))) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(BX * (j,x)),(x * (BX * (A,x)))) is Element of the carrier of AB
mN is Element of the carrier of AB
mN + (x * (BX * (A,x))) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,mN,(x * (BX * (A,x)))) is Element of the carrier of AB
mSN is Element of the carrier of AB
x * mSN is Element of the carrier of AB
K560( the carrier of AB, the multF of AB,x,mSN) is Element of the carrier of AB
mN + (x * mSN) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,mN,(x * mSN)) is Element of the carrier of AB
mSN is Element of the carrier of AB
mN + mSN is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,mN,mSN) is Element of the carrier of AB
((Line (BX,BA)) + (x * (Line (BX,A)))) . x is set
(ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) * (j,x) is Element of the carrier of AB
Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),j) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
(width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) } is set
((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) * (j,x) is Element of the carrier of AB
(ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) * (j,x) is Element of the carrier of AB
((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) * (j,x) is Element of the carrier of AB
(ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) * (j,x) is Element of the carrier of AB
((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) * MV) * (j,x) is Element of the carrier of AB
(ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) * (j,x) is Element of the carrier of AB
len (ReplaceLine (BX,BA,((Line (BX,BA)) + (x * (Line (BX,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg B is finite B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of AB is non empty non trivial V103() set
the carrier of AB * is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
x is Element of the carrier of AB
f is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (f,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
(width f) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width f } is set
Line (f,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
x * (Line (f,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
x multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
[: the carrier of AB, the carrier of AB:] is Relation-like set
bool [: the carrier of AB, the carrier of AB:] is set
the multF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
[:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is Relation-like set
bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is set
id the carrier of AB is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] (x,(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (f,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (f,BA)) + (x * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
the addF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) V224( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (f,BA)),(x * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
X is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,f,X) is set
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width f & width b₁ = width X & f * b₁ = X ) } is set
Line (X,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(width X) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width X } is set
Line (X,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
x * (Line (X,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (X,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (X,BA)) + (x * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),(x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,(ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),(ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) is set
width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) & width b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) & (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) * b₁ = ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))) ) } is set
x is set
x is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f * x is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
x is set
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))))) -tuples_on the carrier of AB
(width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))))) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) } is set
Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))))) -tuples_on the carrier of AB
- x is Element of the carrier of AB
(- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))))) -tuples_on the carrier of AB
(- x) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] ((- x),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A)),((- x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))))) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA)),((- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
(width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) } is set
Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
(- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A)),((- x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA)),((- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA,((Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
ReplaceLine ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA,((Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
len ((Line (X,BA)) + (x * (Line (X,A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
1_ AB is Element of the carrier of AB
1. AB is non zero Element of the carrier of AB
(1_ AB) * x is Element of the carrier of AB
K560( the carrier of AB, the multF of AB,(1_ AB),x) is Element of the carrier of AB
- ((1_ AB) * x) is Element of the carrier of AB
(- ((1_ AB) * x)) * (Line (X,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(- ((1_ AB) * x)) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] ((- ((1_ AB) * x)),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (X,A)),((- ((1_ AB) * x)) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (X,BA)) + (x * (Line (X,A)))) + ((- ((1_ AB) * x)) * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (X,BA)) + (x * (Line (X,A)))),((- ((1_ AB) * x)) * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
- (1_ AB) is Element of the carrier of AB
(- (1_ AB)) * x is Element of the carrier of AB
K560( the carrier of AB, the multF of AB,(- (1_ AB)),x) is Element of the carrier of AB
((- (1_ AB)) * x) * (Line (X,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
((- (1_ AB)) * x) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] (((- (1_ AB)) * x),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (X,A)),(((- (1_ AB)) * x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (X,BA)) + (x * (Line (X,A)))) + (((- (1_ AB)) * x) * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (X,BA)) + (x * (Line (X,A)))),(((- (1_ AB)) * x) * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(- (1_ AB)) * (x * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(- (1_ AB)) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] ((- (1_ AB)),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(x * (Line (X,A))),((- (1_ AB)) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (X,BA)) + (x * (Line (X,A)))) + ((- (1_ AB)) * (x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (X,BA)) + (x * (Line (X,A)))),((- (1_ AB)) * (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
- (x * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
comp AB is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(x * (Line (X,A))),(comp AB)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (X,BA)) + (x * (Line (X,A)))) + (- (x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (X,BA)) + (x * (Line (X,A)))),(- (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(x * (Line (X,A))) + (- (x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(x * (Line (X,A))),(- (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (X,BA)) + ((x * (Line (X,A))) + (- (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),((x * (Line (X,A))) + (- (x * (Line (X,A)))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
0. AB is zero Element of the carrier of AB
(width X) |-> (0. AB) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
(Seg (width X)) --> (0. AB) is Relation-like Seg (width X) -defined Seg (width X) -defined the carrier of AB -valued {(0. AB)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width X)),{(0. AB)}:]
{(0. AB)} is non empty trivial finite 1 -element set
[:(Seg (width X)),{(0. AB)}:] is Relation-like finite set
bool [:(Seg (width X)),{(0. AB)}:] is finite V37() set
(Line (X,BA)) + ((width X) |-> (0. AB)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),((width X) |-> (0. AB))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
len ((Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB *
Replace ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA,x) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
y is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB *
Replace (X,BA,y) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
Replace ((Replace (X,BA,y)),BA,x) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
Replace (X,BA,x) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
ReplaceLine (X,BA,((Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))),A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
len ((Line (f,BA)) + (x * (Line (f,A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(- ((1_ AB) * x)) * (Line (f,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (f,A)),((- ((1_ AB) * x)) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (f,BA)) + (x * (Line (f,A)))) + ((- ((1_ AB) * x)) * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (f,BA)) + (x * (Line (f,A)))),((- ((1_ AB) * x)) * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((- (1_ AB)) * x) * (Line (f,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (f,A)),(((- (1_ AB)) * x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (f,BA)) + (x * (Line (f,A)))) + (((- (1_ AB)) * x) * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (f,BA)) + (x * (Line (f,A)))),(((- (1_ AB)) * x) * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(- (1_ AB)) * (x * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(x * (Line (f,A))),((- (1_ AB)) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (f,BA)) + (x * (Line (f,A)))) + ((- (1_ AB)) * (x * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (f,BA)) + (x * (Line (f,A)))),((- (1_ AB)) * (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
- (x * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(x * (Line (f,A))),(comp AB)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((Line (f,BA)) + (x * (Line (f,A)))) + (- (x * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((Line (f,BA)) + (x * (Line (f,A)))),(- (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(x * (Line (f,A))) + (- (x * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(x * (Line (f,A))),(- (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (f,BA)) + ((x * (Line (f,A))) + (- (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (f,BA)),((x * (Line (f,A))) + (- (x * (Line (f,A)))))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(width f) |-> (0. AB) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
Seg (width f) is finite width f -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width f ) } is set
(Seg (width f)) --> (0. AB) is Relation-like Seg (width f) -defined Seg (width f) -defined the carrier of AB -valued {(0. AB)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width f)),{(0. AB)}:]
[:(Seg (width f)),{(0. AB)}:] is Relation-like finite set
bool [:(Seg (width f)),{(0. AB)}:] is finite V37() set
(Line (f,BA)) + ((width f) |-> (0. AB)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (f,BA)),((width f) |-> (0. AB))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
len ((Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
y is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB *
Replace ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA,y) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
x is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB *
Replace (f,BA,x) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
Replace ((Replace (f,BA,x)),BA,y) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
Replace (f,BA,y) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of the carrier of AB *
ReplaceLine (f,BA,((Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),BA)) + ((- x) * (Line ((ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
L is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
len L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) * L is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg B is finite B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of AB is non empty non trivial V103() set
the carrier of AB * is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
1_ AB is Element of the carrier of AB
1. AB is non zero Element of the carrier of AB
- (1_ AB) is Element of the carrier of AB
x is Element of the carrier of AB
f is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (f,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
(width f) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width f } is set
Line (f,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
x * (Line (f,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
x multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
[: the carrier of AB, the carrier of AB:] is Relation-like set
bool [: the carrier of AB, the carrier of AB:] is set
the multF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
[:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is Relation-like set
bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:] is set
id the carrier of AB is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] (x,(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (f,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (f,BA)) + (x * (Line (f,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
the addF of AB is Relation-like [: the carrier of AB, the carrier of AB:] -defined the carrier of AB -valued Function-like total quasi_total V223( the carrier of AB) V224( the carrier of AB) Element of bool [:[: the carrier of AB, the carrier of AB:], the carrier of AB:]
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (f,BA)),(x * (Line (f,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,n, the carrier of AB
X is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,f,X) is set
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width f & width b₁ = width X & f * b₁ = X ) } is set
Line (X,BA) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(width X) -tuples_on the carrier of AB is functional non empty FinSequence-membered FinSequenceSet of the carrier of AB
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of AB * : len b₁ = width X } is set
Line (X,A) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
x * (Line (X,A)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (X,A)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (X,BA)) + (x * (Line (X,A))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),(x * (Line (X,A)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))) is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of B,K, the carrier of AB
(AB,(ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))),(ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))))) is set
width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of AB * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of AB * : ( len b₁ = width (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) & width b₁ = width (ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A)))))) & (ReplaceLine (f,BA,((Line (f,BA)) + (x * (Line (f,A)))))) * b₁ = ReplaceLine (X,BA,((Line (X,BA)) + (x * (Line (X,A))))) ) } is set
(1_ AB) + x is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(1_ AB),x) is Element of the carrier of AB
0. AB is zero Element of the carrier of AB
- (1. AB) is Element of the carrier of AB
(- (1. AB)) + ((1_ AB) + x) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(- (1. AB)),((1_ AB) + x)) is Element of the carrier of AB
(- (1. AB)) + (1_ AB) is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(- (1. AB)),(1_ AB)) is Element of the carrier of AB
((- (1. AB)) + (1_ AB)) + x is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,((- (1. AB)) + (1_ AB)),x) is Element of the carrier of AB
(0. AB) + x is Element of the carrier of AB
K560( the carrier of AB, the addF of AB,(0. AB),x) is Element of the carrier of AB
x * (Line (X,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (X,BA)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (X,BA)) + (x * (Line (X,BA))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (X,BA)),(x * (Line (X,BA)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(1_ AB) * (Line (X,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
(1_ AB) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] ((1_ AB),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (X,BA)),((1_ AB) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((1_ AB) * (Line (X,BA))) + (x * (Line (X,BA))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((1_ AB) * (Line (X,BA))),(x * (Line (X,BA)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((1_ AB) + x) * (Line (X,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of AB
((1_ AB) + x) multfield is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
the multF of AB [;] (((1_ AB) + x),(id the carrier of AB)) is Relation-like the carrier of AB -defined the carrier of AB -valued Function-like non empty total quasi_total Element of bool [: the carrier of AB, the carrier of AB:]
K501( the carrier of AB, the carrier of AB,(Line (X,BA)),(((1_ AB) + x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
x * (Line (f,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (f,BA)),(x multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(Line (f,BA)) + (x * (Line (f,BA))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,(Line (f,BA)),(x * (Line (f,BA)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
(1_ AB) * (Line (f,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (f,BA)),((1_ AB) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((1_ AB) * (Line (f,BA))) + (x * (Line (f,BA))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K498( the carrier of AB, the carrier of AB, the carrier of AB, the addF of AB,((1_ AB) * (Line (f,BA))),(x * (Line (f,BA)))) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
((1_ AB) + x) * (Line (f,BA)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of AB
K501( the carrier of AB, the carrier of AB,(Line (f,BA)),(((1_ AB) + x) multfield)) is Relation-like NAT -defined the carrier of AB -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AB
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
dom B is finite Element of bool NAT
Line (B,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width B } is set
(width B) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
(Seg (width B)) --> (0. K) is Relation-like Seg (width B) -defined Seg (width B) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width B)),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg (width B)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (width B)),{(0. K)}:] is finite V37() set
BA is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,B,BA) is set
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width B & width b₁ = width BA & B * b₁ = BA ) } is set
Line (BA,n) is Relation-like NAT -defined the carrier of K -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of K
(width BA) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width BA } is set
(width BA) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of K
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
(Seg (width BA)) --> (0. K) is Relation-like Seg (width BA) -defined Seg (width BA) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width BA)),{(0. K)}:]
[:(Seg (width BA)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (width BA)),{(0. K)}:] is finite V37() set
len (Line (BA,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len BA) is finite len BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BA ) } is set
dom BA is finite Element of bool NAT
[n,f] is set
{n,f} is non empty finite V37() set
{n} is non empty trivial finite V37() 1 -element set
{{n,f},{n}} is non empty finite V37() without_zero V103() set
Indices BA is set
[:(dom BA),(Seg (width BA)):] is Relation-like finite set
BA * (n,f) is Element of the carrier of K
Col (A,f) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len A } is set
(Line (B,n)) "*" (Col (A,f)) is Element of the carrier of K
mlt ((Line (B,n)),(Col (A,f))) 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 quasi_total V223( 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
K498( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (B,n)),(Col (A,f))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (B,n)),(Col (A,f)))) 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 quasi_total V223( the carrier of K) V224( the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (mlt ((Line (B,n)),(Col (A,f)))) is Element of the carrier of K
(0. K) * (Col (A,f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
(0. K) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((0. K),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K501( the carrier of K, the carrier of K,(Col (A,f)),((0. K) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum ((0. K) * (Col (A,f))) is Element of the carrier of K
the addF of K $$ ((0. K) * (Col (A,f))) is Element of the carrier of K
Sum (Col (A,f)) is Element of the carrier of K
the addF of K $$ (Col (A,f)) is Element of the carrier of K
(0. K) * (Sum (Col (A,f))) is Element of the carrier of K
K560( the carrier of K, the multF of K,(0. K),(Sum (Col (A,f)))) is Element of the carrier of K
((width BA) |-> (0. K)) . f is set
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(Line (BA,n)) . f is set
len ((width BA) |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
card (Seg (width B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width B))) -tuples_on NAT
(card (Seg (width B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width B)) } is set
BA is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of n -tuples_on NAT
Segm (B,BA,(Sgm (Seg (width B)))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width B)), the carrier of A
Line ((Segm (B,BA,(Sgm (Seg (width B))))),K) is Relation-like NAT -defined the carrier of A -valued Function-like finite width (Segm (B,BA,(Sgm (Seg (width B))))) -element FinSequence-like FinSubsequence-like Element of (width (Segm (B,BA,(Sgm (Seg (width B)))))) -tuples_on the carrier of A
width (Segm (B,BA,(Sgm (Seg (width B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (Segm (B,BA,(Sgm (Seg (width B)))))) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width (Segm (B,BA,(Sgm (Seg (width B))))) } is set
BA . K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (B,(BA . K)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
rng (Sgm (Seg (width B))) is finite V195() V196() V197() V200() set
len (Line (B,(BA . K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (Line (B,(BA . K))) is finite width B -element Element of bool NAT
idseq (width B) is Relation-like NAT -defined Function-like finite width B -element FinSequence-like FinSubsequence-like set
id (Seg (width B)) is Relation-like Seg (width B) -defined Seg (width B) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width B)),(Seg (width B)):]
[:(Seg (width B)),(Seg (width B)):] is Relation-like finite set
bool [:(Seg (width B)),(Seg (width B)):] is finite V37() set
(Line (B,(BA . K))) * (Sgm (Seg (width B))) is Relation-like NAT -defined the carrier of A -valued Function-like finite Element of bool [:NAT, the carrier of A:]
[:NAT, the carrier of A:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of A:] is non empty non trivial non finite V103() set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,A,B) is set
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width A & width b₁ = width B & A * b₁ = B ) } is set
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
card (Seg (width B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width B))) -tuples_on NAT
(card (Seg (width B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width B)) } is set
BA is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of n -tuples_on NAT
rng BA is finite V195() V196() V197() V200() set
Segm (A,BA,(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width A)), the carrier of K
Segm (B,BA,(Sgm (Seg (width B)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width B)), the carrier of K
(K,(Segm (A,BA,(Sgm (Seg (width A))))),(Segm (B,BA,(Sgm (Seg (width B)))))) is set
width (Segm (A,BA,(Sgm (Seg (width A))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (B,BA,(Sgm (Seg (width B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (Segm (A,BA,(Sgm (Seg (width A))))) & width b₁ = width (Segm (B,BA,(Sgm (Seg (width B))))) & (Segm (A,BA,(Sgm (Seg (width A))))) * b₁ = Segm (B,BA,(Sgm (Seg (width B)))) ) } is set
len (Segm (A,BA,(Sgm (Seg (width A))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Segm (B,BA,(Sgm (Seg (width B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is set
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(Segm (A,BA,(Sgm (Seg (width A))))) * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Segm (A,BA,(Sgm (Seg (width A))))) * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,lA] is set
{MV,lA} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,lA},{MV}} is non empty finite V37() without_zero V103() set
Indices ((Segm (A,BA,(Sgm (Seg (width A))))) * X) is set
dom ((Segm (A,BA,(Sgm (Seg (width A))))) * X) is finite Element of bool NAT
width ((Segm (A,BA,(Sgm (Seg (width A))))) * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ((Segm (A,BA,(Sgm (Seg (width A))))) * X)) is finite width ((Segm (A,BA,(Sgm (Seg (width A))))) * X) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((Segm (A,BA,(Sgm (Seg (width A))))) * X) ) } is set
[:(dom ((Segm (A,BA,(Sgm (Seg (width A))))) * X)),(Seg (width ((Segm (A,BA,(Sgm (Seg (width A))))) * X))):] is Relation-like finite set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom BA is finite n -element Element of bool NAT
BA . MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
dom B is finite Element of bool NAT
[(BA . MV),lA] is set
{(BA . MV),lA} is non empty finite V37() set
{(BA . MV)} is non empty trivial finite V37() 1 -element set
{{(BA . MV),lA},{(BA . MV)}} is non empty finite V37() without_zero V103() set
Indices B is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
idseq (width B) is Relation-like NAT -defined Function-like finite width B -element FinSequence-like FinSubsequence-like set
id (Seg (width B)) is Relation-like Seg (width B) -defined Seg (width B) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width B)),(Seg (width B)):]
[:(Seg (width B)),(Seg (width B)):] is Relation-like finite set
bool [:(Seg (width B)),(Seg (width B)):] is finite V37() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm (Seg (width B))) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
dom (Segm (B,BA,(Sgm (Seg (width B))))) is finite Element of bool NAT
Indices (Segm (B,BA,(Sgm (Seg (width B))))) is set
Seg (width (Segm (B,BA,(Sgm (Seg (width B)))))) is finite width (Segm (B,BA,(Sgm (Seg (width B))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,BA,(Sgm (Seg (width B))))) ) } is set
[:(dom (Segm (B,BA,(Sgm (Seg (width B)))))),(Seg (width (Segm (B,BA,(Sgm (Seg (width B))))))):] is Relation-like finite set
Line ((Segm (A,BA,(Sgm (Seg (width A))))),MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (Segm (A,BA,(Sgm (Seg (width A))))) -element FinSequence-like FinSubsequence-like Element of (width (Segm (A,BA,(Sgm (Seg (width A)))))) -tuples_on the carrier of K
(width (Segm (A,BA,(Sgm (Seg (width A)))))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (Segm (A,BA,(Sgm (Seg (width A))))) } is set
Line (A,(BA . MV)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
(width A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width A } is set
((Segm (A,BA,(Sgm (Seg (width A))))) * X) * (MV,lA) is Element of the carrier of K
Col (X,lA) is Relation-like NAT -defined the carrier of K -valued Function-like finite len X -element FinSequence-like FinSubsequence-like Element of (len X) -tuples_on the carrier of K
(len X) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len X } is set
(Line (A,(BA . MV))) "*" (Col (X,lA)) is Element of the carrier of K
mlt ((Line (A,(BA . MV))),(Col (X,lA))) 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 quasi_total V223( 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
K498( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (A,(BA . MV))),(Col (X,lA))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (A,(BA . MV))),(Col (X,lA)))) 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 quasi_total V223( the carrier of K) V224( 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 (A,(BA . MV))),(Col (X,lA)))) is Element of the carrier of K
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA . c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B * ((BA . c₁₃),lA) is Element of the carrier of K
(Segm (B,BA,(Sgm (Seg (width B))))) * (MV,lA) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
(width A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width A } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) --> (0. K) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg (width A)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. K)}:] is finite V37() set
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
dom B is finite Element of bool NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width B) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width B } is set
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
(Seg (width B)) --> (0. K) is Relation-like Seg (width B) -defined Seg (width B) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width B)),{(0. K)}:]
[:(Seg (width B)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (width B)),{(0. K)}:] is finite V37() set
(K,A,B) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width A & width b₁ = width B & A * b₁ = B ) } is set
card (Seg (width B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width B))) -tuples_on NAT
(card (Seg (width B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width B)) } is set
BA is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of n -tuples_on NAT
rng BA is finite V195() V196() V197() V200() set
(dom A) \ (rng BA) is finite Element of bool NAT
Segm (A,BA,(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width A)), the carrier of K
Segm (B,BA,(Sgm (Seg (width B)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width B)), the carrier of K
(K,(Segm (A,BA,(Sgm (Seg (width A))))),(Segm (B,BA,(Sgm (Seg (width B)))))) is set
width (Segm (A,BA,(Sgm (Seg (width A))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (B,BA,(Sgm (Seg (width B))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (Segm (A,BA,(Sgm (Seg (width A))))) & width b₁ = width (Segm (B,BA,(Sgm (Seg (width B))))) & (Segm (A,BA,(Sgm (Seg (width A))))) * b₁ = Segm (B,BA,(Sgm (Seg (width B)))) ) } is set
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
f is set
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Segm (A,BA,(Sgm (Seg (width A))))) * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
A * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width (A * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (A * X) is finite Element of bool NAT
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,lA] is set
{MV,lA} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,lA},{MV}} is non empty finite V37() without_zero V103() set
Indices (A * X) is set
Seg (width (A * X)) is finite width (A * X) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (A * X) ) } is set
[:(dom (A * X)),(Seg (width (A * X))):] is Relation-like finite set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom BA is finite n -element Element of bool NAT
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
idseq (width B) is Relation-like NAT -defined Function-like finite width B -element FinSequence-like FinSubsequence-like set
id (Seg (width B)) is Relation-like Seg (width B) -defined Seg (width B) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width B)),(Seg (width B)):]
[:(Seg (width B)),(Seg (width B)):] is Relation-like finite set
bool [:(Seg (width B)),(Seg (width B)):] is finite V37() set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm (Seg (width B))) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is set
BA . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Indices (Segm (B,BA,(Sgm (Seg (width B))))) is set
dom (Segm (B,BA,(Sgm (Seg (width B))))) is finite Element of bool NAT
Seg (width (Segm (B,BA,(Sgm (Seg (width B)))))) is finite width (Segm (B,BA,(Sgm (Seg (width B))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,BA,(Sgm (Seg (width B))))) ) } is set
[:(dom (Segm (B,BA,(Sgm (Seg (width B)))))),(Seg (width (Segm (B,BA,(Sgm (Seg (width B))))))):] is Relation-like finite set
Seg (card (Seg (width B))) is finite card (Seg (width B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (width B)) ) } is set
[:(Seg n),(Seg (card (Seg (width B)))):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[y,lA] is set
{y,lA} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,lA},{y}} is non empty finite V37() without_zero V103() set
Line ((Segm (A,BA,(Sgm (Seg (width A))))),y) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (Segm (A,BA,(Sgm (Seg (width A))))) -element FinSequence-like FinSubsequence-like Element of (width (Segm (A,BA,(Sgm (Seg (width A)))))) -tuples_on the carrier of K
(width (Segm (A,BA,(Sgm (Seg (width A)))))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (Segm (A,BA,(Sgm (Seg (width A))))) } is set
Line (A,c₁₃) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
(A * X) * (MV,lA) is Element of the carrier of K
Col (X,lA) is Relation-like NAT -defined the carrier of K -valued Function-like finite len X -element FinSequence-like FinSubsequence-like Element of (len X) -tuples_on the carrier of K
(len X) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len X } is set
(Line ((Segm (A,BA,(Sgm (Seg (width A))))),y)) "*" (Col (X,lA)) is Element of the carrier of K
mlt ((Line ((Segm (A,BA,(Sgm (Seg (width A))))),y)),(Col (X,lA))) 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 quasi_total V223( 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
K498( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((Segm (A,BA,(Sgm (Seg (width A))))),y)),(Col (X,lA))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line ((Segm (A,BA,(Sgm (Seg (width A))))),y)),(Col (X,lA)))) 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 quasi_total V223( the carrier of K) V224( 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 ((Segm (A,BA,(Sgm (Seg (width A))))),y)),(Col (X,lA)))) is Element of the carrier of K
(Segm (B,BA,(Sgm (Seg (width B))))) * (y,x) is Element of the carrier of K
B * (MV,lA) is Element of the carrier of K
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (B,MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
Line (A,MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
(A * X) * (MV,lA) is Element of the carrier of K
Col (X,lA) is Relation-like NAT -defined the carrier of K -valued Function-like finite len X -element FinSequence-like FinSubsequence-like Element of (len X) -tuples_on the carrier of K
(len X) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len X } is set
((width A) |-> (0. K)) "*" (Col (X,lA)) is Element of the carrier of K
mlt (((width A) |-> (0. K)),(Col (X,lA))) 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 quasi_total V223( 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
K498( the carrier of K, the carrier of K, the carrier of K, the multF of K,((width A) |-> (0. K)),(Col (X,lA))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt (((width A) |-> (0. K)),(Col (X,lA)))) 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 quasi_total V223( the carrier of K) V224( the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K $$ (mlt (((width A) |-> (0. K)),(Col (X,lA)))) is Element of the carrier of K
(0. K) * (Col (X,lA)) is Relation-like NAT -defined the carrier of K -valued Function-like finite len X -element FinSequence-like FinSubsequence-like Element of (len X) -tuples_on the carrier of K
(0. K) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] ((0. K),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K501( the carrier of K, the carrier of K,(Col (X,lA)),((0. K) multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum ((0. K) * (Col (X,lA))) is Element of the carrier of K
the addF of K $$ ((0. K) * (Col (X,lA))) is Element of the carrier of K
Sum (Col (X,lA)) is Element of the carrier of K
the addF of K $$ (Col (X,lA)) is Element of the carrier of K
(0. K) * (Sum (Col (X,lA))) is Element of the carrier of K
K560( the carrier of K, the multF of K,(0. K),(Sum (Col (X,lA)))) is Element of the carrier of K
(Line (B,MV)) . lA is set
B * (MV,lA) is Element of the carrier of K
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(A * X) * (MV,lA) is Element of the carrier of K
B * (MV,lA) is Element of the carrier of K
(A * X) * (MV,lA) is Element of the carrier of K
B * (MV,lA) is Element of the carrier of K
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,A) is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
B is finite without_zero Element of bool NAT
Segm (K,B,(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width K)), the carrier of n
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite card B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card B) -tuples_on NAT
(card B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card B } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm B),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width K)), the carrier of n
Segm (A,B,(Seg (width A))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width A)), the carrier of n
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (A,(Sgm B),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width A)), the carrier of n
(n,(Segm (K,B,(Seg (width K)))),(Segm (A,B,(Seg (width A))))) is set
width (Segm (K,B,(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (A,B,(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (K,B,(Seg (width K)))) & width b₁ = width (Segm (A,B,(Seg (width A)))) & (Segm (K,B,(Seg (width K)))) * b₁ = Segm (A,B,(Seg (width A))) ) } is set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
rng (Sgm B) is finite V195() V196() V197() V200() set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) --> (0. n) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. n)}:]
[:(Seg (width A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. n)}:] is finite V37() set
(n,K,A) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
B is finite without_zero Element of bool NAT
(dom K) \ B is finite Element of bool NAT
Segm (K,B,(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width K)), the carrier of n
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite card B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card B) -tuples_on NAT
(card B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card B } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm B),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width K)), the carrier of n
Segm (A,B,(Seg (width A))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width A)), the carrier of n
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (A,(Sgm B),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card (Seg (width A)), the carrier of n
(n,(Segm (K,B,(Seg (width K)))),(Segm (A,B,(Seg (width A))))) is set
width (Segm (K,B,(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (A,B,(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (K,B,(Seg (width K)))) & width b₁ = width (Segm (A,B,(Seg (width A)))) & (Segm (K,B,(Seg (width K)))) * b₁ = Segm (A,B,(Seg (width A))) ) } is set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
rng (Sgm B) is finite V195() V196() V197() V200() set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
dom A is finite Element of bool NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
DelLine (A,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,A,B) is set
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width A & width b₁ = width B & A * b₁ = B ) } is set
DelLine (B,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,(DelLine (A,n)),(DelLine (B,n))) is set
width (DelLine (A,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (DelLine (B,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (DelLine (A,n)) & width b₁ = width (DelLine (B,n)) & (DelLine (A,n)) * b₁ = DelLine (B,n) ) } is set
(len A) - 1 is V105() ext-real complex set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
1 - 1 is V105() ext-real complex set
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
card (Seg (len A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{n} is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (len A)) \ {n} is finite without_zero Element of bool NAT
card ((Seg (len A)) \ {n}) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
Segm (A,((Seg (len A)) \ {n}),(Seg (width A))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len A)) \ {n}), card (Seg (width A)), the carrier of K
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg (len A)) \ {n}) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (len A)) \ {n}) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (len A)) \ {n})) -tuples_on NAT
(card ((Seg (len A)) \ {n})) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (len A)) \ {n}) } is set
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (A,(Sgm ((Seg (len A)) \ {n})),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len A)) \ {n}), card (Seg (width A)), the carrier of K
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
Segm (B,((Seg (len A)) \ {n}),(Seg (width B))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len A)) \ {n}), card (Seg (width B)), the carrier of K
card (Seg (width B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width B))) -tuples_on NAT
(card (Seg (width B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width B)) } is set
Segm (B,(Sgm ((Seg (len A)) \ {n})),(Sgm (Seg (width B)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len A)) \ {n}), card (Seg (width B)), the carrier of K
(K,(Segm (A,((Seg (len A)) \ {n}),(Seg (width A)))),(Segm (B,((Seg (len A)) \ {n}),(Seg (width B))))) is set
width (Segm (A,((Seg (len A)) \ {n}),(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (B,((Seg (len A)) \ {n}),(Seg (width B)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (Segm (A,((Seg (len A)) \ {n}),(Seg (width A)))) & width b₁ = width (Segm (B,((Seg (len A)) \ {n}),(Seg (width B)))) & (Segm (A,((Seg (len A)) \ {n}),(Seg (width A)))) * b₁ = Segm (B,((Seg (len A)) \ {n}),(Seg (width B))) ) } is set
AB is set
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Del (A,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
Del (B,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom K is finite Element of bool NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) --> (0. n) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. n)}:]
[:(Seg (width A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. n)}:] is finite V37() set
(n,K,A) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (K,B) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Line (A,B) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
DelLine (K,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
DelLine (A,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,(DelLine (K,B)),(DelLine (A,B))) is set
width (DelLine (K,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (DelLine (A,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (DelLine (K,B)) & width b₁ = width (DelLine (A,B)) & (DelLine (K,B)) * b₁ = DelLine (A,B) ) } is set
(len K) - 1 is V105() ext-real complex set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
1 - 1 is V105() ext-real complex set
AB is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
{B} is non empty trivial finite V37() 1 -element Element of bool NAT
(Seg (len K)) \ {B} is finite without_zero Element of bool NAT
(dom K) \ ((Seg (len K)) \ {B}) is finite Element of bool NAT
(dom K) /\ {B} is finite Element of bool NAT
Line (K,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Line (A,f) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
card ((Seg (len K)) \ {B}) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Del (K,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (Del (K,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Del (A,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (Del (A,B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
Segm (K,((Seg (len K)) \ {B}),(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len K)) \ {B}), card (Seg (width K)), the carrier of n
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg (len K)) \ {B}) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (len K)) \ {B}) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (len K)) \ {B})) -tuples_on NAT
(card ((Seg (len K)) \ {B})) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (len K)) \ {B}) } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm ((Seg (len K)) \ {B})),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len K)) \ {B}), card (Seg (width K)), the carrier of n
Segm (A,((Seg (len K)) \ {B}),(Seg (width A))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len K)) \ {B}), card (Seg (width A)), the carrier of n
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (A,(Sgm ((Seg (len K)) \ {B})),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (len K)) \ {B}), card (Seg (width A)), the carrier of n
(n,(Segm (K,((Seg (len K)) \ {B}),(Seg (width K)))),(Segm (A,((Seg (len K)) \ {B}),(Seg (width A))))) is set
width (Segm (K,((Seg (len K)) \ {B}),(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (A,((Seg (len K)) \ {B}),(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (K,((Seg (len K)) \ {B}),(Seg (width K)))) & width b₁ = width (Segm (A,((Seg (len K)) \ {B}),(Seg (width A)))) & (Segm (K,((Seg (len K)) \ {B}),(Seg (width K)))) * b₁ = Segm (A,((Seg (len K)) \ {B}),(Seg (width A))) ) } is set
(n,(DelLine (K,B)),(Segm (A,((Seg (len K)) \ {B}),(Seg (width A))))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (DelLine (K,B)) & width b₁ = width (Segm (A,((Seg (len K)) \ {B}),(Seg (width A)))) & (DelLine (K,B)) * b₁ = Segm (A,((Seg (len K)) \ {B}),(Seg (width A))) ) } is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is finite V37() set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of B is non empty non trivial V103() set
the carrier of B * is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
len (idseq n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = n } is set
x is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of B
f is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A, the carrier of B
(B,x,f) is set
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B * : ( len b₁ = width x & width b₁ = width f & x * b₁ = f ) } is set
X is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
x * X is Relation-like NAT -defined Seg n -defined the carrier of B * -valued the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of B
f * X is Relation-like NAT -defined Seg n -defined the carrier of B * -valued the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A, the carrier of B
(B,(x * X),(f * X)) is set
width (x * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (f * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B * : ( len b₁ = width (x * X) & width b₁ = width (f * X) & (x * X) * b₁ = f * X ) } is set
rng X is finite set
AB is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of n -tuples_on NAT
dom AB is finite n -element Element of bool NAT
AB * X is Relation-like Seg n -defined NAT -valued RAT -valued Function-like finite V185() V186() V187() V188() Element of bool [:(Seg n),NAT:]
[:(Seg n),NAT:] is Relation-like set
bool [:(Seg n),NAT:] is set
dom X is finite set
BX is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
dom BX is finite Element of bool NAT
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X (#) (idseq n) is Relation-like Seg n -defined Function-like finite set
MV is Relation-like NAT -defined NAT -valued Function-like finite n -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of n -tuples_on NAT
rng MV is finite V195() V196() V197() V200() set
dom x is finite Element of bool NAT
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Seg (width x) is finite width x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
card (Seg (width x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width x)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width x)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width x))) -tuples_on NAT
(card (Seg (width x))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width x)) } is set
Segm (x,MV,(Sgm (Seg (width x)))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width x)), the carrier of B
Seg (len x) is finite len x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
Segm (x,(Seg (len x)),(Seg (width x))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len x)), card (Seg (width x)), the carrier of B
card (Seg (len x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len x)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len x)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len x))) -tuples_on NAT
(card (Seg (len x))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len x)) } is set
Segm (x,(Sgm (Seg (len x))),(Sgm (Seg (width x)))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len x)), card (Seg (width x)), the carrier of B
(Segm (x,(Seg (len x)),(Seg (width x)))) * X is Relation-like Seg n -defined the carrier of B * -valued Function-like finite Function-yielding V147() Element of bool [:(Seg n),( the carrier of B *):]
[:(Seg n),( the carrier of B *):] is Relation-like set
bool [:(Seg n),( the carrier of B *):] is set
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width f) is finite width f -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width f ) } is set
card (Seg (width f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width f)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width f)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width f))) -tuples_on NAT
(card (Seg (width f))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width f)) } is set
Segm (f,MV,(Sgm (Seg (width f)))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, card (Seg (width f)), the carrier of B
Seg (len f) is finite len f -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
Segm (f,(Seg (len f)),(Seg (width f))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len f)), card (Seg (width f)), the carrier of B
card (Seg (len f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len f)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len f)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len f))) -tuples_on NAT
(card (Seg (len f))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len f)) } is set
Segm (f,(Sgm (Seg (len f))),(Sgm (Seg (width f)))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len f)), card (Seg (width f)), the carrier of B
(Segm (f,(Seg (len f)),(Seg (width f)))) * X is Relation-like Seg n -defined the carrier of B * -valued Function-like finite Function-yielding V147() Element of bool [:(Seg n),( the carrier of B *):]
dom f is finite Element of bool NAT
(dom x) \ (rng MV) is finite Element of bool NAT
0. B is zero Element of the carrier of B
(width x) |-> (0. B) is Relation-like NAT -defined the carrier of B -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of B
(width x) -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = width x } is set
(Seg (width x)) --> (0. B) is Relation-like Seg (width x) -defined Seg (width x) -defined the carrier of B -valued {(0. B)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width x)),{(0. B)}:]
{(0. B)} is non empty trivial finite 1 -element set
[:(Seg (width x)),{(0. B)}:] is Relation-like finite set
bool [:(Seg (width x)),{(0. B)}:] is finite V37() set
(width f) |-> (0. B) is Relation-like NAT -defined the carrier of B -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of B
(width f) -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = width f } is set
(Seg (width f)) --> (0. B) is Relation-like Seg (width f) -defined Seg (width f) -defined the carrier of B -valued {(0. B)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width f)),{(0. B)}:]
[:(Seg (width f)),{(0. B)}:] is Relation-like finite set
bool [:(Seg (width f)),{(0. B)}:] is finite V37() set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (x,lA) is Relation-like NAT -defined the carrier of B -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of B
Line (f,lA) is Relation-like NAT -defined the carrier of B -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of B
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
K -' n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (K -' n) is finite K -' n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K -' n ) } is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
1. (A,n) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of A
1. (A,(K -' n)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K -' n, the carrier of A
0. A is zero Element of the carrier of A
K |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
K -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = K } is set
(Seg K) --> (0. A) is Relation-like Seg K -defined Seg K -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. A)}:]
{(0. A)} is non empty trivial finite 1 -element set
[:(Seg K),{(0. A)}:] is Relation-like finite set
bool [:(Seg K),{(0. A)}:] is finite V37() set
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
BA is finite without_zero Element of bool NAT
card BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (B,(Seg n),BA) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card BA, the carrier of A
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Sgm BA is Relation-like NAT -defined NAT -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BA) -tuples_on NAT
(card BA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BA } is set
Segm (B,(Sgm (Seg n)),(Sgm BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card BA, the carrier of A
(Seg K) \ BA is finite without_zero Element of bool NAT
Segm (B,(Seg n),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg K) \ BA), the carrier of A
card ((Seg K) \ BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg K) \ BA) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg K) \ BA) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg K) \ BA)) -tuples_on NAT
(card ((Seg K) \ BA)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg K) \ BA) } is set
Segm (B,(Sgm (Seg n)),(Sgm ((Seg K) \ BA))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg K) \ BA), the carrier of A
(Segm (B,(Seg n),((Seg K) \ BA))) @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
- ((Segm (B,(Seg n),((Seg K) \ BA))) @) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
card (Seg K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K - n is V105() ext-real complex set
0. (A,(K -' n),K) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K, the carrier of A
(K -' n) |-> (K |-> (0. A)) is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite K -' n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (K -' n) -tuples_on (K -tuples_on the carrier of A)
(K -' n) -tuples_on (K -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
(K -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (K -tuples_on the carrier of A) * : len b₁ = K -' n } is set
(Seg (K -' n)) --> (K |-> (0. A)) is Relation-like Seg (K -' n) -defined Seg (K -' n) -defined K -tuples_on the carrier of A -valued {(K |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (K -' n)),{(K |-> (0. A))}:]
{(K |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (K -' n)),{(K |-> (0. A))}:] is Relation-like finite set
bool [:(Seg (K -' n)),{(K |-> (0. A))}:] is finite V37() set
[:(Seg (K -' n)),BA:] is Relation-like finite set
[:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
Indices (0. (A,(K -' n),K)) is set
dom (0. (A,(K -' n),K)) is finite Element of bool NAT
width (0. (A,(K -' n),K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (A,(K -' n),K))) is finite width (0. (A,(K -' n),K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,(K -' n),K)) ) } is set
[:(dom (0. (A,(K -' n),K))),(Seg (width (0. (A,(K -' n),K)))):] is Relation-like finite set
Indices (0. (A,(K -' n),K)) is set
dom (0. (A,(K -' n),K)) is finite Element of bool NAT
width (0. (A,(K -' n),K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (A,(K -' n),K))) is finite width (0. (A,(K -' n),K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,(K -' n),K)) ) } is set
[:(dom (0. (A,(K -' n),K))),(Seg (width (0. (A,(K -' n),K)))):] is Relation-like finite set
[:(Seg (K -' n)),(Seg K):] is Relation-like finite set
[:(Seg (K -' n)),BA:] is Relation-like finite set
[:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
[:(Seg (K -' n)),BA:] is Relation-like finite set
Indices (0. (A,(K -' n),K)) is set
dom (0. (A,(K -' n),K)) is finite Element of bool NAT
width (0. (A,(K -' n),K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (A,(K -' n),K))) is finite width (0. (A,(K -' n),K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,(K -' n),K)) ) } is set
[:(dom (0. (A,(K -' n),K))),(Seg (width (0. (A,(K -' n),K)))):] is Relation-like finite set
[:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
[:(Seg (K -' n)),BA:] is Relation-like finite set
Indices (0. (A,(K -' n),K)) is set
dom (0. (A,(K -' n),K)) is finite Element of bool NAT
width (0. (A,(K -' n),K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (A,(K -' n),K))) is finite width (0. (A,(K -' n),K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,(K -' n),K)) ) } is set
[:(dom (0. (A,(K -' n),K))),(Seg (width (0. (A,(K -' n),K)))):] is Relation-like finite set
[:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
len (Segm (B,(Seg n),((Seg K) \ BA))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Segm (B,(Seg n),((Seg K) \ BA))) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (- ((Segm (B,(Seg n),((Seg K) \ BA))) @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((Segm (B,(Seg n),((Seg K) \ BA))) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (- ((Segm (B,(Seg n),((Seg K) \ BA))) @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (B,(Seg n),((Seg K) \ BA))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:(Seg (K -' n)),BA:] /\ [:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
BA /\ ((Seg K) \ BA) is finite Element of bool NAT
[:(Seg (K -' n)),(BA /\ ((Seg K) \ BA)):] is Relation-like finite set
[:(Seg (K -' n)),{}:] is Relation-like finite set
card (Seg (K -' n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (K -' n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (K -' n)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (K -' n))) -tuples_on NAT
(card (Seg (K -' n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (K -' n)) } is set
(Sgm (Seg (K -' n))) " is Relation-like Function-like set
(Sgm BA) " is Relation-like Function-like set
(Sgm ((Seg K) \ BA)) " is Relation-like Function-like set
X is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,n, the carrier of A
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,lA] is set
{MV,lA} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,lA},{MV}} is non empty finite V37() without_zero V103() set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm (Seg (K -' n))) ") . MV is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm BA) ") . lA is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm ((Seg K) \ BA)) ") . lA is set
X * (c₁₃,x) is Element of the carrier of A
(1. (A,(K -' n))) * (x,y) is Element of the carrier of A
len (0. (A,(K -' n),K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:(Seg (K -' n)),BA:] \/ [:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
MV is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (0. (A,(K -' n),K)), width (0. (A,(K -' n),K)), the carrier of A
Segm (MV,(Seg (K -' n)),BA) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
Segm (MV,(Sgm (Seg (K -' n))),(Sgm BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
Segm (MV,(Seg (K -' n)),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
Segm (MV,(Sgm (Seg (K -' n))),(Sgm ((Seg K) \ BA))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
(Indices MV) \ ([:(Seg (K -' n)),BA:] \/ [:(Seg (K -' n)),((Seg K) \ BA):]) is Element of bool (Indices MV)
bool (Indices MV) is set
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
lA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K, the carrier of A
Segm (lA,(Seg (K -' n)),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
Segm (lA,(Sgm (Seg (K -' n))),(Sgm ((Seg K) \ BA))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
Segm (lA,(Seg (K -' n)),BA) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
Segm (lA,(Sgm (Seg (K -' n))),(Sgm BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg c₁₃ is finite c₁₃ -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= c₁₃ ) } is set
0. (A,n,c₁₃) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,c₁₃, the carrier of A
c₁₃ -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = c₁₃ } is set
c₁₃ |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite c₁₃ -element FinSequence-like FinSubsequence-like Element of c₁₃ -tuples_on the carrier of A
(Seg c₁₃) --> (0. A) is Relation-like Seg c₁₃ -defined Seg c₁₃ -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg c₁₃),{(0. A)}:]
[:(Seg c₁₃),{(0. A)}:] is Relation-like finite set
bool [:(Seg c₁₃),{(0. A)}:] is finite V37() set
n |-> (c₁₃ |-> (0. A)) is Relation-like NAT -defined c₁₃ -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (c₁₃ -tuples_on the carrier of A)
n -tuples_on (c₁₃ -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of c₁₃ -tuples_on the carrier of A
(c₁₃ -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of c₁₃ -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined c₁₃ -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (c₁₃ -tuples_on the carrier of A) * : len b₁ = n } is set
(Seg n) --> (c₁₃ |-> (0. A)) is Relation-like Seg n -defined Seg n -defined c₁₃ -tuples_on the carrier of A -valued {(c₁₃ |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(c₁₃ |-> (0. A))}:]
{(c₁₃ |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(c₁₃ |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{(c₁₃ |-> (0. A))}:] is finite V37() set
(A,B,(0. (A,n,c₁₃))) is set
width (0. (A,n,c₁₃)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width B & width b₁ = width (0. (A,n,c₁₃)) & B * b₁ = 0. (A,n,c₁₃) ) } is set
x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,c₁₃, the carrier of A
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
width (B * x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (B * x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices B is set
dom B is finite Element of bool NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
[:(Seg n),(Seg K):] is Relation-like finite set
[:(Seg n),BA:] is Relation-like finite set
y is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,c₁₃, the carrier of A
Indices y is set
dom y is finite Element of bool NAT
width y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width y) is finite width y -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width y ) } is set
[:(dom y),(Seg (width y)):] is Relation-like finite set
Indices (0. (A,n,c₁₃)) is set
dom (0. (A,n,c₁₃)) is finite Element of bool NAT
Seg (width (0. (A,n,c₁₃))) is finite width (0. (A,n,c₁₃)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,n,c₁₃)) ) } is set
[:(dom (0. (A,n,c₁₃))),(Seg (width (0. (A,n,c₁₃)))):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
[:(Seg n),(Seg c₁₃):] is Relation-like finite set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (x,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of A
(len x) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len x } is set
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (lA,L) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
width lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width lA } is set
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (lA,L) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
width lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width lA } is set
idseq (K -' n) is Relation-like NAT -defined Function-like finite K -' n -element FinSequence-like FinSubsequence-like set
id (Seg (K -' n)) is Relation-like Seg (K -' n) -defined Seg (K -' n) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (K -' n)),(Seg (K -' n)):]
[:(Seg (K -' n)),(Seg (K -' n)):] is Relation-like finite set
bool [:(Seg (K -' n)),(Seg (K -' n)):] is finite V37() set
(idseq (K -' n)) . L is set
(Sgm (Seg (K -' n))) . L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Indices (1. (A,(K -' n))) is set
dom (1. (A,(K -' n))) is finite Element of bool NAT
width (1. (A,(K -' n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (1. (A,(K -' n)))) is finite width (1. (A,(K -' n))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (A,(K -' n))) ) } is set
[:(dom (1. (A,(K -' n)))),(Seg (width (1. (A,(K -' n))))):] is Relation-like finite set
[L,L] is set
{L,L} is non empty finite V37() set
{L} is non empty trivial finite V37() 1 -element set
{{L,L},{L}} is non empty finite V37() without_zero V103() set
rng (Sgm ((Seg K) \ BA)) is finite V195() V196() V197() V200() set
dom (Sgm ((Seg K) \ BA)) is finite card ((Seg K) \ BA) -element Element of bool NAT
(Sgm ((Seg K) \ BA)) . L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (B,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
(Line (B,j)) . ((Sgm ((Seg K) \ BA)) . L) is set
B * (j,((Sgm ((Seg K) \ BA)) . L)) is Element of the carrier of A
(Line (lA,L)) . ((Sgm ((Seg K) \ BA)) . L) is set
lA * (L,((Sgm ((Seg K) \ BA)) . L)) is Element of the carrier of A
(1. (A,(K -' n))) * (L,L) is Element of the carrier of A
1_ A is Element of the carrier of A
1. A is non zero Element of the carrier of A
len (Line (B,j)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is finite V37() set
(idseq n) . j is set
(Sgm (Seg n)) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
len (Line (lA,L)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
mlt ((Line (B,j)),(Line (lA,L))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
K498( the carrier of A, the carrier of A, the carrier of A, the multF of A,(Line (B,j)),(Line (lA,L))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len (mlt ((Line (B,j)),(Line (lA,L)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (mlt ((Line (B,j)),(Line (lA,L)))) is finite Element of bool NAT
(mlt ((Line (B,j)),(Line (lA,L)))) /. ((Sgm ((Seg K) \ BA)) . L) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) . ((Sgm ((Seg K) \ BA)) . L) is set
(B * (j,((Sgm ((Seg K) \ BA)) . L))) * (1_ A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(B * (j,((Sgm ((Seg K) \ BA)) . L))),(1_ A)) is Element of the carrier of A
Indices lA is set
dom lA is finite Element of bool NAT
Seg (width lA) is finite width lA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width lA ) } is set
[:(dom lA),(Seg (width lA)):] is Relation-like finite set
rng (Sgm (Seg (K -' n))) is finite V195() V196() V197() V200() set
rng (Sgm BA) is finite V195() V196() V197() V200() set
rng (Sgm (Seg n)) is finite V195() V196() V197() V200() set
mN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
dom (Sgm BA) is finite card BA -element Element of bool NAT
mSN is set
(Sgm BA) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm BA) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (lA,L)) . ((Sgm BA) . mSN) is set
lA * (L,((Sgm BA) . mSN)) is Element of the carrier of A
[((Sgm (Seg n)) . j),((Sgm BA) . mSN)] is set
{((Sgm (Seg n)) . j),((Sgm BA) . mSN)} is non empty finite V37() set
{((Sgm (Seg n)) . j)} is non empty trivial finite V37() 1 -element set
{{((Sgm (Seg n)) . j),((Sgm BA) . mSN)},{((Sgm (Seg n)) . j)}} is non empty finite V37() without_zero V103() set
[j,mSN] is Element of [:NAT,NAT:]
{j,mSN} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,mSN},{j}} is non empty finite V37() without_zero V103() set
Indices (Segm (B,(Seg n),BA)) is set
dom (Segm (B,(Seg n),BA)) is finite Element of bool NAT
width (Segm (B,(Seg n),BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (B,(Seg n),BA))) is finite width (Segm (B,(Seg n),BA)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,(Seg n),BA)) ) } is set
[:(dom (Segm (B,(Seg n),BA))),(Seg (width (Segm (B,(Seg n),BA)))):] is Relation-like finite set
(Line (B,j)) . ((Sgm BA) . mSN) is set
B * (j,((Sgm BA) . mSN)) is Element of the carrier of A
(Segm (B,(Seg n),BA)) * (j,mSN) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) . mN is set
(0. A) * (lA * (L,((Sgm BA) . mSN))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(0. A),(lA * (L,((Sgm BA) . mSN)))) is Element of the carrier of A
mSN is set
(Sgm ((Seg K) \ BA)) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm ((Seg K) \ BA)) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (B,j)) . ((Sgm ((Seg K) \ BA)) . mSN) is set
B * (j,((Sgm ((Seg K) \ BA)) . mSN)) is Element of the carrier of A
[L,mSN] is set
{L,mSN} is non empty finite V37() set
{{L,mSN},{L}} is non empty finite V37() without_zero V103() set
(Line (lA,L)) . ((Sgm ((Seg K) \ BA)) . mSN) is set
lA * (L,((Sgm ((Seg K) \ BA)) . mSN)) is Element of the carrier of A
(1. (A,(K -' n))) * (L,mSN) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) . mN is set
(B * (j,((Sgm ((Seg K) \ BA)) . mSN))) * (0. A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(B * (j,((Sgm ((Seg K) \ BA)) . mSN))),(0. A)) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) . mN is set
(mlt ((Line (B,j)),(Line (lA,L)))) . mN is set
dom (Sgm BA) is finite card BA -element Element of bool NAT
[((Sgm (Seg n)) . j),((Sgm BA) . j)] is set
{((Sgm (Seg n)) . j),((Sgm BA) . j)} is non empty finite V37() set
{((Sgm (Seg n)) . j)} is non empty trivial finite V37() 1 -element set
{{((Sgm (Seg n)) . j),((Sgm BA) . j)},{((Sgm (Seg n)) . j)}} is non empty finite V37() without_zero V103() set
[j,j] is Element of [:NAT,NAT:]
{j,j} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,j},{j}} is non empty finite V37() without_zero V103() set
Indices (Segm (B,(Seg n),BA)) is set
dom (Segm (B,(Seg n),BA)) is finite Element of bool NAT
width (Segm (B,(Seg n),BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (B,(Seg n),BA))) is finite width (Segm (B,(Seg n),BA)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,(Seg n),BA)) ) } is set
[:(dom (Segm (B,(Seg n),BA))),(Seg (width (Segm (B,(Seg n),BA)))):] is Relation-like finite set
[:(Seg (K -' n)),(Seg K):] is Relation-like finite set
[((Sgm (Seg (K -' n))) . L),((Sgm BA) . j)] is set
{((Sgm (Seg (K -' n))) . L),((Sgm BA) . j)} is non empty finite V37() set
{((Sgm (Seg (K -' n))) . L)} is non empty trivial finite V37() 1 -element set
{{((Sgm (Seg (K -' n))) . L),((Sgm BA) . j)},{((Sgm (Seg (K -' n))) . L)}} is non empty finite V37() without_zero V103() set
[L,j] is set
{L,j} is non empty finite V37() set
{{L,j},{L}} is non empty finite V37() without_zero V103() set
Indices (Segm (lA,(Seg (K -' n)),BA)) is set
dom (Segm (lA,(Seg (K -' n)),BA)) is finite Element of bool NAT
width (Segm (lA,(Seg (K -' n)),BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (lA,(Seg (K -' n)),BA))) is finite width (Segm (lA,(Seg (K -' n)),BA)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (lA,(Seg (K -' n)),BA)) ) } is set
[:(dom (Segm (lA,(Seg (K -' n)),BA))),(Seg (width (Segm (lA,(Seg (K -' n)),BA)))):] is Relation-like finite set
Indices X is set
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
Indices ((Segm (B,(Seg n),((Seg K) \ BA))) @) is set
dom ((Segm (B,(Seg n),((Seg K) \ BA))) @) is finite Element of bool NAT
Seg (width ((Segm (B,(Seg n),((Seg K) \ BA))) @)) is finite width ((Segm (B,(Seg n),((Seg K) \ BA))) @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((Segm (B,(Seg n),((Seg K) \ BA))) @) ) } is set
[:(dom ((Segm (B,(Seg n),((Seg K) \ BA))) @)),(Seg (width ((Segm (B,(Seg n),((Seg K) \ BA))) @))):] is Relation-like finite set
[j,L] is set
{j,L} is non empty finite V37() set
{{j,L},{j}} is non empty finite V37() without_zero V103() set
Indices (Segm (B,(Seg n),((Seg K) \ BA))) is set
dom (Segm (B,(Seg n),((Seg K) \ BA))) is finite Element of bool NAT
Seg (width (Segm (B,(Seg n),((Seg K) \ BA)))) is finite width (Segm (B,(Seg n),((Seg K) \ BA))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,(Seg n),((Seg K) \ BA))) ) } is set
[:(dom (Segm (B,(Seg n),((Seg K) \ BA)))),(Seg (width (Segm (B,(Seg n),((Seg K) \ BA))))):] is Relation-like finite set
(Line (lA,L)) . ((Sgm BA) . j) is set
lA * (L,((Sgm BA) . j)) is Element of the carrier of A
(Segm (lA,(Seg (K -' n)),BA)) * (L,j) is Element of the carrier of A
((Segm (B,(Seg n),((Seg K) \ BA))) @) * (L,j) is Element of the carrier of A
- (((Segm (B,(Seg n),((Seg K) \ BA))) @) * (L,j)) is Element of the carrier of A
(Segm (B,(Seg n),((Seg K) \ BA))) * (j,L) is Element of the carrier of A
- ((Segm (B,(Seg n),((Seg K) \ BA))) * (j,L)) is Element of the carrier of A
- (B * (j,((Sgm ((Seg K) \ BA)) . L))) is Element of the carrier of A
(Line (B,j)) . ((Sgm BA) . j) is set
B * (j,((Sgm BA) . j)) is Element of the carrier of A
(Segm (B,(Seg n),BA)) * (j,j) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) /. ((Sgm BA) . j) is Element of the carrier of A
(mlt ((Line (B,j)),(Line (lA,L)))) . ((Sgm BA) . j) is set
(1_ A) * (- (B * (j,((Sgm ((Seg K) \ BA)) . L)))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(1_ A),(- (B * (j,((Sgm ((Seg K) \ BA)) . L))))) is Element of the carrier of A
Sum (mlt ((Line (B,j)),(Line (lA,L)))) is Element of the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the addF of A $$ (mlt ((Line (B,j)),(Line (lA,L)))) is Element of the carrier of A
(B * (j,((Sgm ((Seg K) \ BA)) . L))) + (- (B * (j,((Sgm ((Seg K) \ BA)) . L)))) is Element of the carrier of A
K560( the carrier of A, the addF of A,(B * (j,((Sgm ((Seg K) \ BA)) . L))),(- (B * (j,((Sgm ((Seg K) \ BA)) . L))))) is Element of the carrier of A
(0. (A,n,c₁₃)) * (y,x) is Element of the carrier of A
(Line (B,j)) "*" (Line (lA,L)) is Element of the carrier of A
y * (y,x) is Element of the carrier of A
Col (x,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of A
(len x) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len x } is set
y * (y,x) is Element of the carrier of A
Line (B,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
(Line (B,j)) "*" (K |-> (0. A)) is Element of the carrier of A
mlt ((Line (B,j)),(K |-> (0. A))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
K498( the carrier of A, the carrier of A, the carrier of A, the multF of A,(Line (B,j)),(K |-> (0. A))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
Sum (mlt ((Line (B,j)),(K |-> (0. A)))) is Element of the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the addF of A $$ (mlt ((Line (B,j)),(K |-> (0. A)))) is Element of the carrier of A
(0. A) * (Line (B,j)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
(0. A) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
bool [: the carrier of A, the carrier of A:] is set
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] ((0. A),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (B,j)),((0. A) multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
Sum ((0. A) * (Line (B,j))) is Element of the carrier of A
the addF of A $$ ((0. A) * (Line (B,j))) is Element of the carrier of A
Sum (Line (B,j)) is Element of the carrier of A
the addF of A $$ (Line (B,j)) is Element of the carrier of A
(0. A) * (Sum (Line (B,j))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(0. A),(Sum (Line (B,j)))) is Element of the carrier of A
(0. (A,n,c₁₃)) * (y,x) is Element of the carrier of A
Col (x,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of A
(len x) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len x } is set
(0. (A,n,c₁₃)) * (y,x) is Element of the carrier of A
y * (y,x) is Element of the carrier of A
(0. (A,n,c₁₃)) * (y,x) is Element of the carrier of A
y * (y,x) is Element of the carrier of A
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
K -' n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg A is finite A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
B is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of B is non empty non trivial V103() set
the carrier of B * is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
1. (B,n) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of B
0. (B,(K -' n),A) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,A, the carrier of B
A -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = A } is set
0. B is zero Element of the carrier of B
A |-> (0. B) is Relation-like NAT -defined the carrier of B -valued Function-like finite A -element FinSequence-like FinSubsequence-like Element of A -tuples_on the carrier of B
(Seg A) --> (0. B) is Relation-like Seg A -defined Seg A -defined the carrier of B -valued {(0. B)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg A),{(0. B)}:]
{(0. B)} is non empty trivial finite 1 -element set
[:(Seg A),{(0. B)}:] is Relation-like finite set
bool [:(Seg A),{(0. B)}:] is finite V37() set
(K -' n) |-> (A |-> (0. B)) is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite K -' n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (K -' n) -tuples_on (A -tuples_on the carrier of B)
(K -' n) -tuples_on (A -tuples_on the carrier of B) is functional non empty FinSequence-membered FinSequenceSet of A -tuples_on the carrier of B
(A -tuples_on the carrier of B) * is functional non empty FinSequence-membered FinSequenceSet of A -tuples_on the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of (A -tuples_on the carrier of B) * : len b₁ = K -' n } is set
Seg (K -' n) is finite K -' n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K -' n ) } is set
(Seg (K -' n)) --> (A |-> (0. B)) is Relation-like Seg (K -' n) -defined Seg (K -' n) -defined A -tuples_on the carrier of B -valued {(A |-> (0. B))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (K -' n)),{(A |-> (0. B))}:]
{(A |-> (0. B))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (K -' n)),{(A |-> (0. B))}:] is Relation-like finite set
bool [:(Seg (K -' n)),{(A |-> (0. B))}:] is finite V37() set
BA is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of B
AB is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A, the carrier of B
(B,BA,AB) is set
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B * : ( len b₁ = width BA & width b₁ = width AB & BA * b₁ = AB ) } is set
x is finite without_zero Element of bool NAT
card x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (BA,(Seg n),x) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card x, the carrier of B
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Sgm x is Relation-like NAT -defined NAT -valued Function-like finite card x -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card x) -tuples_on NAT
(card x) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x } is set
Segm (BA,(Sgm (Seg n)),(Sgm x)) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card x, the carrier of B
(Seg K) \ x is finite without_zero Element of bool NAT
0. (B,K,A) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of B
K |-> (A |-> (0. B)) is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of K -tuples_on (A -tuples_on the carrier of B)
K -tuples_on (A -tuples_on the carrier of B) is functional non empty FinSequence-membered FinSequenceSet of A -tuples_on the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of (A -tuples_on the carrier of B) * : len b₁ = K } is set
(Seg K) --> (A |-> (0. B)) is Relation-like Seg K -defined Seg K -defined A -tuples_on the carrier of B -valued {(A |-> (0. B))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg K),{(A |-> (0. B))}:]
[:(Seg K),{(A |-> (0. B))}:] is Relation-like finite set
bool [:(Seg K),{(A |-> (0. B))}:] is finite V37() set
card (Seg K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K - n is V105() ext-real complex set
card ((Seg K) \ x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:x,(Seg A):] is Relation-like finite set
[:((Seg K) \ x),(Seg A):] is Relation-like finite set
[:x,(Seg A):] /\ [:((Seg K) \ x),(Seg A):] is Relation-like finite set
x /\ ((Seg K) \ x) is finite Element of bool NAT
[:(x /\ ((Seg K) \ x)),(Seg A):] is Relation-like finite set
[:{},(Seg A):] is Relation-like finite set
(Sgm x) " is Relation-like Function-like set
Sgm (Seg A) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg A) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg A)) -tuples_on NAT
card (Seg A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card (Seg A)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg A) } is set
(Sgm (Seg A)) " is Relation-like Function-like set
Sgm ((Seg K) \ x) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg K) \ x) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg K) \ x)) -tuples_on NAT
(card ((Seg K) \ x)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg K) \ x) } is set
(Sgm ((Seg K) \ x)) " is Relation-like Function-like set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,lA] is set
{MV,lA} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,lA},{MV}} is non empty finite V37() without_zero V103() set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm x) ") . MV is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm (Seg A)) ") . lA is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm ((Seg K) \ x)) ") . MV is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB * (c₁₃,x) is Element of the carrier of B
(0. (B,(K -' n),A)) * (x,y) is Element of the carrier of B
Indices BA is set
dom BA is finite Element of bool NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
[:(dom BA),(Seg (width BA)):] is Relation-like finite set
[:(Seg n),(Seg K):] is Relation-like finite set
len (0. (B,K,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (0. (B,K,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices (0. (B,K,A)) is set
dom (0. (B,K,A)) is finite Element of bool NAT
Seg (width (0. (B,K,A))) is finite width (0. (B,K,A)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (B,K,A)) ) } is set
[:(dom (0. (B,K,A))),(Seg (width (0. (B,K,A)))):] is Relation-like finite set
[:(Seg K),(Seg A):] is Relation-like finite set
[:x,(Seg A):] \/ [:((Seg K) \ x),(Seg A):] is Relation-like finite set
MV is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of B
Segm (MV,x,(Seg A)) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card (Seg A), the carrier of B
Segm (MV,(Sgm x),(Sgm (Seg A))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card (Seg A), the carrier of B
Segm (MV,((Seg K) \ x),(Seg A)) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg K) \ x), card (Seg A), the carrier of B
Segm (MV,(Sgm ((Seg K) \ x)),(Sgm (Seg A))) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg K) \ x), card (Seg A), the carrier of B
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
(Indices MV) \ ([:x,(Seg A):] \/ [:((Seg K) \ x),(Seg A):]) is Element of bool (Indices MV)
bool (Indices MV) is set
BA * MV is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B *
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (BA * MV) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (BA * MV) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices AB is set
dom AB is finite Element of bool NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
[:(dom AB),(Seg (width AB)):] is Relation-like finite set
[:(Seg n),(Seg A):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,x] is set
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
c₁₃ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A, the carrier of B
Indices c₁₃ is set
dom c₁₃ is finite Element of bool NAT
width c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width c₁₃) is finite width c₁₃ -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width c₁₃ ) } is set
[:(dom c₁₃),(Seg (width c₁₃)):] is Relation-like finite set
Line (BA,x) is Relation-like NAT -defined the carrier of B -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of B
(width BA) -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = width BA } is set
len (Line (BA,x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (MV,x) is Relation-like NAT -defined the carrier of B -valued Function-like finite len MV -element FinSequence-like FinSubsequence-like Element of (len MV) -tuples_on the carrier of B
(len MV) -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = len MV } is set
len (Col (MV,x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
mlt ((Line (BA,x)),(Col (MV,x))) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like total quasi_total V223( 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
K498( the carrier of B, the carrier of B, the carrier of B, the multF of B,(Line (BA,x)),(Col (MV,x))) is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of B
len (mlt ((Line (BA,x)),(Col (MV,x)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (mlt ((Line (BA,x)),(Col (MV,x)))) is finite Element of bool NAT
rng (Sgm (Seg A)) is finite V195() V196() V197() V200() set
rng (Sgm (Seg n)) is finite V195() V196() V197() V200() set
[:(Seg n),x:] is Relation-like finite set
rng (Sgm x) is finite V195() V196() V197() V200() set
dom (Sgm x) is finite card x -element Element of bool NAT
(Sgm x) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
idseq A is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like set
id (Seg A) is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg A),(Seg A):]
[:(Seg A),(Seg A):] is Relation-like finite set
bool [:(Seg A),(Seg A):] is finite V37() set
(idseq A) . x is set
(Sgm (Seg A)) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm x) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm (Seg A)) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm x) . y),((Sgm (Seg A)) . y)] is set
{((Sgm x) . y),((Sgm (Seg A)) . y)} is non empty finite V37() set
{((Sgm x) . y)} is non empty trivial finite V37() 1 -element set
{{((Sgm x) . y),((Sgm (Seg A)) . y)},{((Sgm x) . y)}} is non empty finite V37() without_zero V103() set
[y,y] is Element of [:NAT,NAT:]
{y,y} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,y},{y}} is non empty finite V37() without_zero V103() set
rng (Sgm ((Seg K) \ x)) is finite V195() V196() V197() V200() set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is finite V37() set
(idseq n) . x is set
(Sgm (Seg n)) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm (Seg n)) . x),((Sgm x) . x)] is set
{((Sgm (Seg n)) . x),((Sgm x) . x)} is non empty finite V37() set
{((Sgm (Seg n)) . x)} is non empty trivial finite V37() 1 -element set
{{((Sgm (Seg n)) . x),((Sgm x) . x)},{((Sgm (Seg n)) . x)}} is non empty finite V37() without_zero V103() set
[y,y] is Element of [:NAT,NAT:]
{y,y} is non empty finite V37() set
{{y,y},{y}} is non empty finite V37() without_zero V103() set
Indices (1. (B,n)) is set
dom (1. (B,n)) is finite Element of bool NAT
width (1. (B,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (1. (B,n))) is finite width (1. (B,n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (B,n)) ) } is set
[:(dom (1. (B,n))),(Seg (width (1. (B,n)))):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
j is set
(Sgm x) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm x) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm (Seg n)) . x),((Sgm x) . x)] is set
{((Sgm (Seg n)) . x),((Sgm x) . x)} is non empty finite V37() set
{{((Sgm (Seg n)) . x),((Sgm x) . x)},{((Sgm (Seg n)) . x)}} is non empty finite V37() without_zero V103() set
[y,x] is Element of [:NAT,NAT:]
{y,x} is non empty finite V37() set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
(Col (MV,x)) . x is set
MV * (x,x) is Element of the carrier of B
(Line (BA,x)) . ((Sgm x) . x) is set
BA * (y,((Sgm x) . x)) is Element of the carrier of B
(Segm (BA,(Seg n),x)) * (y,x) is Element of the carrier of B
(mlt ((Line (BA,x)),(Col (MV,x)))) . x is set
(0. B) * (MV * (x,x)) is Element of the carrier of B
K560( the carrier of B, the multF of B,(0. B),(MV * (x,x))) is Element of the carrier of B
dom (Sgm ((Seg K) \ x)) is finite card ((Seg K) \ x) -element Element of bool NAT
j is set
(Sgm ((Seg K) \ x)) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (BA,x)) . x is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm ((Seg K) \ x)) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA * (y,((Sgm ((Seg K) \ x)) . x)) is Element of the carrier of B
[((Sgm ((Seg K) \ x)) . x),((Sgm (Seg A)) . y)] is set
{((Sgm ((Seg K) \ x)) . x),((Sgm (Seg A)) . y)} is non empty finite V37() set
{((Sgm ((Seg K) \ x)) . x)} is non empty trivial finite V37() 1 -element set
{{((Sgm ((Seg K) \ x)) . x),((Sgm (Seg A)) . y)},{((Sgm ((Seg K) \ x)) . x)}} is non empty finite V37() without_zero V103() set
[x,y] is Element of [:NAT,NAT:]
{x,y} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,y},{x}} is non empty finite V37() without_zero V103() set
Indices (0. (B,(K -' n),A)) is set
dom (0. (B,(K -' n),A)) is finite Element of bool NAT
width (0. (B,(K -' n),A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (B,(K -' n),A))) is finite width (0. (B,(K -' n),A)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (B,(K -' n),A)) ) } is set
[:(dom (0. (B,(K -' n),A))),(Seg (width (0. (B,(K -' n),A)))):] is Relation-like finite set
(Col (MV,x)) . x is set
MV * (((Sgm ((Seg K) \ x)) . x),((Sgm (Seg A)) . x)) is Element of the carrier of B
(0. (B,(K -' n),A)) * (x,y) is Element of the carrier of B
(mlt ((Line (BA,x)),(Col (MV,x)))) . x is set
(BA * (y,((Sgm ((Seg K) \ x)) . x))) * (0. B) is Element of the carrier of B
K560( the carrier of B, the multF of B,(BA * (y,((Sgm ((Seg K) \ x)) . x))),(0. B)) is Element of the carrier of B
(Col (MV,x)) . ((Sgm x) . x) is set
MV * (((Sgm x) . x),x) is Element of the carrier of B
AB * (y,y) is Element of the carrier of B
(Line (BA,x)) . ((Sgm x) . x) is set
BA * (y,((Sgm x) . y)) is Element of the carrier of B
(Segm (BA,(Seg n),x)) * (y,y) is Element of the carrier of B
1_ B is Element of the carrier of B
1. B is non zero Element of the carrier of B
(mlt ((Line (BA,x)),(Col (MV,x)))) . ((Sgm x) . x) is set
(1_ B) * (AB * (y,y)) is Element of the carrier of B
K560( the carrier of B, the multF of B,(1_ B),(AB * (y,y))) is Element of the carrier of B
c₁₃ * (x,x) is Element of the carrier of B
(Line (BA,x)) "*" (Col (MV,x)) is Element of the carrier of B
Sum (mlt ((Line (BA,x)),(Col (MV,x)))) 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 quasi_total V223( the carrier of B) V224( the carrier of B) Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
the addF of B $$ (mlt ((Line (BA,x)),(Col (MV,x)))) is Element of the carrier of B
AB * (x,x) is Element of the carrier of B
{{}} is functional non empty trivial finite V37() 1 -element Element of bool NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of {} ,n, the carrier of A
BA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of {} ,K, the carrier of A
(A,B,BA) is set
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width B & width b₁ = width BA & B * b₁ = BA ) } is set
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is set
x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len (B * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
0. (A,n,K) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
K -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = K } is set
0. A is zero Element of the carrier of A
K |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Seg K) --> (0. A) is Relation-like Seg K -defined Seg K -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. A)}:]
{(0. A)} is non empty trivial finite 1 -element set
[:(Seg K),{(0. A)}:] is Relation-like finite set
bool [:(Seg K),{(0. A)}:] is finite V37() set
n |-> (K |-> (0. A)) is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (K -tuples_on the carrier of A)
n -tuples_on (K -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
(K -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (K -tuples_on the carrier of A) * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (K |-> (0. A)) is Relation-like Seg n -defined Seg n -defined K -tuples_on the carrier of A -valued {(K |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(K |-> (0. A))}:]
{(K |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(K |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{(K |-> (0. A))}:] is finite V37() set
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
(A,(0. (A,n,K)),B) is set
width (0. (A,n,K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width (0. (A,n,K)) & width b₁ = width B & (0. (A,n,K)) * b₁ = B ) } is set
0. (A,n,(width B)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, width B, the carrier of A
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
(width B) |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
(Seg (width B)) --> (0. A) is Relation-like Seg (width B) -defined Seg (width B) -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width B)),{(0. A)}:]
[:(Seg (width B)),{(0. A)}:] is Relation-like finite set
bool [:(Seg (width B)),{(0. A)}:] is finite V37() set
n |-> ((width B) |-> (0. A)) is Relation-like NAT -defined (width B) -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on ((width B) -tuples_on the carrier of A)
n -tuples_on ((width B) -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of (width B) -tuples_on the carrier of A
((width B) -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of (width B) -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined (width B) -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width B) -tuples_on the carrier of A) * : len b₁ = n } is set
(Seg n) --> ((width B) |-> (0. A)) is Relation-like Seg n -defined Seg n -defined (width B) -tuples_on the carrier of A -valued {((width B) |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{((width B) |-> (0. A))}:]
{((width B) |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{((width B) |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{((width B) |-> (0. A))}:] is finite V37() set
x is set
len (0. (A,n,K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom (0. (A,n,K)) is finite Element of bool NAT
len (0. (A,n,(width B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line ((0. (A,n,K)),X) is Relation-like NAT -defined the carrier of A -valued Function-like finite width (0. (A,n,K)) -element FinSequence-like FinSubsequence-like Element of (width (0. (A,n,K))) -tuples_on the carrier of A
(width (0. (A,n,K))) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width (0. (A,n,K)) } is set
(0. (A,n,K)) . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(width (0. (A,n,K))) |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite width (0. (A,n,K)) -element FinSequence-like FinSubsequence-like Element of (width (0. (A,n,K))) -tuples_on the carrier of A
Seg (width (0. (A,n,K))) is finite width (0. (A,n,K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,n,K)) ) } is set
(Seg (width (0. (A,n,K)))) --> (0. A) is Relation-like Seg (width (0. (A,n,K))) -defined Seg (width (0. (A,n,K))) -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width (0. (A,n,K)))),{(0. A)}:]
[:(Seg (width (0. (A,n,K)))),{(0. A)}:] is Relation-like finite set
bool [:(Seg (width (0. (A,n,K)))),{(0. A)}:] is finite V37() set
Line (B,X) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
f is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, width B, the carrier of A
f . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
BX is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(0. (A,n,K)) * BX is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
B . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(0. (A,n,(width B))) . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of BA is non empty non trivial V103() set
the carrier of BA * is functional non empty FinSequence-membered FinSequenceSet of the carrier of BA
AB is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of BA
x is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,B, the carrier of BA
(BA,AB,x) is set
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA * : ( len b₁ = width AB & width b₁ = width x & AB * b₁ = x ) } is set
f is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB * f is Relation-like NAT -defined the carrier of BA * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of BA *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
0. (B,n,K) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of B
the carrier of B is non empty non trivial V103() set
the carrier of B * is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
K -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = K } is set
0. B is zero Element of the carrier of B
K |-> (0. B) is Relation-like NAT -defined the carrier of B -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of B
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Seg K) --> (0. B) is Relation-like Seg K -defined Seg K -defined the carrier of B -valued {(0. B)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. B)}:]
{(0. B)} is non empty trivial finite 1 -element set
[:(Seg K),{(0. B)}:] is Relation-like finite set
bool [:(Seg K),{(0. B)}:] is finite V37() set
n |-> (K |-> (0. B)) is Relation-like NAT -defined K -tuples_on the carrier of B -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (K -tuples_on the carrier of B)
n -tuples_on (K -tuples_on the carrier of B) is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of B
(K -tuples_on the carrier of B) * is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined K -tuples_on the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of (K -tuples_on the carrier of B) * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (K |-> (0. B)) is Relation-like Seg n -defined Seg n -defined K -tuples_on the carrier of B -valued {(K |-> (0. B))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(K |-> (0. B))}:]
{(K |-> (0. B))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(K |-> (0. B))}:] is Relation-like finite set
bool [:(Seg n),{(K |-> (0. B))}:] is finite V37() set
0. (B,n,A) is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,A, the carrier of B
A -tuples_on the carrier of B is functional non empty FinSequence-membered FinSequenceSet of the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of B * : len b₁ = A } is set
A |-> (0. B) is Relation-like NAT -defined the carrier of B -valued Function-like finite A -element FinSequence-like FinSubsequence-like Element of A -tuples_on the carrier of B
Seg A is finite A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
(Seg A) --> (0. B) is Relation-like Seg A -defined Seg A -defined the carrier of B -valued {(0. B)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg A),{(0. B)}:]
[:(Seg A),{(0. B)}:] is Relation-like finite set
bool [:(Seg A),{(0. B)}:] is finite V37() set
n |-> (A |-> (0. B)) is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (A -tuples_on the carrier of B)
n -tuples_on (A -tuples_on the carrier of B) is functional non empty FinSequence-membered FinSequenceSet of A -tuples_on the carrier of B
(A -tuples_on the carrier of B) * is functional non empty FinSequence-membered FinSequenceSet of A -tuples_on the carrier of B
{ b₁ where b₁ is Relation-like NAT -defined A -tuples_on the carrier of B -valued Function-like finite FinSequence-like FinSubsequence-like Element of (A -tuples_on the carrier of B) * : len b₁ = n } is set
(Seg n) --> (A |-> (0. B)) is Relation-like Seg n -defined Seg n -defined A -tuples_on the carrier of B -valued {(A |-> (0. B))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(A |-> (0. B))}:]
{(A |-> (0. B))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(A |-> (0. B))}:] is Relation-like finite set
bool [:(Seg n),{(A |-> (0. B))}:] is finite V37() set
(B,(0. (B,n,K)),(0. (B,n,A))) is set
width (0. (B,n,K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (0. (B,n,A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B * : ( len b₁ = width (0. (B,n,K)) & width b₁ = width (0. (B,n,A)) & (0. (B,n,K)) * b₁ = 0. (B,n,A) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of B : verum } is set
f is set
f is set
X is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K,A, the carrier of B
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (0. (B,n,K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(0. (B,n,K)) * X is Relation-like NAT -defined the carrier of B * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of B *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
0. (A,n,{}) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, {} , the carrier of A
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{} -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = {} } is set
0. A is zero Element of the carrier of A
{} |-> (0. A) is Relation-like non-empty empty-yielding NAT -defined the carrier of A -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of {} -tuples_on the carrier of A
Seg {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
(Seg {}) --> (0. A) is Relation-like non-empty empty-yielding Seg {} -defined Seg {} -defined RAT -valued the carrier of A -valued {(0. A)} -valued Function-like one-to-one constant functional empty total total quasi_total V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool [:(Seg {}),{(0. A)}:]
{(0. A)} is non empty trivial finite 1 -element set
[:(Seg {}),{(0. A)}:] is Relation-like finite set
bool [:(Seg {}),{(0. A)}:] is finite V37() set
n |-> ({} |-> (0. A)) is Relation-like NAT -defined {} -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on ({} -tuples_on the carrier of A)
n -tuples_on ({} -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of {} -tuples_on the carrier of A
({} -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of {} -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined {} -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of ({} -tuples_on the carrier of A) * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> ({} |-> (0. A)) is Relation-like Seg n -defined Seg n -defined RAT -valued {} -tuples_on the carrier of A -valued {({} |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() V185() V186() V187() V188() Element of bool [:(Seg n),{({} |-> (0. A))}:]
{({} |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{({} |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{({} |-> (0. A))}:] is finite V37() set
0. (A,n,K) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
K -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = K } is set
K |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
(Seg K) --> (0. A) is Relation-like Seg K -defined Seg K -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. A)}:]
[:(Seg K),{(0. A)}:] is Relation-like finite set
bool [:(Seg K),{(0. A)}:] is finite V37() set
n |-> (K |-> (0. A)) is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (K -tuples_on the carrier of A)
n -tuples_on (K -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
(K -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of K -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined K -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (K -tuples_on the carrier of A) * : len b₁ = n } is set
(Seg n) --> (K |-> (0. A)) is Relation-like Seg n -defined Seg n -defined K -tuples_on the carrier of A -valued {(K |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(K |-> (0. A))}:]
{(K |-> (0. A))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{(K |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{(K |-> (0. A))}:] is finite V37() set
(A,(0. (A,n,{})),(0. (A,n,K))) is set
width (0. (A,n,{})) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (0. (A,n,K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width (0. (A,n,{})) & width b₁ = width (0. (A,n,K)) & (0. (A,n,{})) * b₁ = 0. (A,n,K) ) } is set
B is set
BA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(0. (A,n,{})) * BA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
0. (K,n,{}) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, {} , the carrier of K
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{} -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = {} } is set
0. K is zero Element of the carrier of K
{} |-> (0. K) is Relation-like non-empty empty-yielding NAT -defined the carrier of K -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of {} -tuples_on the carrier of K
Seg {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
(Seg {}) --> (0. K) is Relation-like non-empty empty-yielding Seg {} -defined Seg {} -defined RAT -valued the carrier of K -valued {(0. K)} -valued Function-like one-to-one constant functional empty total total quasi_total V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool [:(Seg {}),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg {}),{(0. K)}:] is Relation-like finite set
bool [:(Seg {}),{(0. K)}:] is finite V37() set
n |-> ({} |-> (0. K)) is Relation-like NAT -defined {} -tuples_on the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on ({} -tuples_on the carrier of K)
n -tuples_on ({} -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of {} -tuples_on the carrier of K
({} -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of {} -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined {} -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of ({} -tuples_on the carrier of K) * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> ({} |-> (0. K)) is Relation-like Seg n -defined Seg n -defined RAT -valued {} -tuples_on the carrier of K -valued {({} |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() V185() V186() V187() V188() Element of bool [:(Seg n),{({} |-> (0. K))}:]
{({} |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg n),{({} |-> (0. K))}:] is Relation-like finite set
bool [:(Seg n),{({} |-> (0. K))}:] is finite V37() set
(K,(0. (K,n,{})),(0. (K,n,{}))) is set
width (0. (K,n,{})) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (0. (K,n,{})) & width b₁ = width (0. (K,n,{})) & (0. (K,n,{})) * b₁ = 0. (K,n,{}) ) } is set
Indices (0. (K,n,{})) is set
dom (0. (K,n,{})) is finite Element of bool NAT
Seg (width (0. (K,n,{}))) is finite width (0. (K,n,{})) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,n,{})) ) } is set
[:(dom (0. (K,n,{}))),(Seg (width (0. (K,n,{})))):] is Relation-like finite set
BA is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of {} , {} , the carrier of K
(0. (K,n,{})) * BA is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[AB,x] is set
{AB,x} is non empty finite V37() set
{AB} is non empty trivial finite V37() 1 -element set
{{AB,x},{AB}} is non empty finite V37() without_zero V103() set
(0. (K,n,{})) * (AB,x) is Element of the carrier of K
((0. (K,n,{})) * BA) * (AB,x) is Element of the carrier of K
AB is set
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of {} , {} , the carrier of K
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((0. (K,n,{})) * BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (0. (K,n,{})) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((0. (K,n,{})) * BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
F₁() is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of F₁() is non empty non trivial V103() set
the carrier of F₁() * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
F₂() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₃() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₄() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₅() is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
F₆() is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Seg F₃() is finite F₃() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= F₃() ) } is set
the_rank_of F₅() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. F₁() is zero Element of the carrier of F₁()
F₃() |-> (0. F₁()) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite F₃() -element FinSequence-like FinSubsequence-like Element of F₃() -tuples_on the carrier of F₁()
F₃() -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = F₃() } is set
(Seg F₃()) --> (0. F₁()) is Relation-like Seg F₃() -defined Seg F₃() -defined the carrier of F₁() -valued {(0. F₁())} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg F₃()),{(0. F₁())}:]
{(0. F₁())} is non empty trivial finite 1 -element set
[:(Seg F₃()),{(0. F₁())}:] is Relation-like finite set
bool [:(Seg F₃()),{(0. F₁())}:] is finite V37() set
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (K,B) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
(width K) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width K } is set
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT *
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
len (<*> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT
rng (<*> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() V189() V190() V191() V192() V195() V196() V197() V200() set
dom (<*> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(<*> NAT) /. B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (B,((<*> NAT) /. B)) is Element of the carrier of F₁()
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(<*> NAT) /. BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{B} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom (<*> NAT)) \ {B} is Relation-like finite Element of bool NAT
K * (BA,((<*> NAT) /. B)) is Element of the carrier of F₁()
K * (B,BA) is Element of the carrier of F₁()
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
K + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
Seg (K + 1) is non empty finite K + 1 -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K + 1 ) } is set
B is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
BA is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices B is set
dom B is finite Element of bool NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
AB is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng AB is finite V195() V196() V197() V200() set
dom AB is finite Element of bool NAT
B is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
BA is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices B is set
dom B is finite Element of bool NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width B) is finite width B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
AB is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng AB is finite V195() V196() V197() V200() set
dom AB is finite Element of bool NAT
AB is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng AB is finite V195() V196() V197() V200() set
dom AB is finite Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,f] is set
{x,f} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,f},{x}} is non empty finite V37() without_zero V103() set
B * (x,f) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,f] is set
{x,f} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,f},{x}} is non empty finite V37() without_zero V103() set
B * (x,f) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,f] is set
{x,f} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,f},{x}} is non empty finite V37() without_zero V103() set
B * (x,f) is Element of the carrier of F₁()
Seg F₂() is finite F₂() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= F₂() ) } is set
[:(Seg F₂()),(Seg F₃()):] is Relation-like finite set
(len AB) + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{} + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
{((len AB) + 1)} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
Seg (len AB) is finite len AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len AB ) } is set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
MV is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
lA is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
(dom MV) \ {((len AB) + 1)} is finite Element of bool NAT
MV is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
lA is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
(dom MV) \ {((len AB) + 1)} is finite Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
dom c₁₃ is finite Element of bool NAT
(dom c₁₃) \ {((len AB) + 1)} is finite Element of bool NAT
c₁₃ * (x,(K + 1)) is Element of the carrier of F₁()
Line (MV,(X + 1)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width MV -element FinSequence-like FinSubsequence-like Element of (width MV) -tuples_on the carrier of F₁()
(width MV) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width MV } is set
Line (MV,((len AB) + 1)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width MV -element FinSequence-like FinSubsequence-like Element of (width MV) -tuples_on the carrier of F₁()
MV * ((X + 1),(K + 1)) is Element of the carrier of F₁()
(MV * (((len AB) + 1),(K + 1))) " is Element of the carrier of F₁()
(MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) ") is Element of the carrier of F₁()
the multF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
[: the carrier of F₁(), the carrier of F₁():] is Relation-like set
[:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is Relation-like set
bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is set
K560( the carrier of F₁(), the multF of F₁(),(MV * ((X + 1),(K + 1))),((MV * (((len AB) + 1),(K + 1))) ")) is Element of the carrier of F₁()
- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) ")) is Element of the carrier of F₁()
(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width MV -element FinSequence-like FinSubsequence-like Element of (width MV) -tuples_on the carrier of F₁()
(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
bool [: the carrier of F₁(), the carrier of F₁():] is set
id the carrier of F₁() is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (MV,((len AB) + 1))),((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width MV -element FinSequence-like FinSubsequence-like Element of (width MV) -tuples_on the carrier of F₁()
the addF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) V224( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),(Line (MV,(X + 1))),((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
len ((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[((len AB) + 1),(K + 1)] is Element of [:NAT,NAT:]
{((len AB) + 1),(K + 1)} is non empty finite V37() without_zero V103() set
{((len AB) + 1)} is non empty trivial finite V37() 1 -element without_zero V103() set
{{((len AB) + 1),(K + 1)},{((len AB) + 1)}} is non empty finite V37() without_zero V103() set
(ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) is set
dom (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) is finite Element of bool NAT
width (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))))))) is finite width (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) ) } is set
[:(dom (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))))))),(Seg (width (ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))))):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
[((len AB) + 1),x] is set
{((len AB) + 1),x} is non empty finite V37() set
{{((len AB) + 1),x},{((len AB) + 1)}} is non empty finite V37() without_zero V103() set
MV * (((len AB) + 1),x) is Element of the carrier of F₁()
(Line (MV,((len AB) + 1))) . x is set
((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))) . x is set
(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))),(0. F₁())) is Element of the carrier of F₁()
(Line (MV,(X + 1))) . x is set
MV * ((X + 1),x) is Element of the carrier of F₁()
(0. F₁()) + (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),(0. F₁())) is Element of the carrier of F₁()
((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) . x is set
(ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) * (y,x) is Element of the carrier of F₁()
(ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) * (y,x) is Element of the carrier of F₁()
MV * (y,x) is Element of the carrier of F₁()
(ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) * (y,x) is Element of the carrier of F₁()
(ReplaceLine (MV,(X + 1),((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))))) * (y,x) is Element of the carrier of F₁()
F₇(lA,(X + 1),((len AB) + 1),(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) ")))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
x is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len MV) is finite len MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len MV ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
dom x is finite Element of bool NAT
(dom x) \ {((len AB) + 1)} is finite Element of bool NAT
[x,(K + 1)] is set
{x,(K + 1)} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,(K + 1)},{x}} is non empty finite V37() without_zero V103() set
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len x) is finite len x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
x * (x,(K + 1)) is Element of the carrier of F₁()
MV * (x,(K + 1)) is Element of the carrier of F₁()
(Line (MV,((len AB) + 1))) . (K + 1) is set
((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))) . (K + 1) is set
(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (MV * (((len AB) + 1),(K + 1))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))),(MV * (((len AB) + 1),(K + 1)))) is Element of the carrier of F₁()
- (MV * ((X + 1),(K + 1))) is Element of the carrier of F₁()
(- (MV * ((X + 1),(K + 1)))) * ((MV * (((len AB) + 1),(K + 1))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- (MV * ((X + 1),(K + 1)))),((MV * (((len AB) + 1),(K + 1))) ")) is Element of the carrier of F₁()
((- (MV * ((X + 1),(K + 1)))) * ((MV * (((len AB) + 1),(K + 1))) ")) * (MV * (((len AB) + 1),(K + 1))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((- (MV * ((X + 1),(K + 1)))) * ((MV * (((len AB) + 1),(K + 1))) ")),(MV * (((len AB) + 1),(K + 1)))) is Element of the carrier of F₁()
((MV * (((len AB) + 1),(K + 1))) ") * (MV * (((len AB) + 1),(K + 1))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((MV * (((len AB) + 1),(K + 1))) "),(MV * (((len AB) + 1),(K + 1)))) is Element of the carrier of F₁()
(- (MV * ((X + 1),(K + 1)))) * (((MV * (((len AB) + 1),(K + 1))) ") * (MV * (((len AB) + 1),(K + 1)))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- (MV * ((X + 1),(K + 1)))),(((MV * (((len AB) + 1),(K + 1))) ") * (MV * (((len AB) + 1),(K + 1))))) is Element of the carrier of F₁()
1_ F₁() is Element of the carrier of F₁()
1. F₁() is non zero Element of the carrier of F₁()
(- (MV * ((X + 1),(K + 1)))) * (1_ F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- (MV * ((X + 1),(K + 1)))),(1_ F₁())) is Element of the carrier of F₁()
(Line (MV,(X + 1))) . (K + 1) is set
((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) . (K + 1) is set
(MV * ((X + 1),(K + 1))) + (- (MV * ((X + 1),(K + 1)))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(MV * ((X + 1),(K + 1))),(- (MV * ((X + 1),(K + 1))))) is Element of the carrier of F₁()
x * (x,(K + 1)) is Element of the carrier of F₁()
x * (x,(K + 1)) is Element of the carrier of F₁()
x * (x,(K + 1)) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((len AB) + 1),(AB /. x)] is Element of [:NAT,NAT:]
{((len AB) + 1),(AB /. x)} is non empty finite V37() set
{{((len AB) + 1),(AB /. x)},{((len AB) + 1)}} is non empty finite V37() without_zero V103() set
(Line (MV,((len AB) + 1))) . (AB /. x) is set
MV * (((len AB) + 1),(AB /. x)) is Element of the carrier of F₁()
((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))) . (AB /. x) is set
(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),(- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))),(0. F₁())) is Element of the carrier of F₁()
(Line (MV,(X + 1))) . (AB /. x) is set
MV * ((X + 1),(AB /. x)) is Element of the carrier of F₁()
((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) . (AB /. x) is set
(MV * ((X + 1),(AB /. x))) + (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(MV * ((X + 1),(AB /. x))),(0. F₁())) is Element of the carrier of F₁()
[x,(AB /. x)] is set
{x,(AB /. x)} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,(AB /. x)},{x}} is non empty finite V37() without_zero V103() set
x * (x,(AB /. x)) is Element of the carrier of F₁()
MV * (x,(AB /. x)) is Element of the carrier of F₁()
x * (x,(AB /. x)) is Element of the carrier of F₁()
MV * (x,(AB /. x)) is Element of the carrier of F₁()
x * (x,(AB /. x)) is Element of the carrier of F₁()
x * (x,(AB /. x)) is Element of the carrier of F₁()
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{x} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom AB) \ {x} is finite Element of bool NAT
x * (L,(AB /. x)) is Element of the carrier of F₁()
[L,(AB /. x)] is set
{L,(AB /. x)} is non empty finite V37() set
{L} is non empty trivial finite V37() 1 -element set
{{L,(AB /. x)},{L}} is non empty finite V37() without_zero V103() set
MV * (L,(AB /. x)) is Element of the carrier of F₁()
Seg (width x) is finite width x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
x * (x,L) is Element of the carrier of F₁()
[((len AB) + 1),L] is set
{((len AB) + 1),L} is non empty finite V37() set
{{((len AB) + 1),L},{((len AB) + 1)}} is non empty finite V37() without_zero V103() set
[x,L] is set
{x,L} is non empty finite V37() set
{{x,L},{x}} is non empty finite V37() without_zero V103() set
MV * (x,L) is Element of the carrier of F₁()
MV * (((len AB) + 1),L) is Element of the carrier of F₁()
(Line (MV,((len AB) + 1))) . L is set
((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1)))) . L is set
(Line (MV,(X + 1))) . L is set
MV * (x,L) is Element of the carrier of F₁()
((Line (MV,(X + 1))) + ((- ((MV * ((X + 1),(K + 1))) * ((MV * (((len AB) + 1),(K + 1))) "))) * (Line (MV,((len AB) + 1))))) . L is set
(0. F₁()) + (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),(0. F₁())) is Element of the carrier of F₁()
the_rank_of x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
j is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
x is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
y is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices x is set
dom x is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width x) is finite width x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
[:(dom x),(Seg (width x)):] is Relation-like finite set
(dom x) \ {((len AB) + 1)} is finite Element of bool NAT
B * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
B * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
1_ F₁() is Element of the carrier of F₁()
1. F₁() is non zero Element of the carrier of F₁()
F₇(BA,((len AB) + 1),x,(1_ F₁())) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Line (B,x) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of F₁()
(width B) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width B } is set
Line (B,((len AB) + 1)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of F₁()
(1_ F₁()) * (Line (B,x)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of F₁()
(1_ F₁()) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
[: the carrier of F₁(), the carrier of F₁():] is Relation-like set
bool [: the carrier of F₁(), the carrier of F₁():] is set
the multF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
[:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is Relation-like set
bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is set
id the carrier of F₁() is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] ((1_ F₁()),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (B,x)),((1_ F₁()) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of F₁()
the addF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) V224( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),(Line (B,((len AB) + 1))),((1_ F₁()) * (Line (B,x)))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x))))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) is set
dom (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) is finite Element of bool NAT
width (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x))))))) is finite width (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) ) } is set
[:(dom (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x))))))),(Seg (width (ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))))):] is Relation-like finite set
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
(Line (B,((len AB) + 1))) + (Line (B,x)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of F₁()
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),(Line (B,((len AB) + 1))),(Line (B,x))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
len ((Line (B,((len AB) + 1))) + (Line (B,x))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[((len AB) + 1),f] is set
{((len AB) + 1),f} is non empty finite V37() set
{((len AB) + 1)} is non empty trivial finite V37() 1 -element without_zero V103() set
{{((len AB) + 1),f},{((len AB) + 1)}} is non empty finite V37() without_zero V103() set
((Line (B,((len AB) + 1))) + (Line (B,x))) . (K + 1) is set
(Line (B,x)) . (K + 1) is set
B * (x,(K + 1)) is Element of the carrier of F₁()
(Line (B,((len AB) + 1))) . (K + 1) is set
(0. F₁()) + (B * (x,(K + 1))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),(B * (x,(K + 1)))) is Element of the carrier of F₁()
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[c₁₃,x] is set
{c₁₃,x} is non empty finite V37() set
{c₁₃} is non empty trivial finite V37() 1 -element set
{{c₁₃,x},{c₁₃}} is non empty finite V37() without_zero V103() set
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
B * (c₁₃,x) is Element of the carrier of F₁()
[x,x] is set
{x,x} is non empty finite V37() set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
(Line (B,((len AB) + 1))) . x is set
B * (((len AB) + 1),x) is Element of the carrier of F₁()
(Line (B,x)) . x is set
B * (x,x) is Element of the carrier of F₁()
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
((Line (B,((len AB) + 1))) + (Line (B,x))) . x is set
(B * (((len AB) + 1),x)) + (B * (x,x)) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(B * (((len AB) + 1),x)),(B * (x,x))) is Element of the carrier of F₁()
(0. F₁()) + (B * (x,x)) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),(B * (x,x))) is Element of the carrier of F₁()
(0. F₁()) + (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),(0. F₁())) is Element of the carrier of F₁()
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[c₁₃,x] is set
{c₁₃,x} is non empty finite V37() set
{c₁₃} is non empty trivial finite V37() 1 -element set
{{c₁₃,x},{c₁₃}} is non empty finite V37() without_zero V103() set
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,(AB /. c₁₃)) is Element of the carrier of F₁()
{c₁₃} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom AB) \ {c₁₃} is finite Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (x,(AB /. c₁₃)) is Element of the carrier of F₁()
(ReplaceLine (B,((len AB) + 1),((Line (B,((len AB) + 1))) + ((1_ F₁()) * (Line (B,x)))))) * (c₁₃,x) is Element of the carrier of F₁()
AB . c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[c₁₃,(AB /. c₁₃)] is set
{c₁₃,(AB /. c₁₃)} is non empty finite V37() set
{c₁₃} is non empty trivial finite V37() 1 -element set
{{c₁₃,(AB /. c₁₃)},{c₁₃}} is non empty finite V37() without_zero V103() set
B * (c₁₃,(AB /. c₁₃)) is Element of the carrier of F₁()
[x,(AB /. c₁₃)] is set
{x,(AB /. c₁₃)} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,(AB /. c₁₃)},{x}} is non empty finite V37() without_zero V103() set
B * (x,(AB /. c₁₃)) is Element of the carrier of F₁()
[c₁₃,x] is set
{c₁₃,x} is non empty finite V37() set
{{c₁₃,x},{c₁₃}} is non empty finite V37() without_zero V103() set
B * (c₁₃,x) is Element of the carrier of F₁()
B * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
X is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
BX is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices X is set
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
X is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
BX is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices X is set
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
(dom X) \ {((len AB) + 1)} is finite Element of bool NAT
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X * (MV,(K + 1)) is Element of the carrier of F₁()
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,lA] is set
{MV,lA} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,lA},{MV}} is non empty finite V37() without_zero V103() set
X * (MV,lA) is Element of the carrier of F₁()
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X * (c₁₃,(AB /. c₁₃)) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{c₁₃} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom AB) \ {c₁₃} is finite Element of bool NAT
X * (x,(AB /. c₁₃)) is Element of the carrier of F₁()
X * (c₁₃,x) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X * (x,(K + 1)) is Element of the carrier of F₁()
MV is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
lA is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (((len AB) + 1),(K + 1)) is Element of the carrier of F₁()
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
(dom MV) \ {((len AB) + 1)} is finite Element of bool NAT
<*(K + 1)*> is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() V189() V190() V191() V192() Element of NAT *
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
[1,(K + 1)] is set
{1,(K + 1)} is non empty finite V37() without_zero V103() set
{{1,(K + 1)},{1}} is non empty finite V37() without_zero V103() set
{[1,(K + 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
AB ^ <*(K + 1)*> is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
c₁₃ is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len c₁₃ is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
rng c₁₃ is non empty finite V195() V196() V197() V200() set
dom c₁₃ is non empty finite Element of bool NAT
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len MV) is finite len MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len MV ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,x] is set
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
MV * (x,x) is Element of the carrier of F₁()
MV * (x,x) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg ((len AB) + 1) is non empty finite (len AB) + 1 -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (len AB) + 1 ) } is set
(dom AB) \/ {((len AB) + 1)} is non empty finite Element of bool NAT
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (x,(c₁₃ /. x)) is Element of the carrier of F₁()
c₁₃ /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV * (x,(c₁₃ /. x)) is Element of the carrier of F₁()
c₁₃ /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (x,(c₁₃ /. x)) is Element of the carrier of F₁()
c₁₃ /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (x,(c₁₃ /. x)) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len c₁₃) is non empty finite len c₁₃ -element without_zero V103() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len c₁₃ ) } is set
{x} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom c₁₃) \ {x} is finite Element of bool NAT
MV * (x,(c₁₃ /. x)) is Element of the carrier of F₁()
c₁₃ . ((len AB) + 1) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(dom MV) \ {x} is finite Element of bool NAT
MV * (x,(K + 1)) is Element of the carrier of F₁()
AB . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[x,(c₁₃ /. x)] is set
{x,(c₁₃ /. x)} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,(c₁₃ /. x)},{x}} is non empty finite V37() without_zero V103() set
(dom AB) \ {x} is finite Element of bool NAT
MV * (x,(AB /. x)) is Element of the carrier of F₁()
MV * (x,x) is Element of the carrier of F₁()
AB /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,x] is set
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
rng <*(K + 1)*> is non empty trivial finite 1 -element V195() V196() V197() V200() set
{(K + 1)} is non empty trivial finite V37() 1 -element without_zero V103() Element of bool NAT
(rng AB) \/ {(K + 1)} is non empty finite set
(Seg K) \/ {(K + 1)} is non empty finite without_zero V103() Element of bool NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,x] is set
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
MV * (x,x) is Element of the carrier of F₁()
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV * (y,(c₁₃ /. y)) is Element of the carrier of F₁()
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{y} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom c₁₃) \ {y} is finite Element of bool NAT
MV * (y,(c₁₃ /. y)) is Element of the carrier of F₁()
MV * (y,y) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,f] is set
{x,f} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,f},{x}} is non empty finite V37() without_zero V103() set
B * (x,f) is Element of the carrier of F₁()
Seg {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT *
B is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT *
len B is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT
rng B is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() V189() V190() V191() V192() V195() V196() V197() V200() set
dom B is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BA,AB] is set
{BA,AB} is non empty finite V37() set
{BA} is non empty trivial finite V37() 1 -element set
{{BA,AB},{BA}} is non empty finite V37() without_zero V103() set
K * (BA,AB) is Element of the carrier of F₁()
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BA,AB] is set
{BA,AB} is non empty finite V37() set
{BA} is non empty trivial finite V37() 1 -element set
{{BA,AB},{BA}} is non empty finite V37() without_zero V103() set
K * (BA,AB) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (x,(B /. x)) is Element of the carrier of F₁()
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{x} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom B) \ {x} is Relation-like finite Element of bool NAT
K * (f,(B /. x)) is Element of the carrier of F₁()
K * (x,f) is Element of the carrier of F₁()
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices K is set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng B is finite V195() V196() V197() V200() set
dom B is finite Element of bool NAT
B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng B is finite V195() V196() V197() V200() set
dom B is finite Element of bool NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
Seg F₂() is finite F₂() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= F₂() ) } is set
idseq (len B) is Relation-like NAT -defined Function-like finite len B -element FinSequence-like FinSubsequence-like set
id (Seg (len B)) is Relation-like Seg (len B) -defined Seg (len B) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len B)),(Seg (len B)):]
[:(Seg (len B)),(Seg (len B)):] is Relation-like finite set
bool [:(Seg (len B)),(Seg (len B)):] is finite V37() set
len (idseq (len B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len B } is set
AB is Relation-like NAT -defined NAT -valued Function-like finite len B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (len B) -tuples_on NAT
x is Relation-like NAT -defined NAT -valued Function-like finite len B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (len B) -tuples_on NAT
Segm (K,AB,x) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len B, len B, the carrier of F₁()
diagonal_of_Matrix (Segm (K,AB,x)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
Indices (Segm (K,AB,x)) is set
dom (Segm (K,AB,x)) is finite Element of bool NAT
width (Segm (K,AB,x)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (K,AB,x))) is finite width (Segm (K,AB,x)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (K,AB,x)) ) } is set
[:(dom (Segm (K,AB,x))),(Seg (width (Segm (K,AB,x)))):] is Relation-like finite set
dom (diagonal_of_Matrix (Segm (K,AB,x))) is finite Element of bool NAT
len (diagonal_of_Matrix (Segm (K,AB,x))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(diagonal_of_Matrix (Segm (K,AB,x))) . BX is set
[BX,BX] is Element of [:NAT,NAT:]
{BX,BX} is non empty finite V37() set
{BX} is non empty trivial finite V37() 1 -element set
{{BX,BX},{BX}} is non empty finite V37() without_zero V103() set
(Segm (K,AB,x)) * (BX,BX) is Element of the carrier of F₁()
AB . BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B . BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * ((AB . BX),(B . BX)) is Element of the carrier of F₁()
K * (BX,(B . BX)) is Element of the carrier of F₁()
B /. BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (BX,(B /. BX)) is Element of the carrier of F₁()
Product (diagonal_of_Matrix (Segm (K,AB,x))) is Element of the carrier of F₁()
the multF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
[: the carrier of F₁(), the carrier of F₁():] is Relation-like set
[:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is Relation-like set
bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is set
the multF of F₁() $$ (diagonal_of_Matrix (Segm (K,AB,x))) is Element of the carrier of F₁()
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BX,MV] is set
{BX,MV} is non empty finite V37() set
{BX} is non empty trivial finite V37() 1 -element set
{{BX,MV},{BX}} is non empty finite V37() without_zero V103() set
{MV} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom B) \ {MV} is finite Element of bool NAT
(Segm (K,AB,x)) * (BX,MV) is Element of the carrier of F₁()
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB . lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B . c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * ((AB . lA),(B . c₁₃)) is Element of the carrier of F₁()
B . MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * (BX,(B . MV)) is Element of the carrier of F₁()
B /. MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (BX,(B /. MV)) is Element of the carrier of F₁()
Det (Segm (K,AB,x)) is Element of the carrier of F₁()
Permutations (len B) is set
the addF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) V224( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
FinOmega (Permutations (len B)) is Element of K96((Permutations (len B)))
K96((Permutations (len B))) is V24() set
Path_product (Segm (K,AB,x)) is Relation-like Permutations (len B) -defined the carrier of F₁() -valued Function-like total quasi_total Element of bool [:(Permutations (len B)), the carrier of F₁():]
[:(Permutations (len B)), the carrier of F₁():] is Relation-like set
bool [:(Permutations (len B)), the carrier of F₁():] is set
K103((Permutations (len B)), the carrier of F₁(), the addF of F₁(),(FinOmega (Permutations (len B))),(Path_product (Segm (K,AB,x)))) is Element of the carrier of F₁()
Segm (K,(Seg (len B)),(Seg (width K))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len B)), card (Seg (width K)), the carrier of F₁()
card (Seg (len B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len B))) -tuples_on NAT
(card (Seg (len B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len B)) } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm (Seg (len B))),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len B)), card (Seg (width K)), the carrier of F₁()
len (Segm (K,(Seg (len B)),(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng AB is finite V195() V196() V197() V200() set
rng x is finite V195() V196() V197() V200() set
[:(rng AB),(rng x):] is Relation-like RAT -valued finite V185() V186() V187() V188() set
(width K) |-> (0. F₁()) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
(width K) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width K } is set
(Seg (width K)) --> (0. F₁()) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of F₁() -valued {(0. F₁())} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. F₁())}:]
[:(Seg (width K)),{(0. F₁())}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. F₁())}:] is finite V37() set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(dom K) \ (Seg (len B)) is finite Element of bool NAT
Line (K,MV) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[MV,c₁₃] is set
{MV,c₁₃} is non empty finite V37() set
{MV} is non empty trivial finite V37() 1 -element set
{{MV,c₁₃},{MV}} is non empty finite V37() without_zero V103() set
(Line (K,MV)) . c₁₃ is set
K * (MV,c₁₃) is Element of the carrier of F₁()
((width K) |-> (0. F₁())) . c₁₃ is set
len (Line (K,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((width K) |-> (0. F₁())) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of (Segm (K,(Seg (len B)),(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (K,BX) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (BX,(B /. BX)) is Element of the carrier of F₁()
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{BX} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom B) \ {BX} is finite Element of bool NAT
K * (MV,(B /. BX)) is Element of the carrier of F₁()
K * (BX,MV) is Element of the carrier of F₁()
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom K is finite Element of bool NAT
B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng B is finite V195() V196() V197() V200() set
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
dom B is finite Element of bool NAT
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng B is finite V195() V196() V197() V200() set
dom B is finite Element of bool NAT
B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
rng B is finite V195() V196() V197() V200() set
dom B is finite Element of bool NAT
BA is finite without_zero Element of bool NAT
card BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (card BA) is finite card BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card BA ) } is set
Segm (K,(Seg (card BA)),BA) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card BA)), card BA, the carrier of F₁()
card (Seg (card BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (card BA)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card BA)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card BA))) -tuples_on NAT
(card (Seg (card BA))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card BA)) } is set
Sgm BA is Relation-like NAT -defined NAT -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BA) -tuples_on NAT
(card BA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BA } is set
Segm (K,(Sgm (Seg (card BA))),(Sgm BA)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card BA)), card BA, the carrier of F₁()
Seg (the_rank_of F₅()) is finite the_rank_of F₅() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= the_rank_of F₅() ) } is set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (dom B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (the_rank_of F₅())) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B . f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B /. f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices (Segm (K,(Seg (card BA)),BA)) is set
dom (Segm (K,(Seg (card BA)),BA)) is finite Element of bool NAT
width (Segm (K,(Seg (card BA)),BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (K,(Seg (card BA)),BA))) is finite width (Segm (K,(Seg (card BA)),BA)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (K,(Seg (card BA)),BA)) ) } is set
[:(dom (Segm (K,(Seg (card BA)),BA))),(Seg (width (Segm (K,(Seg (card BA)),BA)))):] is Relation-like finite set
[:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{f} is non empty trivial finite V37() 1 -element Element of bool NAT
(dom B) \ {f} is finite Element of bool NAT
idseq (the_rank_of F₅()) is Relation-like NAT -defined Function-like finite the_rank_of F₅() -element FinSequence-like FinSubsequence-like set
id (Seg (the_rank_of F₅())) is Relation-like Seg (the_rank_of F₅()) -defined Seg (the_rank_of F₅()) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):]
bool [:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):] is finite V37() set
(idseq (the_rank_of F₅())) . x is set
[x,f] is Element of [:NAT,NAT:]
{x,f} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,f},{x}} is non empty finite V37() without_zero V103() set
(Segm (K,(Seg (card BA)),BA)) * (x,f) is Element of the carrier of F₁()
Sgm (Seg (the_rank_of F₅())) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (the_rank_of F₅())) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (the_rank_of F₅()))) -tuples_on NAT
(card (Seg (the_rank_of F₅()))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (the_rank_of F₅())) } is set
(Sgm (Seg (the_rank_of F₅()))) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B . f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * (((Sgm (Seg (the_rank_of F₅()))) . x),(B . f)) is Element of the carrier of F₁()
K * (x,(B . f)) is Element of the carrier of F₁()
B /. f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (x,(B /. f)) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (x,((Sgm BA) /. x)) is Element of the carrier of F₁()
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (K,x) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
(width K) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width K } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm BA) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K * (x,f) is Element of the carrier of F₁()
B /. x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
F₁() is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of F₁() is non empty non trivial V103() set
the carrier of F₁() * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
F₂() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₃() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₄() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
F₅() is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
F₆() is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
1_ F₁() is Element of the carrier of F₁()
1. F₁() is non zero Element of the carrier of F₁()
- (1_ F₁()) is Element of the carrier of F₁()
Seg F₃() is finite F₃() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= F₃() ) } is set
the_rank_of F₅() is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. F₁() is zero Element of the carrier of F₁()
F₃() |-> (0. F₁()) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite F₃() -element FinSequence-like FinSubsequence-like Element of F₃() -tuples_on the carrier of F₁()
F₃() -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = F₃() } is set
(Seg F₃()) --> (0. F₁()) is Relation-like Seg F₃() -defined Seg F₃() -defined the carrier of F₁() -valued {(0. F₁())} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg F₃()),{(0. F₁())}:]
{(0. F₁())} is non empty trivial finite 1 -element set
[:(Seg F₃()),{(0. F₁())}:] is Relation-like finite set
bool [:(Seg F₃()),{(0. F₁())}:] is finite V37() set
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (K,B) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
(width K) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width K } is set
Line (K,BA) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
AB is Element of the carrier of F₁()
AB * (Line (K,BA)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
AB multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
[: the carrier of F₁(), the carrier of F₁():] is Relation-like set
bool [: the carrier of F₁(), the carrier of F₁():] is set
the multF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
[:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is Relation-like set
bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is set
id the carrier of F₁() is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] (AB,(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (K,BA)),(AB multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(Line (K,B)) + (AB * (Line (K,BA))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of F₁()
the addF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) V224( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),(Line (K,B)),(AB * (Line (K,BA)))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
ReplaceLine (K,B,((Line (K,B)) + (AB * (Line (K,BA))))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
F₇(A,B,BA,AB) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
B is finite without_zero Element of bool NAT
K is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Seg (card B) is finite card B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card B ) } is set
Segm (K,(Seg (card B)),B) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
card (Seg (card B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (card B)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card B)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card B))) -tuples_on NAT
(card (Seg (card B))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card B)) } is set
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite card B -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card B) -tuples_on NAT
(card B) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card B } is set
Segm (K,(Sgm (Seg (card B))),(Sgm B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
1. (F₁(),(card B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B, card B, the carrier of F₁()
Indices (1. (F₁(),(card B))) is set
dom (1. (F₁(),(card B))) is finite Element of bool NAT
width (1. (F₁(),(card B))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (1. (F₁(),(card B)))) is finite width (1. (F₁(),(card B))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (F₁(),(card B))) ) } is set
[:(dom (1. (F₁(),(card B)))),(Seg (width (1. (F₁(),(card B))))):] is Relation-like finite set
Seg (the_rank_of F₅()) is finite the_rank_of F₅() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= the_rank_of F₅() ) } is set
[:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x + 1 is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
X is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Segm (X,(Seg (card B)),B) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
Segm (X,(Sgm (Seg (card B))),(Sgm B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
X is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Segm (X,(Seg (card B)),B) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
Segm (X,(Sgm (Seg (card B))),(Sgm B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
Line (X,(x + 1)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
(width X) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width X } is set
(Sgm B) /. (x + 1) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
X * ((x + 1),((Sgm B) /. (x + 1))) is Element of the carrier of F₁()
(X * ((x + 1),((Sgm B) /. (x + 1)))) " is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁()) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") + (- (1_ F₁())) is Element of the carrier of F₁()
the addF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) V224( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
[: the carrier of F₁(), the carrier of F₁():] is Relation-like set
[:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is Relation-like set
bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():] is set
K560( the carrier of F₁(), the addF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(- (1_ F₁()))) is Element of the carrier of F₁()
(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
bool [: the carrier of F₁(), the carrier of F₁():] is set
the multF of F₁() is Relation-like [: the carrier of F₁(), the carrier of F₁():] -defined the carrier of F₁() -valued Function-like total quasi_total V223( the carrier of F₁()) Element of bool [:[: the carrier of F₁(), the carrier of F₁():], the carrier of F₁():]
id the carrier of F₁() is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),(Line (X,(x + 1))),((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Sgm (Seg (card B))),(Sgm B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
dom (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) is finite Element of bool NAT
width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))))) is finite width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) ) } is set
F₇(BX,(x + 1),(x + 1),(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁()))) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
x is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg F₂() is finite F₂() -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= F₂() ) } is set
1 + x is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
- (1. F₁()) is Element of the carrier of F₁()
(1_ F₁()) + (- (1_ F₁())) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(1_ F₁()),(- (1_ F₁()))) is Element of the carrier of F₁()
((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),((1_ F₁()) + (- (1_ F₁()))),((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
(0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
(1_ F₁()) * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
(1_ F₁()) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] ((1_ F₁()),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),((1_ F₁()) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
((1_ F₁()) * (Line (X,(x + 1)))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
K498( the carrier of F₁(), the carrier of F₁(), the carrier of F₁(), the addF of F₁(),((1_ F₁()) * (Line (X,(x + 1)))),((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(- (1_ F₁())),((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
(1_ F₁()) + ((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(1_ F₁()),((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) "))) is Element of the carrier of F₁()
((1_ F₁()) + ((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) "))) * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
((1_ F₁()) + ((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) "))) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] (((1_ F₁()) + ((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) "))),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),(((1_ F₁()) + ((- (1_ F₁())) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) "))) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(1_ F₁()) + (- (1_ F₁())) is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(1_ F₁()),(- (1_ F₁()))) is Element of the carrier of F₁()
((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),((1_ F₁()) + (- (1_ F₁()))),((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
(((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
(((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] ((((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),((((1_ F₁()) + (- (1_ F₁()))) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ") is Element of the carrier of F₁()
K560( the carrier of F₁(), the addF of F₁(),(0. F₁()),((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) is Element of the carrier of F₁()
((0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
((0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] (((0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),(((0. F₁()) + ((X * ((x + 1),((Sgm B) /. (x + 1)))) ")) multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] (((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
rng (Sgm B) is finite V195() V196() V197() V200() set
dom (Sgm B) is finite card B -element Element of bool NAT
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) * (y,((Sgm B) /. y)) is Element of the carrier of F₁()
(Sgm B) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,((Sgm B) /. y)] is set
{y,((Sgm B) /. y)} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,((Sgm B) /. y)},{y}} is non empty finite V37() without_zero V103() set
Indices X is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
X * (y,((Sgm B) /. y)) is Element of the carrier of F₁()
X * (y,((Sgm B) /. y)) is Element of the carrier of F₁()
(X * (y,((Sgm B) /. y))) " is Element of the carrier of F₁()
((X * (y,((Sgm B) /. y))) ") * (Line (X,(x + 1))) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
((X * (y,((Sgm B) /. y))) ") multfield is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
the multF of F₁() [;] (((X * (y,((Sgm B) /. y))) "),(id the carrier of F₁())) is Relation-like the carrier of F₁() -defined the carrier of F₁() -valued Function-like non empty total quasi_total Element of bool [: the carrier of F₁(), the carrier of F₁():]
K501( the carrier of F₁(), the carrier of F₁(),(Line (X,(x + 1))),(((X * (y,((Sgm B) /. y))) ") multfield)) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
(((X * (y,((Sgm B) /. y))) ") * (Line (X,(x + 1)))) . ((Sgm B) /. y) is set
(Line (X,(x + 1))) . ((Sgm B) /. y) is set
((X * (y,((Sgm B) /. y))) ") * (X * (y,((Sgm B) /. y))) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * (y,((Sgm B) /. y))) "),(X * (y,((Sgm B) /. y)))) is Element of the carrier of F₁()
Indices X is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
Indices (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) is set
[:(dom (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))))),(Seg (width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))))):] is Relation-like finite set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Indices (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) is set
dom (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) is finite Element of bool NAT
width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B))) is finite width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) ) } is set
[:(dom (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B))),(Seg (width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)))):] is Relation-like finite set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
(Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)) * (y,x) is Element of the carrier of F₁()
Seg (card (Seg (card B))) is finite card (Seg (card B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (card B)) ) } is set
[:(Seg (card (Seg (card B)))),(Seg (width (Segm ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),(Seg (card B)),B)))):] is Relation-like finite set
card (card B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (card (card B)) is finite card (card B) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (card B) ) } is set
idseq (card B) is Relation-like NAT -defined Function-like finite card B -element FinSequence-like FinSubsequence-like set
id (Seg (card B)) is Relation-like Seg (card B) -defined Seg (card B) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (card B)),(Seg (card B)):]
[:(Seg (card B)),(Seg (card B)):] is Relation-like finite set
bool [:(Seg (card B)),(Seg (card B)):] is finite V37() set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(idseq (card B)) . j is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm B) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) * (j,((Sgm B) . x)) is Element of the carrier of F₁()
rng (Sgm (Seg (card B))) is finite V195() V196() V197() V200() set
[:(Seg (card B)),B:] is Relation-like finite set
[j,((Sgm B) . x)] is set
{j,((Sgm B) . x)} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,((Sgm B) . x)},{j}} is non empty finite V37() without_zero V103() set
(Sgm B) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Indices (Segm (X,(Seg (card B)),B)) is set
dom (Segm (X,(Seg (card B)),B)) is finite Element of bool NAT
width (Segm (X,(Seg (card B)),B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (X,(Seg (card B)),B))) is finite width (Segm (X,(Seg (card B)),B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (X,(Seg (card B)),B)) ) } is set
[:(dom (Segm (X,(Seg (card B)),B))),(Seg (width (Segm (X,(Seg (card B)),B)))):] is Relation-like finite set
L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (Line (X,(x + 1)))) . L is set
(Line (X,(x + 1))) . L is set
X * (j,L) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (X * (j,L)) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(X * (j,L))) is Element of the carrier of F₁()
(Segm (X,(Seg (card B)),B)) * (y,x) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * ((Segm (X,(Seg (card B)),B)) * (y,x)) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),((Segm (X,(Seg (card B)),B)) * (y,x))) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(0. F₁())) is Element of the carrier of F₁()
(Sgm (Seg (card B))) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X * (((Sgm (Seg (card B))) . j),((Sgm B) . x)) is Element of the carrier of F₁()
(Segm (X,(Seg (card B)),B)) * (y,x) is Element of the carrier of F₁()
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) /. y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) * (y,((Sgm B) /. y)) is Element of the carrier of F₁()
(Sgm B) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[y,((Sgm B) /. y)] is set
{y,((Sgm B) /. y)} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,((Sgm B) /. y)},{y}} is non empty finite V37() without_zero V103() set
Indices X is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
X * (y,((Sgm B) /. y)) is Element of the carrier of F₁()
len (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))))) is finite len (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) ) } is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line ((ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))),y) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) -element FinSequence-like FinSubsequence-like Element of (width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))))) -tuples_on the carrier of F₁()
(width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1)))))))) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width (ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) } is set
Line (X,y) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of F₁()
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(ReplaceLine (X,(x + 1),((Line (X,(x + 1))) + ((((X * ((x + 1),((Sgm B) /. (x + 1)))) ") - (1_ F₁())) * (Line (X,(x + 1))))))) * (y,x) is Element of the carrier of F₁()
[y,x] is set
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
Indices X is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
(((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (Line (X,(x + 1)))) . x is set
(Line (X,(x + 1))) . x is set
X * (y,x) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (X * (y,x)) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(X * (y,x))) is Element of the carrier of F₁()
((X * ((x + 1),((Sgm B) /. (x + 1)))) ") * (0. F₁()) is Element of the carrier of F₁()
K560( the carrier of F₁(), the multF of F₁(),((X * ((x + 1),((Sgm B) /. (x + 1)))) "),(0. F₁())) is Element of the carrier of F₁()
X * (y,x) is Element of the carrier of F₁()
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) /. AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (AB,((Sgm B) /. AB)) is Element of the carrier of F₁()
AB is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₃(), the carrier of F₁()
the_rank_of AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of F₂(),F₄(), the carrier of F₁()
Segm (AB,(Seg (card B)),B) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
Segm (AB,(Sgm (Seg (card B))),(Sgm B)) is Relation-like NAT -defined the carrier of F₁() * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card B)), card B, the carrier of F₁()
dom AB is finite Element of bool NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
Indices (Segm (AB,(Seg (card B)),B)) is set
dom (Segm (AB,(Seg (card B)),B)) is finite Element of bool NAT
width (Segm (AB,(Seg (card B)),B)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (AB,(Seg (card B)),B))) is finite width (Segm (AB,(Seg (card B)),B)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (AB,(Seg (card B)),B)) ) } is set
[:(dom (Segm (AB,(Seg (card B)),B))),(Seg (width (Segm (AB,(Seg (card B)),B)))):] is Relation-like finite set
dom (Sgm B) is finite card B -element Element of bool NAT
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[X,BX] is set
{X,BX} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,BX},{X}} is non empty finite V37() without_zero V103() set
idseq (the_rank_of F₅()) is Relation-like NAT -defined Function-like finite the_rank_of F₅() -element FinSequence-like FinSubsequence-like set
id (Seg (the_rank_of F₅())) is Relation-like Seg (the_rank_of F₅()) -defined Seg (the_rank_of F₅()) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):]
bool [:(Seg (the_rank_of F₅())),(Seg (the_rank_of F₅())):] is finite V37() set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(idseq (the_rank_of F₅())) . MV is set
(Segm (AB,(Seg (card B)),B)) * (X,BX) is Element of the carrier of F₁()
Sgm (Seg (the_rank_of F₅())) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (the_rank_of F₅())) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (the_rank_of F₅()))) -tuples_on NAT
card (Seg (the_rank_of F₅())) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card (Seg (the_rank_of F₅()))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (the_rank_of F₅())) } is set
(Sgm (Seg (the_rank_of F₅()))) . MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm B) . lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB * (((Sgm (Seg (the_rank_of F₅()))) . MV),((Sgm B) . lA)) is Element of the carrier of F₁()
AB * (MV,((Sgm B) . lA)) is Element of the carrier of F₁()
(Sgm B) /. lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB * (MV,((Sgm B) /. lA)) is Element of the carrier of F₁()
(1. (F₁(),(card B))) * (X,BX) is Element of the carrier of F₁()
(1. (F₁(),(card B))) * (X,BX) is Element of the carrier of F₁()
(1. (F₁(),(card B))) * (X,BX) is Element of the carrier of F₁()
(1. (F₁(),(card B))) * (X,BX) is Element of the carrier of F₁()
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (AB,X) is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of F₁()
(width AB) -tuples_on the carrier of F₁() is functional non empty FinSequence-membered FinSequenceSet of the carrier of F₁()
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F₁() -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of F₁() * : len b₁ = width AB } is set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm B) . BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
AB * (BX,MV) is Element of the carrier of F₁()
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^^ A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K ^^ A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,K,A) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width A & K * b₁ = A ) } is set
f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),f,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width f) + (width AB), the carrier of n
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width f) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),f,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,f,AB) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width f & width b₁ = width AB & f * b₁ = AB ) } is set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
- (1_ n) is Element of the carrier of n
X is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
MV is Element of the carrier of n
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (X,lA) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
(width X) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width X } is set
Line (BX,lA) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
(width BX) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width BX } is set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (X,c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
MV * (Line (X,c₁₃)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
MV multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (MV,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,(Line (X,c₁₃)),(MV multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Line (X,lA)) + (MV * (Line (X,c₁₃))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (X,lA)),(MV * (Line (X,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
Line (BX,c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
MV * (Line (BX,c₁₃)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Line (BX,c₁₃)),(MV multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Line (BX,lA)) + (MV * (Line (BX,c₁₃))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (BX,lA)),(MV * (Line (BX,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (BX,lA,((Line (BX,lA)) + (MV * (Line (BX,c₁₃))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(n,X,BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width X & width b₁ = width BX & X * b₁ = BX ) } is set
(n,(ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))),H₁(BX,lA,c₁₃,MV)) is set
width (ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (ReplaceLine (BX,lA,((Line (BX,lA)) + (MV * (Line (BX,c₁₃)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))) & width b₁ = width (ReplaceLine (BX,lA,((Line (BX,lA)) + (MV * (Line (BX,c₁₃)))))) & (ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))) * b₁ = ReplaceLine (BX,lA,((Line (BX,lA)) + (MV * (Line (BX,c₁₃))))) ) } is set
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
L is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),x,L) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width x) + (width L), the carrier of n
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width x) + (width L) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),x,L) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,x,L) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width x & width b₁ = width L & x * b₁ = L ) } is set
( the carrier of n,(len K),(width K),(width A),X,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (MV * (Line (X,c₁₃))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (MV * (Line (BX,c₁₃))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Line (X,lA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (Line (BX,lA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Line (X,lA)) + (MV * (Line (X,c₁₃)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((Line (BX,lA)) + (MV * (Line (BX,c₁₃)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,(len K),(width K),(width A),(ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))),H₁(BX,lA,c₁₃,MV)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width (ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃))))))) + (width H₁(BX,lA,c₁₃,MV)), the carrier of n
width H₁(BX,lA,c₁₃,MV) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃))))))) + (width H₁(BX,lA,c₁₃,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
((Line (X,lA)) + (MV * (Line (X,c₁₃)))) ^ ((Line (BX,lA)) + (MV * (Line (BX,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width X) + (width BX) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
ReplaceLine (( the carrier of n,(len K),(width K),(width A),X,BX),lA,(((Line (X,lA)) + (MV * (Line (X,c₁₃)))) ^ ((Line (BX,lA)) + (MV * (Line (BX,c₁₃)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
(Line (X,lA)) ^ (Line (BX,lA)) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width X) + (width BX) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(MV * (Line (X,c₁₃))) ^ (MV * (Line (BX,c₁₃))) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width X) + (width BX) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((Line (X,lA)) ^ (Line (BX,lA))) + ((MV * (Line (X,c₁₃))) ^ (MV * (Line (BX,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,((Line (X,lA)) ^ (Line (BX,lA))),((MV * (Line (X,c₁₃))) ^ (MV * (Line (BX,c₁₃))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (( the carrier of n,(len K),(width K),(width A),X,BX),lA,(((Line (X,lA)) ^ (Line (BX,lA))) + ((MV * (Line (X,c₁₃))) ^ (MV * (Line (BX,c₁₃)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
(Line (X,c₁₃)) ^ (Line (BX,c₁₃)) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width X) + (width BX) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K501( the carrier of n, the carrier of n,((Line (X,c₁₃)) ^ (Line (BX,c₁₃))),(MV multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((Line (X,lA)) ^ (Line (BX,lA))) + (MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,((Line (X,lA)) ^ (Line (BX,lA))),(MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (( the carrier of n,(len K),(width K),(width A),X,BX),lA,(((Line (X,lA)) ^ (Line (BX,lA))) + (MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
width ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,(len K),(width K),(width A),X,BX) } is set
(Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)) + (MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)),(MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃))))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (( the carrier of n,(len K),(width K),(width A),X,BX),lA,((Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)) + (MV * ((Line (X,c₁₃)) ^ (Line (BX,c₁₃)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
MV * (Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,(Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃)),(MV multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)) + (MV * (Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)),(MV * (Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (( the carrier of n,(len K),(width K),(width A),X,BX),lA,((Line (( the carrier of n,(len K),(width K),(width A),X,BX),lA)) + (MV * (Line (( the carrier of n,(len K),(width K),(width A),X,BX),c₁₃))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
the_rank_of ( the carrier of n,(len K),(width K),(width A),(ReplaceLine (X,lA,((Line (X,lA)) + (MV * (Line (X,c₁₃)))))),H₁(BX,lA,c₁₃,MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
L is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),x,L) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width x) + (width L), the carrier of n
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width L is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width x) + (width L) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),x,L) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,x,L) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width x & width b₁ = width L & x * b₁ = L ) } is set
X is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
BX is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
( the carrier of n,(len K),(width K),(width A),X,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,X,BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width X & width b₁ = width BX & X * b₁ = BX ) } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
the_rank_of f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. n is zero Element of the carrier of n
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
MV is finite without_zero Element of bool NAT
X is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
the_rank_of X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
Seg (card MV) is finite card MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card MV ) } is set
Segm (X,(Seg (card MV)),MV) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card MV, the carrier of n
card (Seg (card MV)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (card MV)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card MV)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card MV))) -tuples_on NAT
(card (Seg (card MV))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card MV)) } is set
Sgm MV is Relation-like NAT -defined NAT -valued Function-like finite card MV -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card MV) -tuples_on NAT
(card MV) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card MV } is set
Segm (X,(Sgm (Seg (card MV))),(Sgm MV)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card MV, the carrier of n
1. (n,(card MV)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card MV, card MV, the carrier of n
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width X) is finite width X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
len ( the carrier of n,(len K),(width K),(width A),f,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) \ (Seg (card MV)) is finite without_zero Element of bool NAT
(Sgm (Seg (card MV))) " (Seg (card MV)) is finite set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
(Sgm (Seg (width K))) " MV is finite set
dom (Sgm (Seg (card MV))) is finite card (Seg (card MV)) -element Element of bool NAT
Seg (card (Seg (card MV))) is finite card (Seg (card MV)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (card MV)) ) } is set
rng (Sgm (Seg (card MV))) is finite V195() V196() V197() V200() set
x is finite without_zero Element of bool NAT
rng (Sgm (Seg (width K))) is finite V195() V196() V197() V200() set
x is finite without_zero Element of bool NAT
(Sgm (Seg (width K))) .: x is finite V195() V196() V197() V200() set
dom (Sgm (Seg (width K))) is finite card (Seg (width K)) -element Element of bool NAT
Seg (card (Seg (width K))) is finite card (Seg (width K)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (width K)) ) } is set
card x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (X,y) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
(width X) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width X } is set
(width X) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
(Seg (width X)) --> (0. n) is Relation-like Seg (width X) -defined Seg (width X) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width X)),{(0. n)}:]
[:(Seg (width X)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width X)),{(0. n)}:] is finite V37() set
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len BX) is finite len BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BX ) } is set
dom BX is finite Element of bool NAT
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,(len K),(width K),(width A),X,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(dom X) \ (Seg (card MV)) is finite Element of bool NAT
(width BX) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
(width BX) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width BX } is set
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
(Seg (width BX)) --> (0. n) is Relation-like Seg (width BX) -defined Seg (width BX) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width BX)),{(0. n)}:]
[:(Seg (width BX)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width BX)),{(0. n)}:] is finite V37() set
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (X,y) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
Line (BX,y) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
(n,X,BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width X & width b₁ = width BX & X * b₁ = BX ) } is set
Segm (X,(Seg (card MV)),(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width K)), the carrier of n
Segm (X,(Sgm (Seg (card MV))),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width K)), the carrier of n
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
Segm (BX,(Seg (card MV)),(Seg (width A))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width A)), the carrier of n
card (Seg (width A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width A))) -tuples_on NAT
(card (Seg (width A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width A)) } is set
Segm (BX,(Sgm (Seg (card MV))),(Sgm (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width A)), the carrier of n
(n,(Segm (X,(Seg (card MV)),(Seg (width K)))),(Segm (BX,(Seg (card MV)),(Seg (width A))))) is set
width (Segm (X,(Seg (card MV)),(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (Segm (BX,(Seg (card MV)),(Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (X,(Seg (card MV)),(Seg (width K)))) & width b₁ = width (Segm (BX,(Seg (card MV)),(Seg (width A)))) & (Segm (X,(Seg (card MV)),(Seg (width K)))) * b₁ = Segm (BX,(Seg (card MV)),(Seg (width A))) ) } is set
Segm ((Segm (X,(Seg (card MV)),(Seg (width K)))),x,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card x, the carrier of n
card x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm x is Relation-like NAT -defined NAT -valued Function-like finite card x -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card x) -tuples_on NAT
(card x) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x } is set
Sgm x is Relation-like NAT -defined NAT -valued Function-like finite card x -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card x) -tuples_on NAT
(card x) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card x } is set
Segm ((Segm (X,(Seg (card MV)),(Seg (width K)))),(Sgm x),(Sgm x)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card x, the carrier of n
(Seg (card (Seg (width K)))) \ x is finite without_zero Element of bool NAT
Seg (card (Seg (width A))) is finite card (Seg (width A)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (width A)) ) } is set
(card (Seg (width K))) -' (card (Seg (card MV))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. (n,((card (Seg (width K))) -' (card (Seg (card MV)))),(card (Seg (width A)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of (card (Seg (width K))) -' (card (Seg (card MV))), card (Seg (width A)), the carrier of n
(card (Seg (width A))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = card (Seg (width A)) } is set
(card (Seg (width A))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite card (Seg (width A)) -element FinSequence-like FinSubsequence-like Element of (card (Seg (width A))) -tuples_on the carrier of n
(Seg (card (Seg (width A)))) --> (0. n) is Relation-like Seg (card (Seg (width A))) -defined Seg (card (Seg (width A))) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card (Seg (width A)))),{(0. n)}:]
[:(Seg (card (Seg (width A)))),{(0. n)}:] is Relation-like finite set
bool [:(Seg (card (Seg (width A)))),{(0. n)}:] is finite V37() set
((card (Seg (width K))) -' (card (Seg (card MV)))) |-> ((card (Seg (width A))) |-> (0. n)) is Relation-like NAT -defined (card (Seg (width A))) -tuples_on the carrier of n -valued Function-like finite (card (Seg (width K))) -' (card (Seg (card MV))) -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of ((card (Seg (width K))) -' (card (Seg (card MV)))) -tuples_on ((card (Seg (width A))) -tuples_on the carrier of n)
((card (Seg (width K))) -' (card (Seg (card MV)))) -tuples_on ((card (Seg (width A))) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (card (Seg (width A))) -tuples_on the carrier of n
((card (Seg (width A))) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (card (Seg (width A))) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (card (Seg (width A))) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((card (Seg (width A))) -tuples_on the carrier of n) * : len b₁ = (card (Seg (width K))) -' (card (Seg (card MV))) } is set
Seg ((card (Seg (width K))) -' (card (Seg (card MV)))) is finite (card (Seg (width K))) -' (card (Seg (card MV))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (card (Seg (width K))) -' (card (Seg (card MV))) ) } is set
(Seg ((card (Seg (width K))) -' (card (Seg (card MV))))) --> ((card (Seg (width A))) |-> (0. n)) is Relation-like Seg ((card (Seg (width K))) -' (card (Seg (card MV)))) -defined Seg ((card (Seg (width K))) -' (card (Seg (card MV)))) -defined (card (Seg (width A))) -tuples_on the carrier of n -valued {((card (Seg (width A))) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg ((card (Seg (width K))) -' (card (Seg (card MV))))),{((card (Seg (width A))) |-> (0. n))}:]
{((card (Seg (width A))) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg ((card (Seg (width K))) -' (card (Seg (card MV))))),{((card (Seg (width A))) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg ((card (Seg (width K))) -' (card (Seg (card MV))))),{((card (Seg (width A))) |-> (0. n))}:] is finite V37() set
y is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (width K)), card (Seg (width A)), the carrier of n
Segm (y,((Seg (card (Seg (width K)))) \ x),(Seg (card (Seg (width A))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (card (Seg (width K)))) \ x), card (Seg (card (Seg (width A)))), the carrier of n
card ((Seg (card (Seg (width K)))) \ x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (card (Seg (width A)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg (card (Seg (width K)))) \ x) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (card (Seg (width K)))) \ x) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (card (Seg (width K)))) \ x)) -tuples_on NAT
(card ((Seg (card (Seg (width K)))) \ x)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (card (Seg (width K)))) \ x) } is set
Sgm (Seg (card (Seg (width A)))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card (Seg (width A)))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card (Seg (width A))))) -tuples_on NAT
(card (Seg (card (Seg (width A))))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card (Seg (width A)))) } is set
Segm (y,(Sgm ((Seg (card (Seg (width K)))) \ x)),(Sgm (Seg (card (Seg (width A)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card ((Seg (card (Seg (width K)))) \ x), card (Seg (card (Seg (width A)))), the carrier of n
Segm (y,x,(Seg (card (Seg (width A))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card (Seg (card (Seg (width A)))), the carrier of n
Segm (y,(Sgm x),(Sgm (Seg (card (Seg (width A)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card x, card (Seg (card (Seg (width A)))), the carrier of n
(n,X,BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width X & width b₁ = width BX & X * b₁ = BX ) } is set
y is set
( the carrier of n,(len K),(width K),(width A),X,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,(width X) + (width BX), the carrier of n
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)) is finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,(len K),(width K),(width A),X,BX) ) } is set
Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Seg (card MV)),(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))), the carrier of n
card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)))) -tuples_on NAT
(card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))) } is set
Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Sgm (Seg (card MV))),(Sgm (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card MV)), card (Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))), the carrier of n
len (Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Seg (card MV)),(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom ( the carrier of n,(len K),(width K),(width A),X,BX) is finite Element of bool NAT
len ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len ( the carrier of n,(len K),(width K),(width A),X,BX)) is finite len ( the carrier of n,(len K),(width K),(width A),X,BX) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len ( the carrier of n,(len K),(width K),(width A),X,BX) ) } is set
Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Seg (len K)),(Seg (width X))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width X)), the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width X)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm (Seg (width X)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width X)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width X))) -tuples_on NAT
(card (Seg (width X))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width X)) } is set
Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Sgm (Seg (len K))),(Sgm (Seg (width X)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card (Seg (width X)), the carrier of n
the_rank_of (Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Seg (len K)),(Seg (width X)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (X,j) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K, width A, the carrier of n
Line (BX,j) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
(width BX) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width BX } is set
(width BX) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of n
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
(Seg (width BX)) --> (0. n) is Relation-like Seg (width BX) -defined Seg (width BX) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width BX)),{(0. n)}:]
[:(Seg (width BX)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width BX)),{(0. n)}:] is finite V37() set
Line (( the carrier of n,(len K),(width K),(width A),X,BX),j) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
(width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,(len K),(width K),(width A),X,BX) } is set
(Line (X,j)) ^ (Line (BX,j)) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width X) + (width BX) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(width X) + (width BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width ( the carrier of n,(len K),(width K),(width A),X,BX)) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),X,BX) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -tuples_on the carrier of n
(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))) --> (0. n) is Relation-like Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -defined Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX)) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))),{(0. n)}:]
[:(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))),{(0. n)}:] is finite V37() set
the_rank_of (Segm (( the carrier of n,(len K),(width K),(width A),X,BX),(Seg (card MV)),(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of ( the carrier of n,(len K),(width K),(width A),X,BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:(Seg (len K)),(Seg (width X)):] is Relation-like finite set
Indices ( the carrier of n,(len K),(width K),(width A),X,BX) is set
[:(dom ( the carrier of n,(len K),(width K),(width A),X,BX)),(Seg (width ( the carrier of n,(len K),(width K),(width A),X,BX))):] is Relation-like finite set
0. (n,(len K),(width K)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(len K) |-> ((width K) |-> (0. n)) is Relation-like NAT -defined (width K) -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on ((width K) -tuples_on the carrier of n)
(len K) -tuples_on ((width K) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (width K) -tuples_on the carrier of n
((width K) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (width K) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (width K) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width K) -tuples_on the carrier of n) * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> ((width K) |-> (0. n)) is Relation-like Seg (len K) -defined Seg (len K) -defined (width K) -tuples_on the carrier of n -valued {((width K) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{((width K) |-> (0. n))}:]
{((width K) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len K)),{((width K) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{((width K) |-> (0. n))}:] is finite V37() set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
(0. (n,(len K),(width K))) . c₁₃ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (X,c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width X -element FinSequence-like FinSubsequence-like Element of (width X) -tuples_on the carrier of n
(width X) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width X } is set
X . c₁₃ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (0. (n,(len K),(width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices ( the carrier of n,(len K),(width K),(width A),f,AB) is set
dom ( the carrier of n,(len K),(width K),(width A),f,AB) is finite Element of bool NAT
width ( the carrier of n,(len K),(width K),(width A),f,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ( the carrier of n,(len K),(width K),(width A),f,AB)) is finite width ( the carrier of n,(len K),(width K),(width A),f,AB) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,(len K),(width K),(width A),f,AB) ) } is set
[:(dom ( the carrier of n,(len K),(width K),(width A),f,AB)),(Seg (width ( the carrier of n,(len K),(width K),(width A),f,AB))):] is Relation-like finite set
(width K) + (width A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg ((width K) + (width A)) is finite (width K) + (width A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (width K) + (width A) ) } is set
[:(Seg (len K)),(Seg ((width K) + (width A))):] is Relation-like finite set
the Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K, width A, the carrier of n is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K, width A, the carrier of n
(Seg ((width K) + (width A))) \ (Seg (width K)) is finite without_zero Element of bool NAT
[:(Seg (len K)),((Seg ((width K) + (width A))) \ (Seg (width K))):] is Relation-like finite set
Indices (K ^^ A) is set
dom (K ^^ A) is finite Element of bool NAT
width (K ^^ A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (K ^^ A)) is finite width (K ^^ A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (K ^^ A) ) } is set
[:(dom (K ^^ A)),(Seg (width (K ^^ A))):] is Relation-like finite set
Segm (( the carrier of n,(len K),(width K),(width A),f,AB),(Seg (len K)),((Seg ((width K) + (width A))) \ (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg ((width K) + (width A))) \ (Seg (width K))), the carrier of n
card (Seg (len K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card ((Seg ((width K) + (width A))) \ (Seg (width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (len K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (len K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (len K))) -tuples_on NAT
(card (Seg (len K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (len K)) } is set
Sgm ((Seg ((width K) + (width A))) \ (Seg (width K))) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg ((width K) + (width A))) \ (Seg (width K))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg ((width K) + (width A))) \ (Seg (width K)))) -tuples_on NAT
(card ((Seg ((width K) + (width A))) \ (Seg (width K)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg ((width K) + (width A))) \ (Seg (width K))) } is set
Segm (( the carrier of n,(len K),(width K),(width A),f,AB),(Sgm (Seg (len K))),(Sgm ((Seg ((width K) + (width A))) \ (Seg (width K))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (len K)), card ((Seg ((width K) + (width A))) \ (Seg (width K))), the carrier of n
the_rank_of A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. (n,(len K),(width A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
(width A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
(Seg (width A)) --> (0. n) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. n)}:]
[:(Seg (width A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. n)}:] is finite V37() set
(len K) |-> ((width A) |-> (0. n)) is Relation-like NAT -defined (width A) -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on ((width A) -tuples_on the carrier of n)
(len K) -tuples_on ((width A) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (width A) -tuples_on the carrier of n
((width A) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (width A) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (width A) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width A) -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> ((width A) |-> (0. n)) is Relation-like Seg (len K) -defined Seg (len K) -defined (width A) -tuples_on the carrier of n -valued {((width A) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{((width A) |-> (0. n))}:]
{((width A) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len K)),{((width A) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{((width A) |-> (0. n))}:] is finite V37() set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K, width A, the carrier of n : verum } is set
Indices AB is set
dom AB is finite Element of bool NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
[:(dom AB),(Seg (width AB)):] is Relation-like finite set
Seg (width A) is finite width A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(Seg (len K)),(Seg (width A)):] is Relation-like finite set
Indices (0. (n,(len K),(width K))) is set
dom (0. (n,(len K),(width K))) is finite Element of bool NAT
width (0. (n,(len K),(width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (n,(len K),(width K)))) is finite width (0. (n,(len K),(width K))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (n,(len K),(width K))) ) } is set
[:(dom (0. (n,(len K),(width K)))),(Seg (width (0. (n,(len K),(width K))))):] is Relation-like finite set
[:(Seg (len K)),(Seg (width K)):] is Relation-like finite set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[c₁₃,x] is set
{c₁₃,x} is non empty finite V37() set
{c₁₃} is non empty trivial finite V37() 1 -element set
{{c₁₃,x},{c₁₃}} is non empty finite V37() without_zero V103() set
( the carrier of n,(len K),(width K),(width A),f,AB) * (c₁₃,x) is Element of the carrier of n
Line (( the carrier of n,(len K),(width K),(width A),f,AB),c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,(len K),(width K),(width A),f,AB) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,(len K),(width K),(width A),f,AB)) -tuples_on the carrier of n
(width ( the carrier of n,(len K),(width K),(width A),f,AB)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,(len K),(width K),(width A),f,AB) } is set
dom (Line (( the carrier of n,(len K),(width K),(width A),f,AB),c₁₃)) is finite width ( the carrier of n,(len K),(width K),(width A),f,AB) -element Element of bool NAT
Line (f,c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width f -element FinSequence-like FinSubsequence-like Element of (width f) -tuples_on the carrier of n
(width f) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width f } is set
len (Line (f,c₁₃)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (AB,c₁₃) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
(width AB) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width AB } is set
dom (Line (AB,c₁₃)) is finite width AB -element Element of bool NAT
dom (Line (f,c₁₃)) is finite width f -element Element of bool NAT
(Line (f,c₁₃)) ^ (Line (AB,c₁₃)) is Relation-like NAT -defined the carrier of n -valued Function-like finite (width f) + (width AB) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(width f) + (width AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Line (( the carrier of n,(len K),(width K),(width A),f,AB),c₁₃)) . x is set
(Line (f,c₁₃)) . x is set
f * (c₁₃,x) is Element of the carrier of n
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width K) + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width K) + x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[c₁₃,x] is set
{c₁₃,x} is non empty finite V37() set
{{c₁₃,x},{c₁₃}} is non empty finite V37() without_zero V103() set
Indices A is set
dom A is finite Element of bool NAT
[:(dom A),(Seg (width A)):] is Relation-like finite set
0. (n,(len K),(width A)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width A } is set
(width A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(Seg (width A)) --> (0. n) is Relation-like Seg (width A) -defined Seg (width A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width A)),{(0. n)}:]
[:(Seg (width A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width A)),{(0. n)}:] is finite V37() set
(len K) |-> ((width A) |-> (0. n)) is Relation-like NAT -defined (width A) -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on ((width A) -tuples_on the carrier of n)
(len K) -tuples_on ((width A) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (width A) -tuples_on the carrier of n
((width A) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (width A) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (width A) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width A) -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> ((width A) |-> (0. n)) is Relation-like Seg (len K) -defined Seg (len K) -defined (width A) -tuples_on the carrier of n -valued {((width A) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{((width A) |-> (0. n))}:]
{((width A) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len K)),{((width A) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{((width A) |-> (0. n))}:] is finite V37() set
(Line (( the carrier of n,(len K),(width K),(width A),f,AB),c₁₃)) . x is set
(Line (AB,c₁₃)) . x is set
AB * (c₁₃,x) is Element of the carrier of n
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,A)) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
B is set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
{} -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = {} } is set
0. n is zero Element of the carrier of n
{} |-> (0. n) is Relation-like non-empty empty-yielding NAT -defined the carrier of n -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of {} -tuples_on the carrier of n
Seg {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered without_zero V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
(Seg {}) --> (0. n) is Relation-like non-empty empty-yielding Seg {} -defined Seg {} -defined RAT -valued the carrier of n -valued {(0. n)} -valued Function-like one-to-one constant functional empty total total quasi_total V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of bool [:(Seg {}),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg {}),{(0. n)}:] is Relation-like finite set
bool [:(Seg {}),{(0. n)}:] is finite V37() set
AB is Relation-like non-empty empty-yielding NAT -defined the carrier of n -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of {} -tuples_on the carrier of n
( the carrier of n,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of n
len AB is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of n
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ( the carrier of n,AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Col (BA,1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on the carrier of n
(len BA) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len BA } is set
( the carrier of n,(Col (BA,1))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (Col (BA,1)),1, the carrier of n
len (Col (BA,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(Col (BA,1))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (Col (BA,1)), the carrier of n
[1,(Col (BA,1))] is set
{1,(Col (BA,1))} is non empty finite V37() set
{{1,(Col (BA,1))},{1}} is non empty finite V37() without_zero V103() set
{[1,(Col (BA,1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(Col (BA,1))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,B) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len B,1, the carrier of n
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*B*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len B, the carrier of n
[1,B] is set
{1,B} is non empty finite V37() set
{{1,B},{1}} is non empty finite V37() without_zero V103() set
{[1,B]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*B*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len ( the carrier of n,B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*BA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BA, the carrier of n
[1,BA] is set
{1,BA} is non empty finite V37() set
{{1,BA},{1}} is non empty finite V37() without_zero V103() set
{[1,BA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,K,A) is set
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,A)) } is set
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
B is set
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*BA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BA, the carrier of n
[1,BA] is set
{1,BA} is non empty finite V37() set
{{1,BA},{1}} is non empty finite V37() without_zero V103() set
{[1,BA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. n is zero Element of the carrier of n
A |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite A -element FinSequence-like FinSubsequence-like Element of A -tuples_on the carrier of n
A -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = A } is set
Seg A is finite A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
(Seg A) --> (0. n) is Relation-like Seg A -defined Seg A -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg A),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg A),{(0. n)}:] is Relation-like finite set
bool [:(Seg A),{(0. n)}:] is finite V37() set
(n,K,(A |-> (0. n))) is Element of bool the carrier of ((width K) -VectSp_over n)
( the carrier of n,(A |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (A |-> (0. n)),1, the carrier of n
len (A |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(A |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (A |-> (0. n)), the carrier of n
[1,(A |-> (0. n))] is set
{1,(A |-> (0. n))} is non empty finite V37() set
{{1,(A |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,(A |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(A |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,(A |-> (0. n)))) is set
width ( the carrier of n,(A |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,(A |-> (0. n))) & K * b₁ = ( the carrier of n,(A |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,(A |-> (0. n)))) } is set
0. (n,A,1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
A |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite A -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of A -tuples_on (1 -tuples_on the carrier of n)
A -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = A } is set
(Seg A) --> (1 |-> (0. n)) is Relation-like non-empty Seg A -defined Seg A -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg A),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg A),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg A),{(1 |-> (0. n))}:] is finite V37() set
f is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
X is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,X) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len X,1, the carrier of n
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*X*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len X, the carrier of n
[1,X] is set
{1,X} is non empty finite V37() set
{{1,X},{1}} is non empty finite V37() without_zero V103() set
{[1,X]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*X*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
x is Element of the carrier of n
x * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
len (x * (0. (n,A,1))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (0. (n,A,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (0. (n,A,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (0. (n,A,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Element of the carrier of n
x * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
(0. n) * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
x * ((0. n) * (0. (n,A,1))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
x * (0. n) is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( 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
K560( the carrier of n, the multF of n,x,(0. n)) is Element of the carrier of n
(x * (0. n)) * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
x is Element of the carrier of n
x * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
x is Element of the carrier of n
x * (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of A,1, the carrier of n
BX is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of n,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BX,1, the carrier of n
<*BX*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BX, the carrier of n
[1,BX] is set
{1,BX} is non empty finite V37() set
{{1,BX},{1}} is non empty finite V37() without_zero V103() set
{[1,BX]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BX*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x * ( the carrier of n,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BX,1, the carrier of n
(n,K,(0. (n,A,1))) is set
width (0. (n,A,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width (0. (n,A,1)) & K * b₁ = 0. (n,A,1) ) } is set
x * BX is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
x multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (x,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,BX,(x multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(x * BX)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (x * BX),1, the carrier of n
len (x * BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(x * BX)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (x * BX), the carrier of n
[1,(x * BX)] is set
{1,(x * BX)} is non empty finite V37() set
{{1,(x * BX)},{1}} is non empty finite V37() without_zero V103() set
{[1,(x * BX)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(x * BX)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x * f is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
x is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
f is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
X is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,X) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len X,1, the carrier of n
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*X*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len X, the carrier of n
[1,X] is set
{1,X} is non empty finite V37() set
{{1,X},{1}} is non empty finite V37() without_zero V103() set
{[1,X]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*X*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BX is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,BX) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BX,1, the carrier of n
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*BX*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BX, the carrier of n
[1,BX] is set
{1,BX} is non empty finite V37() set
{{1,BX},{1}} is non empty finite V37() without_zero V103() set
{[1,BX]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BX*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
MV is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
( the carrier of n,MV) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len MV,1, the carrier of n
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*MV*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len MV, the carrier of n
[1,MV] is set
{1,MV} is non empty finite V37() set
{{1,MV},{1}} is non empty finite V37() without_zero V103() set
{[1,MV]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*MV*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
lA is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
( the carrier of n,lA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len lA,1, the carrier of n
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*lA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len lA, the carrier of n
[1,lA] is set
{1,lA} is non empty finite V37() set
{{1,lA},{1}} is non empty finite V37() without_zero V103() set
{[1,lA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*lA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,MV) + ( the carrier of n,lA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(0. (n,A,1)) + (0. (n,A,1)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,((0. (n,A,1)) + (0. (n,A,1)))) is set
width ((0. (n,A,1)) + (0. (n,A,1))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ((0. (n,A,1)) + (0. (n,A,1))) & K * b₁ = (0. (n,A,1)) + (0. (n,A,1)) ) } is set
MV + lA is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,MV,lA) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(MV + lA)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (MV + lA),1, the carrier of n
len (MV + lA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(MV + lA)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (MV + lA), the carrier of n
[1,(MV + lA)] is set
{1,(MV + lA)} is non empty finite V37() set
{{1,(MV + lA)},{1}} is non empty finite V37() without_zero V103() set
{[1,(MV + lA)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(MV + lA)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x + f is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
the addF of ((width K) -VectSp_over n) is Relation-like [: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] -defined the carrier of ((width K) -VectSp_over n) -valued Function-like total quasi_total V223( the carrier of ((width K) -VectSp_over n)) V224( the carrier of ((width K) -VectSp_over n)) Element of bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):]
[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] is Relation-like set
[:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is Relation-like set
bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is set
K560( the carrier of ((width K) -VectSp_over n), the addF of ((width K) -VectSp_over n),x,f) is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
x is Element of bool the carrier of ((width K) -VectSp_over n)
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,K,A) is Element of bool the carrier of ((width K) -VectSp_over n)
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,A)) } is set
BA is set
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of n
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width (K * x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
B is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(n,B,( the carrier of n,((len K) |-> (0. n)))) is set
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width B & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & B * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*BA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BA, the carrier of n
[1,BA] is set
{1,BA} is non empty finite V37() set
{{1,BA},{1}} is non empty finite V37() without_zero V103() set
{[1,BA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
len ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K ^^ ( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K ^^ ( the carrier of n,((len K) |-> (0. n)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA is set
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of n
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
0. n is zero Element of the carrier of n
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
the carrier of A is non empty set
B is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
the carrier of B is non empty set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
(n,K) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,K,A) is Element of bool the carrier of ((width K) -VectSp_over n)
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
( the carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*A*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len A, the carrier of n
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() without_zero V103() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*A*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,A)) is set
width ( the carrier of n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,A) & K * b₁ = ( the carrier of n,A) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,A)) } is set
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len A } is set
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*BA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BA, the carrier of n
[1,BA] is set
{1,BA} is non empty finite V37() set
{{1,BA},{1}} is non empty finite V37() without_zero V103() set
{[1,BA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,BA) is Element of bool the carrier of ((width K) -VectSp_over n)
(n,K,( the carrier of n,BA)) is set
width ( the carrier of n,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,BA) & K * b₁ = ( the carrier of n,BA) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,BA)) } is set
x is set
f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len f,1, the carrier of n
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*f*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len f, the carrier of n
[1,f] is set
{1,f} is non empty finite V37() set
{{1,f},{1}} is non empty finite V37() without_zero V103() set
{[1,f]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*f*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
BX is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
len ( the carrier of n,BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the carrier of (n,K) is non empty set
0. n is zero Element of the carrier of n
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
MV is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
MV + (n,K) is Element of bool the carrier of ((width K) -VectSp_over n)
{ (MV + b₁) where b₁ is Element of the carrier of ((width K) -VectSp_over n) : b₁ in (n,K) } is set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
- (1_ n) is Element of the carrier of n
(- (1_ n)) * BA is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(- (1_ n)) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] ((- (1_ n)),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,BA,((- (1_ n)) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len ((- (1_ n)) * BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(- (1_ n)) * ( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
( the carrier of n,((- (1_ n)) * BA)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((- (1_ n)) * BA),1, the carrier of n
<*((- (1_ n)) * BA)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((- (1_ n)) * BA), the carrier of n
[1,((- (1_ n)) * BA)] is set
{1,((- (1_ n)) * BA)} is non empty finite V37() set
{{1,((- (1_ n)) * BA)},{1}} is non empty finite V37() without_zero V103() set
{[1,((- (1_ n)) * BA)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((- (1_ n)) * BA)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,BA) + ((- (1_ n)) * ( the carrier of n,BA)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BA + ((- (1_ n)) * BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BA,((- (1_ n)) * BA)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(BA + ((- (1_ n)) * BA))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (BA + ((- (1_ n)) * BA)),1, the carrier of n
len (BA + ((- (1_ n)) * BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(BA + ((- (1_ n)) * BA))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (BA + ((- (1_ n)) * BA)), the carrier of n
[1,(BA + ((- (1_ n)) * BA))] is set
{1,(BA + ((- (1_ n)) * BA))} is non empty finite V37() set
{{1,(BA + ((- (1_ n)) * BA))},{1}} is non empty finite V37() without_zero V103() set
{[1,(BA + ((- (1_ n)) * BA))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(BA + ((- (1_ n)) * BA))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
- BA is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
comp n is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,BA,(comp n)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
BA + (- BA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BA,(- BA)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(BA + (- BA))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (BA + (- BA)),1, the carrier of n
len (BA + (- BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(BA + (- BA))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (BA + (- BA)), the carrier of n
[1,(BA + (- BA))] is set
{1,(BA + (- BA))} is non empty finite V37() set
{{1,(BA + (- BA))},{1}} is non empty finite V37() without_zero V103() set
{[1,(BA + (- BA))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(BA + (- BA))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
lA is set
c₁₃ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,c₁₃) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len c₁₃,1, the carrier of n
len c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*c₁₃*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len c₁₃, the carrier of n
[1,c₁₃] is set
{1,c₁₃} is non empty finite V37() set
{{1,c₁₃},{1}} is non empty finite V37() without_zero V103() set
{[1,c₁₃]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*c₁₃*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
( the carrier of n,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len x,1, the carrier of n
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*x*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len x, the carrier of n
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() without_zero V103() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*x*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(- (1_ n)) * BX is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,BX,((- (1_ n)) multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
x + ((- (1_ n)) * BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,x,((- (1_ n)) * BX)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len ((- (1_ n)) * BX) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width ((- (1_ n)) * ( the carrier of n,BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(- (1_ n)) * ( the carrier of n,f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len f,1, the carrier of n
(n,K,((- (1_ n)) * ( the carrier of n,BA))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ((- (1_ n)) * ( the carrier of n,BA)) & K * b₁ = (- (1_ n)) * ( the carrier of n,BA) ) } is set
( the carrier of n,x) + ((- (1_ n)) * ( the carrier of n,f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,((- (1_ n)) * BX)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((- (1_ n)) * BX),1, the carrier of n
<*((- (1_ n)) * BX)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((- (1_ n)) * BX), the carrier of n
[1,((- (1_ n)) * BX)] is set
{1,((- (1_ n)) * BX)} is non empty finite V37() set
{{1,((- (1_ n)) * BX)},{1}} is non empty finite V37() without_zero V103() set
{[1,((- (1_ n)) * BX)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((- (1_ n)) * BX)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,x) + ( the carrier of n,((- (1_ n)) * BX)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,(x + ((- (1_ n)) * BX))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (x + ((- (1_ n)) * BX)),1, the carrier of n
len (x + ((- (1_ n)) * BX)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(x + ((- (1_ n)) * BX))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (x + ((- (1_ n)) * BX)), the carrier of n
[1,(x + ((- (1_ n)) * BX))] is set
{1,(x + ((- (1_ n)) * BX))} is non empty finite V37() set
{{1,(x + ((- (1_ n)) * BX))},{1}} is non empty finite V37() without_zero V103() set
{[1,(x + ((- (1_ n)) * BX))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(x + ((- (1_ n)) * BX))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
y is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
BX + (x + ((- (1_ n)) * BX)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BX,(x + ((- (1_ n)) * BX))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((- (1_ n)) * BX) + x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,((- (1_ n)) * BX),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
BX + (((- (1_ n)) * BX) + x) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BX,(((- (1_ n)) * BX) + x)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
- BX is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K501( the carrier of n, the carrier of n,BX,(comp n)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(- BX) + x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(- BX),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
BX + ((- BX) + x) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BX,((- BX) + x)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
BX + (- BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BX,(- BX)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(BX + (- BX)) + x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(BX + (- BX)),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
((width K) |-> (0. n)) + x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,((width K) |-> (0. n)),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
MV + y is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
the addF of ((width K) -VectSp_over n) is Relation-like [: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] -defined the carrier of ((width K) -VectSp_over n) -valued Function-like total quasi_total V223( the carrier of ((width K) -VectSp_over n)) V224( the carrier of ((width K) -VectSp_over n)) Element of bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):]
[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] is Relation-like set
[:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is Relation-like set
bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is set
K560( the carrier of ((width K) -VectSp_over n), the addF of ((width K) -VectSp_over n),MV,y) is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
lA is set
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
( the carrier of n,BA) + ( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
c₁₃ is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
MV + c₁₃ is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
the addF of ((width K) -VectSp_over n) is Relation-like [: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] -defined the carrier of ((width K) -VectSp_over n) -valued Function-like total quasi_total V223( the carrier of ((width K) -VectSp_over n)) V224( the carrier of ((width K) -VectSp_over n)) Element of bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):]
[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):] is Relation-like set
[:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is Relation-like set
bool [:[: the carrier of ((width K) -VectSp_over n), the carrier of ((width K) -VectSp_over n):], the carrier of ((width K) -VectSp_over n):] is set
K560( the carrier of ((width K) -VectSp_over n), the addF of ((width K) -VectSp_over n),MV,c₁₃) is Relation-like Function-like Element of the carrier of ((width K) -VectSp_over n)
x is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len x,1, the carrier of n
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*x*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len x, the carrier of n
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() without_zero V103() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*x*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BX + x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,BX,x) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,(BX + x)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (BX + x),1, the carrier of n
len (BX + x) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(BX + x)*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (BX + x), the carrier of n
[1,(BX + x)] is set
{1,(BX + x)} is non empty finite V37() set
{{1,(BX + x)},{1}} is non empty finite V37() without_zero V103() set
{[1,(BX + x)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(BX + x)*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
( the carrier of n,f) + ( the carrier of n,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,K) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
0. n is zero Element of the carrier of n
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
the carrier of ((width K) -VectSp_over n) is non empty set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
bool the carrier of ((width K) -VectSp_over n) is set
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
B is set
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
0. (n,(len K),(width K)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
(len K) |-> ((width K) |-> (0. n)) is Relation-like NAT -defined (width K) -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on ((width K) -tuples_on the carrier of n)
(len K) -tuples_on ((width K) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (width K) -tuples_on the carrier of n
((width K) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (width K) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (width K) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width K) -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> ((width K) |-> (0. n)) is Relation-like Seg (len K) -defined Seg (len K) -defined (width K) -tuples_on the carrier of n -valued {((width K) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{((width K) |-> (0. n))}:]
{((width K) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len K)),{((width K) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{((width K) |-> (0. n))}:] is finite V37() set
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of n
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
{} -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = {} } is set
<*> the carrier of n is Relation-like non-empty empty-yielding NAT -defined the carrier of n -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V105() Function-yielding V147() ext-real non positive non negative complex V185() V186() V187() V188() Element of the carrier of n *
{(<*> the carrier of n)} is functional non empty trivial finite V37() 1 -element Element of bool NAT
BA is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K,1, the carrier of n : verum } is set
( the carrier of n,BA) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of n
<*BA*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len BA, the carrier of n
[1,BA] is set
{1,BA} is non empty finite V37() set
{{1,BA},{1}} is non empty finite V37() without_zero V103() set
{[1,BA]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*BA*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the carrier of (n,K) is non empty set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,K) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices K is set
dom K is finite Element of bool NAT
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
[:(dom K),(Seg (width K)):] is Relation-like finite set
0. n is zero Element of the carrier of n
A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[A,B] is set
{A,B} is non empty finite V37() set
{A} is non empty trivial finite V37() 1 -element set
{{A,B},{A}} is non empty finite V37() without_zero V103() set
K * (A,B) is Element of the carrier of n
1. (n,(width K)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of width K, width K, the carrier of n
Line ((1. (n,(width K))),B) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (1. (n,(width K))) -element FinSequence-like FinSubsequence-like Element of (width (1. (n,(width K)))) -tuples_on the carrier of n
width (1. (n,(width K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (1. (n,(width K)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width (1. (n,(width K))) } is set
dom (Line ((1. (n,(width K))),B)) is finite width (1. (n,(width K))) -element Element of bool NAT
Indices (1. (n,(width K))) is set
dom (1. (n,(width K))) is finite Element of bool NAT
Seg (width (1. (n,(width K)))) is finite width (1. (n,(width K))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (n,(width K))) ) } is set
[:(dom (1. (n,(width K)))),(Seg (width (1. (n,(width K))))):] is Relation-like finite set
[:(Seg (width K)),(Seg (width K)):] is Relation-like finite set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[B,AB] is set
{B,AB} is non empty finite V37() set
{B} is non empty trivial finite V37() 1 -element set
{{B,AB},{B}} is non empty finite V37() without_zero V103() set
(1. (n,(width K))) * (B,AB) is Element of the carrier of n
(Line ((1. (n,(width K))),B)) . AB is set
Line (K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
dom (Line (K,A)) is finite width K -element Element of bool NAT
[B,B] is set
{B,B} is non empty finite V37() set
{{B,B},{B}} is non empty finite V37() without_zero V103() set
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
(1. (n,(width K))) * (B,B) is Element of the carrier of n
(Line ((1. (n,(width K))),B)) . B is set
mlt ((Line ((1. (n,(width K))),B)),(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( 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
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line ((1. (n,(width K))),B)),(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line ((1. (n,(width K))),B)),(Line (K,A)))) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the addF of n $$ (mlt ((Line ((1. (n,(width K))),B)),(Line (K,A)))) is Element of the carrier of n
(Line (K,A)) . B is set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
the carrier of (n,K) is non empty set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,AB) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of n
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
Indices ( the carrier of n,((len K) |-> (0. n))) is set
dom ( the carrier of n,((len K) |-> (0. n))) is finite Element of bool NAT
Seg (width ( the carrier of n,((len K) |-> (0. n)))) is finite width ( the carrier of n,((len K) |-> (0. n))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,((len K) |-> (0. n))) ) } is set
[:(dom ( the carrier of n,((len K) |-> (0. n)))),(Seg (width ( the carrier of n,((len K) |-> (0. n))))):] is Relation-like finite set
[:(Seg (len K)),(Seg 1):] is Relation-like finite set
[A,1] is set
{A,1} is non empty finite V37() set
{{A,1},{A}} is non empty finite V37() without_zero V103() set
Col (x,1) is Relation-like NAT -defined the carrier of n -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of n
(len x) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len x } is set
(Line (K,A)) "*" (Col (x,1)) is Element of the carrier of n
mlt ((Line (K,A)),(Col (x,1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Line (K,A)),(Col (x,1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Line (K,A)),(Col (x,1)))) is Element of the carrier of n
the addF of n $$ (mlt ((Line (K,A)),(Col (x,1)))) is Element of the carrier of n
(0. (n,(len K),1)) * (A,1) is Element of the carrier of n
(Col (x,1)) "*" (Line (K,A)) is Element of the carrier of n
mlt ((Col (x,1)),(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,(Col (x,1)),(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((Col (x,1)),(Line (K,A)))) is Element of the carrier of n
the addF of n $$ (mlt ((Col (x,1)),(Line (K,A)))) is Element of the carrier of n
mlt (AB,(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
K498( the carrier of n, the carrier of n, the carrier of n, the multF of n,AB,(Line (K,A))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt (AB,(Line (K,A)))) is Element of the carrier of n
the addF of n $$ (mlt (AB,(Line (K,A)))) is Element of the carrier of n
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
1_ A is Element of the carrier of A
1. A is non zero Element of the carrier of A
- (1_ A) is Element of the carrier of A
B is Element of the carrier of A
BA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
(A,BA) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width BA) -VectSp_over A
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) -VectSp_over A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over A
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (BA,AB) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of A
(width BA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width BA } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (BA,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of A
B * (Line (BA,x)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of A
B multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
bool [: the carrier of A, the carrier of A:] is set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] (B,(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (BA,x)),(B multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Line (BA,AB)) + (B * (Line (BA,x))) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BA -element FinSequence-like FinSubsequence-like Element of (width BA) -tuples_on the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
K498( the carrier of A, the carrier of A, the carrier of A, the addF of A,(Line (BA,AB)),(B * (Line (BA,x)))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
(A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) -VectSp_over A
width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) -VectSp_over A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over A
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. A is zero Element of the carrier of A
(len BA) |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on the carrier of A
(len BA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len BA } is set
Seg (len BA) is finite len BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BA ) } is set
(Seg (len BA)) --> (0. A) is Relation-like Seg (len BA) -defined Seg (len BA) -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len BA)),{(0. A)}:]
{(0. A)} is non empty trivial finite 1 -element set
[:(Seg (len BA)),{(0. A)}:] is Relation-like finite set
bool [:(Seg (len BA)),{(0. A)}:] is finite V37() set
len ((len BA) |-> (0. A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
( the carrier of A,((len BA) |-> (0. A))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len BA) |-> (0. A)),1, the carrier of A
<*((len BA) |-> (0. A))*> is Relation-like NAT -defined the carrier of A * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len BA) |-> (0. A)), the carrier of A
[1,((len BA) |-> (0. A))] is set
{1,((len BA) |-> (0. A))} is non empty finite V37() set
{{1,((len BA) |-> (0. A))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len BA) |-> (0. A))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len BA) |-> (0. A))*> @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
BX is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,1, the carrier of A
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (BX,AB) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of A
(width BX) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width BX } is set
Line (BX,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of A
B * (Line (BX,x)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of A
K501( the carrier of A, the carrier of A,(Line (BX,x)),(B multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Line (BX,AB)) + (B * (Line (BX,x))) is Relation-like NAT -defined the carrier of A -valued Function-like finite width BX -element FinSequence-like FinSubsequence-like Element of (width BX) -tuples_on the carrier of A
K498( the carrier of A, the carrier of A, the carrier of A, the addF of A,(Line (BX,AB)),(B * (Line (BX,x)))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x))))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,1, the carrier of A
0. (A,(len BA),1) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len BA,1, the carrier of A
1 -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = 1 } is set
1 |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of A
(Seg 1) --> (0. A) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of A -valued {(0. A)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. A)}:]
[:(Seg 1),{(0. A)}:] is Relation-like finite set
bool [:(Seg 1),{(0. A)}:] is finite V37() set
(len BA) |-> (1 |-> (0. A)) is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len BA) -tuples_on (1 -tuples_on the carrier of A)
(len BA) -tuples_on (1 -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
(1 -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of A) * : len b₁ = len BA } is set
(Seg (len BA)) --> (1 |-> (0. A)) is Relation-like non-empty Seg (len BA) -defined Seg (len BA) -defined 1 -tuples_on the carrier of A -valued {(1 |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len BA)),{(1 |-> (0. A))}:]
{(1 |-> (0. A))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len BA)),{(1 |-> (0. A))}:] is Relation-like finite set
bool [:(Seg (len BA)),{(1 |-> (0. A))}:] is finite V37() set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[lA,c₁₃] is set
{lA,c₁₃} is non empty finite V37() set
{lA} is non empty trivial finite V37() 1 -element set
{{lA,c₁₃},{lA}} is non empty finite V37() without_zero V103() set
Indices BX is set
dom BX is finite Element of bool NAT
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
[:(dom BX),(Seg (width BX)):] is Relation-like finite set
len ((Line (BX,AB)) + (B * (Line (BX,x)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Line (BX,x)) . c₁₃ is set
BX * (x,c₁₃) is Element of the carrier of A
(B * (Line (BX,x))) . c₁₃ is set
B * (BX * (x,c₁₃)) is Element of the carrier of A
K560( the carrier of A, the multF of A,B,(BX * (x,c₁₃))) is Element of the carrier of A
[:(Seg n),(Seg 1):] is Relation-like finite set
[x,c₁₃] is set
{x,c₁₃} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,c₁₃},{x}} is non empty finite V37() without_zero V103() set
(Line (BX,AB)) . c₁₃ is set
BX * (AB,c₁₃) is Element of the carrier of A
((Line (BX,AB)) + (B * (Line (BX,x)))) . c₁₃ is set
(BX * (AB,c₁₃)) + (B * (BX * (x,c₁₃))) is Element of the carrier of A
K560( the carrier of A, the addF of A,(BX * (AB,c₁₃)),(B * (BX * (x,c₁₃)))) is Element of the carrier of A
(0. A) + (B * (BX * (x,c₁₃))) is Element of the carrier of A
K560( the carrier of A, the addF of A,(0. A),(B * (BX * (x,c₁₃)))) is Element of the carrier of A
B * (0. A) is Element of the carrier of A
K560( the carrier of A, the multF of A,B,(0. A)) is Element of the carrier of A
(0. A) + (B * (0. A)) is Element of the carrier of A
K560( the carrier of A, the addF of A,(0. A),(B * (0. A))) is Element of the carrier of A
(0. A) + (0. A) is Element of the carrier of A
K560( the carrier of A, the addF of A,(0. A),(0. A)) is Element of the carrier of A
BX * (lA,c₁₃) is Element of the carrier of A
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX * (x,x) is Element of the carrier of A
(ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x)))))) * (x,x) is Element of the carrier of A
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX * (x,x) is Element of the carrier of A
(ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x)))))) * (x,x) is Element of the carrier of A
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX * (x,x) is Element of the carrier of A
(ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x)))))) * (x,x) is Element of the carrier of A
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BX * (x,x) is Element of the carrier of A
(ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x)))))) * (x,x) is Element of the carrier of A
BX * (lA,c₁₃) is Element of the carrier of A
(ReplaceLine (BX,AB,((Line (BX,AB)) + (B * (Line (BX,x)))))) * (lA,c₁₃) is Element of the carrier of A
(A,BA,BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width BA & width b₁ = width BX & BA * b₁ = BX ) } is set
(A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))),BX) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) & width b₁ = width BX & (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) * b₁ = BX ) } is set
len (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the carrier of (A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) is non empty set
(A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))),((len BA) |-> (0. A))) is linearly-closed Element of bool the carrier of ((width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) -VectSp_over A)
the carrier of ((width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) -VectSp_over A) is non empty set
bool the carrier of ((width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x))))))) -VectSp_over A) is set
(A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))),( the carrier of A,((len BA) |-> (0. A)))) is set
width ( the carrier of A,((len BA) |-> (0. A))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) & width b₁ = width ( the carrier of A,((len BA) |-> (0. A))) & (ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))) * b₁ = ( the carrier of A,((len BA) |-> (0. A))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A : ( the carrier of A,b₁) in (A,(ReplaceLine (BA,AB,((Line (BA,AB)) + (B * (Line (BA,x)))))),( the carrier of A,((len BA) |-> (0. A)))) } is set
the carrier of (A,BA) is non empty set
(A,BA,((len BA) |-> (0. A))) is linearly-closed Element of bool the carrier of ((width BA) -VectSp_over A)
the carrier of ((width BA) -VectSp_over A) is non empty set
bool the carrier of ((width BA) -VectSp_over A) is set
(A,BA,( the carrier of A,((len BA) |-> (0. A)))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width BA & width b₁ = width ( the carrier of A,((len BA) |-> (0. A))) & BA * b₁ = ( the carrier of A,((len BA) |-> (0. A))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A : ( the carrier of A,b₁) in (A,BA,( the carrier of A,((len BA) |-> (0. A)))) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is zero Element of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
dom K is finite Element of bool NAT
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
(n,K) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
A is finite without_zero Element of bool NAT
(dom K) \ A is finite Element of bool NAT
Segm (K,A,(Seg (width K))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width K)), the carrier of n
card A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (width K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm A is Relation-like NAT -defined NAT -valued Function-like finite card A -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card A) -tuples_on NAT
(card A) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card A } is set
Sgm (Seg (width K)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width K)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width K))) -tuples_on NAT
(card (Seg (width K))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width K)) } is set
Segm (K,(Sgm A),(Sgm (Seg (width K)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width K)), the carrier of n
(n,(Segm (K,A,(Seg (width K))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (Segm (K,A,(Seg (width K))))) -VectSp_over n
width (Segm (K,A,(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (Segm (K,A,(Seg (width K))))) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (0. n) is Relation-like Seg (len K) -defined Seg (len K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len K)),{(0. n)}:]
[:(Seg (len K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len K)),{(0. n)}:] is finite V37() set
( the carrier of n,((len K) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len K) |-> (0. n)),1, the carrier of n
len ((len K) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len K) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len K) |-> (0. n)), the carrier of n
[1,((len K) |-> (0. n))] is set
{1,((len K) |-> (0. n))} is non empty finite V37() set
{{1,((len K) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len K) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len K) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (Segm (K,A,(Seg (width K)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the carrier of (n,(Segm (K,A,(Seg (width K))))) is non empty set
(card A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite card A -element FinSequence-like FinSubsequence-like Element of (card A) -tuples_on the carrier of n
(card A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = card A } is set
Seg (card A) is finite card A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card A ) } is set
(Seg (card A)) --> (0. n) is Relation-like Seg (card A) -defined Seg (card A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (card A)),{(0. n)}:]
[:(Seg (card A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (card A)),{(0. n)}:] is finite V37() set
(n,(Segm (K,A,(Seg (width K)))),((card A) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width (Segm (K,A,(Seg (width K))))) -VectSp_over n)
the carrier of ((width (Segm (K,A,(Seg (width K))))) -VectSp_over n) is non empty set
bool the carrier of ((width (Segm (K,A,(Seg (width K))))) -VectSp_over n) is set
( the carrier of n,((card A) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((card A) |-> (0. n)),1, the carrier of n
len ((card A) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((card A) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((card A) |-> (0. n)), the carrier of n
[1,((card A) |-> (0. n))] is set
{1,((card A) |-> (0. n))} is non empty finite V37() set
{{1,((card A) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((card A) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((card A) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,(Segm (K,A,(Seg (width K)))),( the carrier of n,((card A) |-> (0. n)))) is set
width ( the carrier of n,((card A) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (K,A,(Seg (width K)))) & width b₁ = width ( the carrier of n,((card A) |-> (0. n))) & (Segm (K,A,(Seg (width K)))) * b₁ = ( the carrier of n,((card A) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,(Segm (K,A,(Seg (width K)))),( the carrier of n,((card A) |-> (0. n)))) } is set
the carrier of (n,K) is non empty set
(n,K,((len K) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width K) -VectSp_over n)
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
(n,K,( the carrier of n,((len K) |-> (0. n)))) is set
width ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width K & width b₁ = width ( the carrier of n,((len K) |-> (0. n))) & K * b₁ = ( the carrier of n,((len K) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,K,( the carrier of n,((len K) |-> (0. n)))) } is set
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
len ( the carrier of n,((len K) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom ( the carrier of n,((len K) |-> (0. n))) is finite Element of bool NAT
Seg (width ( the carrier of n,((len K) |-> (0. n)))) is finite width ( the carrier of n,((len K) |-> (0. n))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of n,((len K) |-> (0. n))) ) } is set
card (Seg (width ( the carrier of n,((len K) |-> (0. n))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg 1) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg 1) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg 1)) -tuples_on NAT
card (Seg 1) is non empty V26() V27() V28() V32() finite cardinal V105() ext-real positive non negative complex Element of NAT
(card (Seg 1)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg 1) } is set
rng (Sgm (Seg 1)) is finite V195() V196() V197() V200() set
Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n)))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width ( the carrier of n,((len K) |-> (0. n))))), the carrier of n
Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n))))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width ( the carrier of n,((len K) |-> (0. n))))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) -tuples_on NAT
(card (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width ( the carrier of n,((len K) |-> (0. n))))) } is set
Segm (( the carrier of n,((len K) |-> (0. n))),(Sgm A),(Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n))))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A, card (Seg (width ( the carrier of n,((len K) |-> (0. n))))), the carrier of n
Indices (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) is set
dom (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) is finite Element of bool NAT
width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n)))))))) is finite width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) ) } is set
[:(dom (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n)))))))),(Seg (width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))))):] is Relation-like finite set
0. (n,(card A),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card A,1, the carrier of n
(card A) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite card A -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (card A) -tuples_on (1 -tuples_on the carrier of n)
(card A) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = card A } is set
(Seg (card A)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (card A) -defined Seg (card A) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (card A)),{(1 |-> (0. n))}:]
[:(Seg (card A)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (card A)),{(1 |-> (0. n))}:] is finite V37() set
Indices (0. (n,(card A),1)) is set
dom (0. (n,(card A),1)) is finite Element of bool NAT
width (0. (n,(card A),1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (n,(card A),1))) is finite width (0. (n,(card A),1)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (n,(card A),1)) ) } is set
[:(dom (0. (n,(card A),1))),(Seg (width (0. (n,(card A),1)))):] is Relation-like finite set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[X,BX] is set
{X,BX} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,BX},{X}} is non empty finite V37() without_zero V103() set
[:A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))):] is Relation-like finite set
Indices ( the carrier of n,((len K) |-> (0. n))) is set
[:(dom ( the carrier of n,((len K) |-> (0. n)))),(Seg (width ( the carrier of n,((len K) |-> (0. n))))):] is Relation-like finite set
rng (Sgm A) is finite V195() V196() V197() V200() set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm A) . MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) . lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[((Sgm A) . MV),((Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) . lA)] is set
{((Sgm A) . MV),((Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) . lA)} is non empty finite V37() set
{((Sgm A) . MV)} is non empty trivial finite V37() 1 -element set
{{((Sgm A) . MV),((Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) . lA)},{((Sgm A) . MV)}} is non empty finite V37() without_zero V103() set
(Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) * (X,BX) is Element of the carrier of n
( the carrier of n,((len K) |-> (0. n))) * (((Sgm A) . MV),((Sgm (Seg (width ( the carrier of n,((len K) |-> (0. n)))))) . lA)) is Element of the carrier of n
(0. (n,(card A),1)) * (X,BX) is Element of the carrier of n
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (( the carrier of n,((len K) |-> (0. n))),X) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,((len K) |-> (0. n))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n
(width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width ( the carrier of n,((len K) |-> (0. n))) } is set
( the carrier of n,((len K) |-> (0. n))) . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width ( the carrier of n,((len K) |-> (0. n))) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n
(Seg (width ( the carrier of n,((len K) |-> (0. n))))) --> (0. n) is Relation-like Seg (width ( the carrier of n,((len K) |-> (0. n)))) -defined Seg (width ( the carrier of n,((len K) |-> (0. n)))) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width ( the carrier of n,((len K) |-> (0. n))))),{(0. n)}:]
[:(Seg (width ( the carrier of n,((len K) |-> (0. n))))),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width ( the carrier of n,((len K) |-> (0. n))))),{(0. n)}:] is finite V37() set
(len K) |-> ((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n)) is Relation-like NAT -defined (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on ((width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n)
(len K) -tuples_on ((width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n
((width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of ((width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n) * : len b₁ = len K } is set
(Seg (len K)) --> ((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n)) is Relation-like Seg (len K) -defined Seg (len K) -defined (width ( the carrier of n,((len K) |-> (0. n)))) -tuples_on the carrier of n -valued {((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))}:]
{((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (len K)),{((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))}:] is finite V37() set
((len K) |-> ((width ( the carrier of n,((len K) |-> (0. n)))) |-> (0. n))) . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (K,X) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(n,(Segm (K,A,(Seg (width K)))),(Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n)))))))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width (Segm (K,A,(Seg (width K)))) & width b₁ = width (Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n))))))) & (Segm (K,A,(Seg (width K)))) * b₁ = Segm (( the carrier of n,((len K) |-> (0. n))),A,(Seg (width ( the carrier of n,((len K) |-> (0. n)))))) ) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom K is finite Element of bool NAT
Sum K is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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 addF of n $$ K is Element of the carrier of n
rng K is finite set
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
idseq (len K) is Relation-like NAT -defined Function-like finite len K -element FinSequence-like FinSubsequence-like set
id (Seg (len K)) is Relation-like Seg (len K) -defined Seg (len K) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (len K)),(Seg (len K)):]
[:(Seg (len K)),(Seg (len K)):] is Relation-like finite set
bool [:(Seg (len K)),(Seg (len K)):] is finite V37() set
rng (idseq (len K)) is finite set
dom (idseq (len K)) is finite len K -element Element of bool NAT
B is set
Sgm B is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
K * (Sgm B) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of n:] is non empty non trivial non finite V103() set
(dom K) \ B is finite Element of bool NAT
Sgm ((dom K) \ B) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
K * (Sgm ((dom K) \ B)) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
rng (Sgm B) is finite V195() V196() V197() V200() set
(idseq (len K)) " B is finite set
rng (Sgm ((dom K) \ B)) is finite V195() V196() V197() V200() set
(idseq (len K)) " ((dom K) \ B) is finite set
(Sgm B) ^ (Sgm ((dom K) \ B)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V185() V186() V187() V188() FinSequence of NAT
[:(dom K),(dom K):] is Relation-like finite set
bool [:(dom K),(dom K):] is finite V37() set
K * ((Sgm B) ^ (Sgm ((dom K) \ B))) is Relation-like NAT -defined the carrier of n -valued Function-like finite Element of bool [:NAT, the carrier of n:]
BA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng BA is finite set
AB is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng AB is finite set
f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum f is Element of the carrier of n
the addF of n $$ f is Element of the carrier of n
X is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum X is Element of the carrier of n
the addF of n $$ X is Element of the carrier of n
(Sum f) + (Sum X) is Element of the carrier of n
K560( the carrier of n, the addF of n,(Sum f),(Sum X)) is Element of the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n $$ x is Element of the carrier of n
f ^ X is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n $$ (f ^ X) is Element of the carrier of n
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg K is finite K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
K -' n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (K -' n) is finite K -' n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= K -' n ) } is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
1. (A,n) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of A
1. (A,(K -' n)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K -' n, the carrier of A
K -VectSp_over A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over A
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
(A,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width B) -VectSp_over A
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width B) -VectSp_over A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over A
BA is finite without_zero Element of bool NAT
card BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (B,(Seg n),BA) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card BA, the carrier of A
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Sgm BA is Relation-like NAT -defined NAT -valued Function-like finite card BA -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card BA) -tuples_on NAT
(card BA) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card BA } is set
Segm (B,(Sgm (Seg n)),(Sgm BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card BA, the carrier of A
(Seg K) \ BA is finite without_zero Element of bool NAT
Segm (B,(Seg n),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg K) \ BA), the carrier of A
card ((Seg K) \ BA) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg K) \ BA) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg K) \ BA) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg K) \ BA)) -tuples_on NAT
(card ((Seg K) \ BA)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg K) \ BA) } is set
Segm (B,(Sgm (Seg n)),(Sgm ((Seg K) \ BA))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg n), card ((Seg K) \ BA), the carrier of A
(Segm (B,(Seg n),((Seg K) \ BA))) @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
- ((Segm (B,(Seg n),((Seg K) \ BA))) @) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
0. A is zero Element of the carrier of A
K |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
K -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = K } is set
(Seg K) --> (0. A) is Relation-like Seg K -defined Seg K -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg K),{(0. A)}:]
{(0. A)} is non empty trivial finite 1 -element set
[:(Seg K),{(0. A)}:] is Relation-like finite set
bool [:(Seg K),{(0. A)}:] is finite V37() set
AB is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K, the carrier of A
Segm (AB,(Seg (K -' n)),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
card (Seg (K -' n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (K -' n)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (K -' n)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (K -' n))) -tuples_on NAT
(card (Seg (K -' n))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (K -' n)) } is set
Segm (AB,(Sgm (Seg (K -' n))),(Sgm ((Seg K) \ BA))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card ((Seg K) \ BA), the carrier of A
Segm (AB,(Seg (K -' n)),BA) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
Segm (AB,(Sgm (Seg (K -' n))),(Sgm BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card BA, the carrier of A
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices AB is set
dom AB is finite Element of bool NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
[:(dom AB),(Seg (width AB)):] is Relation-like finite set
[:(Seg (K -' n)),(Seg K):] is Relation-like finite set
[:(Seg (K -' n)),((Seg K) \ BA):] is Relation-like finite set
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
lines AB is finite Element of bool the carrier of (K -VectSp_over A)
the carrier of (K -VectSp_over A) is non empty set
bool the carrier of (K -VectSp_over A) is set
(len B) |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of A
(len B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len B } is set
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
(Seg (len B)) --> (0. A) is Relation-like Seg (len B) -defined Seg (len B) -defined the carrier of A -valued {(0. A)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len B)),{(0. A)}:]
[:(Seg (len B)),{(0. A)}:] is Relation-like finite set
bool [:(Seg (len B)),{(0. A)}:] is finite V37() set
(A,B,((len B) |-> (0. A))) is linearly-closed Element of bool the carrier of ((width B) -VectSp_over A)
the carrier of ((width B) -VectSp_over A) is non empty set
bool the carrier of ((width B) -VectSp_over A) is set
( the carrier of A,((len B) |-> (0. A))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len B) |-> (0. A)),1, the carrier of A
len ((len B) |-> (0. A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len B) |-> (0. A))*> is Relation-like NAT -defined the carrier of A * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len B) |-> (0. A)), the carrier of A
[1,((len B) |-> (0. A))] is set
{1,((len B) |-> (0. A))} is non empty finite V37() set
{{1,((len B) |-> (0. A))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len B) |-> (0. A))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len B) |-> (0. A))*> @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
(A,B,( the carrier of A,((len B) |-> (0. A)))) is set
width ( the carrier of A,((len B) |-> (0. A))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width B & width b₁ = width ( the carrier of A,((len B) |-> (0. A))) & B * b₁ = ( the carrier of A,((len B) |-> (0. A))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A : ( the carrier of A,b₁) in (A,B,( the carrier of A,((len B) |-> (0. A)))) } is set
x is set
f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (AB,f) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(width AB) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width AB } is set
( the carrier of A,(Line (AB,f))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (Line (AB,f)),1, the carrier of A
len (Line (AB,f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(Line (AB,f))*> is Relation-like NAT -defined the carrier of A * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (Line (AB,f)), the carrier of A
[1,(Line (AB,f))] is set
{1,(Line (AB,f))} is non empty finite V37() set
{{1,(Line (AB,f))},{1}} is non empty finite V37() without_zero V103() set
{[1,(Line (AB,f))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(Line (AB,f))*> @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Col (( the carrier of A,(Line (AB,f))),1) is Relation-like NAT -defined the carrier of A -valued Function-like finite len ( the carrier of A,(Line (AB,f))) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of A,(Line (AB,f)))) -tuples_on the carrier of A
len ( the carrier of A,(Line (AB,f))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len ( the carrier of A,(Line (AB,f)))) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len ( the carrier of A,(Line (AB,f))) } is set
Col (( the carrier of A,(Line (AB,f))),BX) is Relation-like NAT -defined the carrier of A -valued Function-like finite len ( the carrier of A,(Line (AB,f))) -element FinSequence-like FinSubsequence-like Element of (len ( the carrier of A,(Line (AB,f)))) -tuples_on the carrier of A
0. (A,n,1) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,1, the carrier of A
1 -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = 1 } is set
1 |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of A
(Seg 1) --> (0. A) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of A -valued {(0. A)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. A)}:]
[:(Seg 1),{(0. A)}:] is Relation-like finite set
bool [:(Seg 1),{(0. A)}:] is finite V37() set
n |-> (1 |-> (0. A)) is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite n -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of n -tuples_on (1 -tuples_on the carrier of A)
n -tuples_on (1 -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
(1 -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of A) * : len b₁ = n } is set
(Seg n) --> (1 |-> (0. A)) is Relation-like non-empty Seg n -defined Seg n -defined 1 -tuples_on the carrier of A -valued {(1 |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg n),{(1 |-> (0. A))}:]
{(1 |-> (0. A))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg n),{(1 |-> (0. A))}:] is Relation-like finite set
bool [:(Seg n),{(1 |-> (0. A))}:] is finite V37() set
(A,B,(0. (A,n,1))) is set
width (0. (A,n,1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A * : ( len b₁ = width B & width b₁ = width (0. (A,n,1)) & B * b₁ = 0. (A,n,1) ) } is set
Lin (lines AB) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of K -VectSp_over A
Lin (A,B,((len B) |-> (0. A))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width B) -VectSp_over A
the carrier of (Lin (lines AB)) is non empty set
the carrier of (Lin (A,B,((len B) |-> (0. A)))) is non empty set
(K -' n) + {} is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Det (1. (A,(K -' n))) is Element of the carrier of A
Permutations (K -' n) is set
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
FinOmega (Permutations (K -' n)) is Element of K96((Permutations (K -' n)))
K96((Permutations (K -' n))) is V24() set
Path_product (1. (A,(K -' n))) is Relation-like Permutations (K -' n) -defined the carrier of A -valued Function-like total quasi_total Element of bool [:(Permutations (K -' n)), the carrier of A:]
[:(Permutations (K -' n)), the carrier of A:] is Relation-like set
bool [:(Permutations (K -' n)), the carrier of A:] is set
K103((Permutations (K -' n)), the carrier of A, the addF of A,(FinOmega (Permutations (K -' n))),(Path_product (1. (A,(K -' n))))) is Element of the carrier of A
1_ A is Element of the carrier of A
1. A is non zero Element of the carrier of A
card (Seg K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K - n is V105() ext-real complex set
EqSegm (AB,(Seg (K -' n)),((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (K -' n)), card (Seg (K -' n)), the carrier of A
the carrier of (A,B) is non empty set
f is set
X is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
( the carrier of A,X) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len X,1, the carrier of A
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*X*> is Relation-like NAT -defined the carrier of A * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len X, the carrier of A
[1,X] is set
{1,X} is non empty finite V37() set
{{1,X},{1}} is non empty finite V37() without_zero V103() set
{[1,X]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*X*> @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
(width AB) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width AB } is set
dom X is finite Element of bool NAT
Seg (len X) is finite len X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
BX is Relation-like NAT -defined (width AB) -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() FinSequence of (width AB) -tuples_on the carrier of A
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dom BX is finite Element of bool NAT
MV is Relation-like NAT -defined the carrier of (K -VectSp_over A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over A)
FinS2MX MV is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len MV,K, the carrier of A
len MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Sgm ((Seg K) \ BA)) . c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
X /. ((Sgm ((Seg K) \ BA)) . c₁₃) is Element of the carrier of A
lA is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of K -' n,K, the carrier of A
Line (lA,c₁₃) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
width lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width lA } is set
lA . c₁₃ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (AB,c₁₃) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
x is Element of the carrier of A
x * (Line (AB,c₁₃)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
x multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
bool [: the carrier of A, the carrier of A:] is set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] (x,(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (AB,c₁₃)),(x multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
MX2FinS AB is Relation-like NAT -defined the carrier of (K -VectSp_over A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over A)
c₁₃ is Relation-like the carrier of (K -VectSp_over A) -defined the carrier of A -valued Function-like total quasi_total Linear_Combination of lines AB
c₁₃ (#) (MX2FinS AB) is Relation-like NAT -defined the carrier of (K -VectSp_over A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over A)
Sum c₁₃ is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
( the carrier of A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len x,1, the carrier of A
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*x*> is Relation-like NAT -defined the carrier of A * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len x, the carrier of A
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() without_zero V103() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*x*> @ is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
y is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
len y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * y is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of A *
Col (y,1) is Relation-like NAT -defined the carrier of A -valued Function-like finite len y -element FinSequence-like FinSubsequence-like Element of (len y) -tuples_on the carrier of A
(len y) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len y } is set
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
rng (Sgm (Seg (K -' n))) is finite V195() V196() V197() V200() set
rng (Sgm BA) is finite V195() V196() V197() V200() set
dom (Sgm BA) is finite card BA -element Element of bool NAT
j is set
(Sgm BA) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (B,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
x is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
mlt ((Line (B,x)),x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
K498( the carrier of A, the carrier of A, the carrier of A, the multF of A,(Line (B,x)),x) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
dom (mlt ((Line (B,x)),x)) is finite width B -element Element of bool NAT
(mlt ((Line (B,x)),x)) * (Sgm BA) is Relation-like NAT -defined the carrier of A -valued Function-like finite Element of bool [:NAT, the carrier of A:]
[:NAT, the carrier of A:] is Relation-like non empty non trivial non finite V103() set
bool [:NAT, the carrier of A:] is non empty non trivial non finite V103() set
(mlt ((Line (B,x)),x)) * (Sgm ((Seg K) \ BA)) is Relation-like NAT -defined the carrier of A -valued Function-like finite Element of bool [:NAT, the carrier of A:]
Sum (mlt ((Line (B,x)),x)) is Element of the carrier of A
the addF of A $$ (mlt ((Line (B,x)),x)) is Element of the carrier of A
mN is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
mSN is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
Sum mN is Element of the carrier of A
the addF of A $$ mN is Element of the carrier of A
Sum mSN is Element of the carrier of A
the addF of A $$ mSN is Element of the carrier of A
(Sum mN) + (Sum mSN) is Element of the carrier of A
K560( the carrier of A, the addF of A,(Sum mN),(Sum mSN)) is Element of the carrier of A
Seg (card BA) is finite card BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card BA ) } is set
dom mN is finite Element of bool NAT
Indices (1. (A,n)) is set
dom (1. (A,n)) is finite Element of bool NAT
width (1. (A,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (1. (A,n))) is finite width (1. (A,n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (A,n)) ) } is set
[:(dom (1. (A,n))),(Seg (width (1. (A,n)))):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
[x,x] is Element of [:NAT,NAT:]
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
(Sgm BA) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x . ((Sgm BA) . x) is set
x /. ((Sgm BA) . x) is Element of the carrier of A
(Line (B,x)) . ((Sgm BA) . x) is set
B * (x,((Sgm BA) . x)) is Element of the carrier of A
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg n),(Seg n):]
bool [:(Seg n),(Seg n):] is finite V37() set
(idseq n) . x is set
(Sgm (Seg n)) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[x,mSN] is set
{x,mSN} is non empty finite V37() set
{{x,mSN},{x}} is non empty finite V37() without_zero V103() set
(Sgm BA) . mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x . ((Sgm BA) . mSN) is set
x /. ((Sgm BA) . mSN) is Element of the carrier of A
(Line (B,x)) . ((Sgm BA) . mSN) is set
B * (x,((Sgm BA) . mSN)) is Element of the carrier of A
mN . mSN is set
(mlt ((Line (B,x)),x)) . ((Sgm BA) . mSN) is set
B * (((Sgm (Seg n)) . x),((Sgm BA) . mSN)) is Element of the carrier of A
(x /. ((Sgm BA) . mSN)) * (B * (((Sgm (Seg n)) . x),((Sgm BA) . mSN))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . mSN)),(B * (((Sgm (Seg n)) . x),((Sgm BA) . mSN)))) is Element of the carrier of A
(1. (A,n)) * (x,mSN) is Element of the carrier of A
(x /. ((Sgm BA) . mSN)) * ((1. (A,n)) * (x,mSN)) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . mSN)),((1. (A,n)) * (x,mSN))) is Element of the carrier of A
(x /. ((Sgm BA) . mSN)) * (0. A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . mSN)),(0. A)) is Element of the carrier of A
mN . x is set
(mlt ((Line (B,x)),x)) . ((Sgm BA) . x) is set
B * (((Sgm (Seg n)) . x),((Sgm BA) . x)) is Element of the carrier of A
(x /. ((Sgm BA) . x)) * (B * (((Sgm (Seg n)) . x),((Sgm BA) . x))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . x)),(B * (((Sgm (Seg n)) . x),((Sgm BA) . x)))) is Element of the carrier of A
(1. (A,n)) * (x,x) is Element of the carrier of A
(x /. ((Sgm BA) . x)) * ((1. (A,n)) * (x,x)) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . x)),((1. (A,n)) * (x,x))) is Element of the carrier of A
(x /. ((Sgm BA) . x)) * (1_ A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm BA) . x)),(1_ A)) is Element of the carrier of A
x /. y is Element of the carrier of A
x . y is set
dom (Sgm ((Seg K) \ BA)) is finite card ((Seg K) \ BA) -element Element of bool NAT
Seg (card ((Seg K) \ BA)) is finite card ((Seg K) \ BA) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card ((Seg K) \ BA) ) } is set
rng (Sgm ((Seg K) \ BA)) is finite V195() V196() V197() V200() set
dom mSN is finite Element of bool NAT
len mSN is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. (A,(len B),1) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len B,1, the carrier of A
1 -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = 1 } is set
1 |-> (0. A) is Relation-like NAT -defined the carrier of A -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of A
(Seg 1) --> (0. A) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of A -valued {(0. A)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. A)}:]
[:(Seg 1),{(0. A)}:] is Relation-like finite set
bool [:(Seg 1),{(0. A)}:] is finite V37() set
(len B) |-> (1 |-> (0. A)) is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len B) -tuples_on (1 -tuples_on the carrier of A)
(len B) -tuples_on (1 -tuples_on the carrier of A) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
(1 -tuples_on the carrier of A) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of A) * : len b₁ = len B } is set
(Seg (len B)) --> (1 |-> (0. A)) is Relation-like non-empty Seg (len B) -defined Seg (len B) -defined 1 -tuples_on the carrier of A -valued {(1 |-> (0. A))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len B)),{(1 |-> (0. A))}:]
{(1 |-> (0. A))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len B)),{(1 |-> (0. A))}:] is Relation-like finite set
bool [:(Seg (len B)),{(1 |-> (0. A))}:] is finite V37() set
Indices (0. (A,(len B),1)) is set
dom (0. (A,(len B),1)) is finite Element of bool NAT
width (0. (A,(len B),1)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (0. (A,(len B),1))) is finite width (0. (A,(len B),1)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (A,(len B),1)) ) } is set
[:(dom (0. (A,(len B),1))),(Seg (width (0. (A,(len B),1)))):] is Relation-like finite set
[:(Seg (len B)),(Seg 1):] is Relation-like finite set
[x,1] is Element of [:NAT,NAT:]
{x,1} is non empty finite V37() set
{{x,1},{x}} is non empty finite V37() without_zero V103() set
( the carrier of A,((len B) |-> (0. A))) * (x,1) is Element of the carrier of A
(Line (B,x)) "*" (Col (y,1)) is Element of the carrier of A
mlt ((Line (B,x)),(Col (y,1))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
K498( the carrier of A, the carrier of A, the carrier of A, the multF of A,(Line (B,x)),(Col (y,1))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
Sum (mlt ((Line (B,x)),(Col (y,1)))) is Element of the carrier of A
the addF of A $$ (mlt ((Line (B,x)),(Col (y,1)))) is Element of the carrier of A
(K -' n) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = K -' n } is set
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
idseq (K -' n) is Relation-like NAT -defined Function-like finite K -' n -element FinSequence-like FinSubsequence-like set
id (Seg (K -' n)) is Relation-like Seg (K -' n) -defined Seg (K -' n) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (K -' n)),(Seg (K -' n)):]
[:(Seg (K -' n)),(Seg (K -' n)):] is Relation-like finite set
bool [:(Seg (K -' n)),(Seg (K -' n)):] is finite V37() set
(idseq (K -' n)) . j is set
(Sgm (Seg (K -' n))) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (lA,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
lA . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (AB,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(Sgm ((Seg K) \ BA)) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x /. ((Sgm ((Seg K) \ BA)) . j) is Element of the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . j)) * (Line (AB,j)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . j)) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] ((x /. ((Sgm ((Seg K) \ BA)) . j)),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (AB,j)),((x /. ((Sgm ((Seg K) \ BA)) . j)) multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
x . ((Sgm ((Seg K) \ BA)) . j) is set
B * (x,((Sgm ((Seg K) \ BA)) . j)) is Element of the carrier of A
(Line (B,x)) . ((Sgm ((Seg K) \ BA)) . j) is set
(x /. ((Sgm ((Seg K) \ BA)) . j)) * (B * (x,((Sgm ((Seg K) \ BA)) . j))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm ((Seg K) \ BA)) . j)),(B * (x,((Sgm ((Seg K) \ BA)) . j)))) is Element of the carrier of A
(mlt ((Line (B,x)),x)) . ((Sgm ((Seg K) \ BA)) . j) is set
mSN is Relation-like NAT -defined the carrier of A -valued Function-like finite K -' n -element FinSequence-like FinSubsequence-like Element of (K -' n) -tuples_on the carrier of A
mSN . j is set
[:(Seg (K -' n)),BA:] is Relation-like finite set
[j,y] is set
{j,y} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,y},{j}} is non empty finite V37() without_zero V103() set
[j,x] is set
{j,x} is non empty finite V37() set
{{j,x},{j}} is non empty finite V37() without_zero V103() set
Indices (Segm (AB,(Seg (K -' n)),BA)) is set
dom (Segm (AB,(Seg (K -' n)),BA)) is finite Element of bool NAT
width (Segm (AB,(Seg (K -' n)),BA)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (AB,(Seg (K -' n)),BA))) is finite width (Segm (AB,(Seg (K -' n)),BA)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (AB,(Seg (K -' n)),BA)) ) } is set
[:(dom (Segm (AB,(Seg (K -' n)),BA))),(Seg (width (Segm (AB,(Seg (K -' n)),BA)))):] is Relation-like finite set
Indices ((Segm (B,(Seg n),((Seg K) \ BA))) @) is set
dom ((Segm (B,(Seg n),((Seg K) \ BA))) @) is finite Element of bool NAT
width ((Segm (B,(Seg n),((Seg K) \ BA))) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ((Segm (B,(Seg n),((Seg K) \ BA))) @)) is finite width ((Segm (B,(Seg n),((Seg K) \ BA))) @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((Segm (B,(Seg n),((Seg K) \ BA))) @) ) } is set
[:(dom ((Segm (B,(Seg n),((Seg K) \ BA))) @)),(Seg (width ((Segm (B,(Seg n),((Seg K) \ BA))) @))):] is Relation-like finite set
[x,j] is set
{x,j} is non empty finite V37() set
{{x,j},{x}} is non empty finite V37() without_zero V103() set
Indices (Segm (B,(Seg n),((Seg K) \ BA))) is set
dom (Segm (B,(Seg n),((Seg K) \ BA))) is finite Element of bool NAT
width (Segm (B,(Seg n),((Seg K) \ BA))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (B,(Seg n),((Seg K) \ BA)))) is finite width (Segm (B,(Seg n),((Seg K) \ BA))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (B,(Seg n),((Seg K) \ BA))) ) } is set
[:(dom (Segm (B,(Seg n),((Seg K) \ BA)))),(Seg (width (Segm (B,(Seg n),((Seg K) \ BA))))):] is Relation-like finite set
(Line (AB,j)) . y is set
AB * (((Sgm (Seg (K -' n))) . j),((Sgm BA) . x)) is Element of the carrier of A
(- ((Segm (B,(Seg n),((Seg K) \ BA))) @)) * (j,x) is Element of the carrier of A
((Segm (B,(Seg n),((Seg K) \ BA))) @) * (j,x) is Element of the carrier of A
- (((Segm (B,(Seg n),((Seg K) \ BA))) @) * (j,x)) is Element of the carrier of A
(Segm (B,(Seg n),((Seg K) \ BA))) * (x,j) is Element of the carrier of A
- ((Segm (B,(Seg n),((Seg K) \ BA))) * (x,j)) is Element of the carrier of A
- (B * (x,((Sgm ((Seg K) \ BA)) . j))) is Element of the carrier of A
dom lA is finite Element of bool NAT
Col (lA,y) is Relation-like NAT -defined the carrier of A -valued Function-like finite len lA -element FinSequence-like FinSubsequence-like Element of (len lA) -tuples_on the carrier of A
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len lA } is set
(Col (lA,y)) . j is set
lA * (j,y) is Element of the carrier of A
(Line (lA,j)) . y is set
(x /. ((Sgm ((Seg K) \ BA)) . j)) * (- (B * (x,((Sgm ((Seg K) \ BA)) . j)))) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm ((Seg K) \ BA)) . j)),(- (B * (x,((Sgm ((Seg K) \ BA)) . j))))) is Element of the carrier of A
- ((x /. ((Sgm ((Seg K) \ BA)) . j)) * (B * (x,((Sgm ((Seg K) \ BA)) . j)))) is Element of the carrier of A
- mSN is Relation-like NAT -defined the carrier of A -valued Function-like finite K -' n -element FinSequence-like FinSubsequence-like Element of (K -' n) -tuples_on the carrier of A
comp A is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,mSN,(comp A)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(- mSN) . j is set
Sum (Col (lA,y)) is Element of the carrier of A
the addF of A $$ (Col (lA,y)) is Element of the carrier of A
Sum mSN is Element of the carrier of A
the addF of A $$ mSN is Element of the carrier of A
- (Sum mSN) is Element of the carrier of A
(Sum mSN) + (Sum mN) is Element of the carrier of A
K560( the carrier of A, the addF of A,(Sum mSN),(Sum mN)) is Element of the carrier of A
(- (Sum mSN)) + ((Sum mSN) + (Sum mN)) is Element of the carrier of A
K560( the carrier of A, the addF of A,(- (Sum mSN)),((Sum mSN) + (Sum mN))) is Element of the carrier of A
(- (Sum mSN)) + (Sum mSN) is Element of the carrier of A
K560( the carrier of A, the addF of A,(- (Sum mSN)),(Sum mSN)) is Element of the carrier of A
((- (Sum mSN)) + (Sum mSN)) + (Sum mN) is Element of the carrier of A
K560( the carrier of A, the addF of A,((- (Sum mSN)) + (Sum mSN)),(Sum mN)) is Element of the carrier of A
(0. A) + (Sum mN) is Element of the carrier of A
K560( the carrier of A, the addF of A,(0. A),(Sum mN)) is Element of the carrier of A
rng (Sgm ((Seg K) \ BA)) is finite V195() V196() V197() V200() set
dom (Sgm ((Seg K) \ BA)) is finite card ((Seg K) \ BA) -element Element of bool NAT
y is set
(Sgm ((Seg K) \ BA)) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (card ((Seg K) \ BA)) is finite card ((Seg K) \ BA) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card ((Seg K) \ BA) ) } is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Line (lA,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
lA . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (AB,x) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
x is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
(Sgm ((Seg K) \ BA)) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x /. ((Sgm ((Seg K) \ BA)) . x) is Element of the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . x)) * (Line (AB,x)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . x)) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] ((x /. ((Sgm ((Seg K) \ BA)) . x)),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (AB,x)),((x /. ((Sgm ((Seg K) \ BA)) . x)) multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
[x,x] is Element of [:NAT,NAT:]
{x,x} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,x},{x}} is non empty finite V37() without_zero V103() set
[:(Seg (K -' n)),(Seg (K -' n)):] is Relation-like finite set
Indices (1. (A,(K -' n))) is set
dom (1. (A,(K -' n))) is finite Element of bool NAT
width (1. (A,(K -' n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (1. (A,(K -' n)))) is finite width (1. (A,(K -' n))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (A,(K -' n))) ) } is set
[:(dom (1. (A,(K -' n)))),(Seg (width (1. (A,(K -' n))))):] is Relation-like finite set
idseq (K -' n) is Relation-like NAT -defined Function-like finite K -' n -element FinSequence-like FinSubsequence-like set
id (Seg (K -' n)) is Relation-like Seg (K -' n) -defined Seg (K -' n) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (K -' n)),(Seg (K -' n)):]
bool [:(Seg (K -' n)),(Seg (K -' n)):] is finite V37() set
(idseq (K -' n)) . x is set
(Sgm (Seg (K -' n))) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (AB,x)) . y is set
AB * (((Sgm (Seg (K -' n))) . x),((Sgm ((Seg K) \ BA)) . x)) is Element of the carrier of A
(1. (A,(K -' n))) * (x,x) is Element of the carrier of A
Col (lA,y) is Relation-like NAT -defined the carrier of A -valued Function-like finite len lA -element FinSequence-like FinSubsequence-like Element of (len lA) -tuples_on the carrier of A
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len lA } is set
dom (Col (lA,y)) is finite len lA -element Element of bool NAT
Seg (len lA) is finite len lA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len lA ) } is set
dom lA is finite Element of bool NAT
j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (lA,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width lA -element FinSequence-like FinSubsequence-like Element of (width lA) -tuples_on the carrier of A
lA . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (AB,j) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(Sgm ((Seg K) \ BA)) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x /. ((Sgm ((Seg K) \ BA)) . j) is Element of the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . j)) * (Line (AB,j)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of A
(x /. ((Sgm ((Seg K) \ BA)) . j)) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] ((x /. ((Sgm ((Seg K) \ BA)) . j)),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (AB,j)),((x /. ((Sgm ((Seg K) \ BA)) . j)) multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
[j,x] is set
{j,x} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,x},{j}} is non empty finite V37() without_zero V103() set
(idseq (K -' n)) . j is set
(Sgm (Seg (K -' n))) . j is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(Line (AB,j)) . y is set
AB * (((Sgm (Seg (K -' n))) . j),((Sgm ((Seg K) \ BA)) . x)) is Element of the carrier of A
(1. (A,(K -' n))) * (j,x) is Element of the carrier of A
(Col (lA,y)) . j is set
lA * (j,y) is Element of the carrier of A
((x /. ((Sgm ((Seg K) \ BA)) . j)) * (Line (AB,j))) . y is set
(x /. ((Sgm ((Seg K) \ BA)) . j)) * (0. A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm ((Seg K) \ BA)) . j)),(0. A)) is Element of the carrier of A
(Col (lA,y)) . x is set
lA * (x,y) is Element of the carrier of A
(Line (lA,x)) . y is set
(x /. ((Sgm ((Seg K) \ BA)) . x)) * (1_ A) is Element of the carrier of A
K560( the carrier of A, the multF of A,(x /. ((Sgm ((Seg K) \ BA)) . x)),(1_ A)) is Element of the carrier of A
x /. y is Element of the carrier of A
x . y is set
Sum (Col (lA,y)) is Element of the carrier of A
the addF of A $$ (Col (lA,y)) is Element of the carrier of A
Col (lA,y) is Relation-like NAT -defined the carrier of A -valued Function-like finite len lA -element FinSequence-like FinSubsequence-like Element of (len lA) -tuples_on the carrier of A
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len lA } is set
Sum (Col (lA,y)) is Element of the carrier of A
the addF of A $$ (Col (lA,y)) is Element of the carrier of A
x is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
x . y is set
Col (lA,y) is Relation-like NAT -defined the carrier of A -valued Function-like finite len lA -element FinSequence-like FinSubsequence-like Element of (len lA) -tuples_on the carrier of A
len lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len lA) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = len lA } is set
Sum (Col (lA,y)) is Element of the carrier of A
the addF of A $$ (Col (lA,y)) is Element of the carrier of A
x is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
x . y is set
Carrier c₁₃ is finite Element of bool the carrier of (K -VectSp_over A)
x is Relation-like NAT -defined the carrier of A -valued Function-like finite K -element FinSequence-like FinSubsequence-like Element of K -tuples_on the carrier of A
x . y is set
{ (Sum b₁) where b₁ is Relation-like the carrier of (K -VectSp_over A) -defined the carrier of A -valued Function-like total quasi_total Linear_Combination of lines AB : verum } is set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(K,(1. (K,n))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (1. (K,n))) -VectSp_over K
width (1. (K,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (1. (K,n))) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim (K,(1. (K,n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
the carrier of (K,(1. (K,n))) is non empty set
(0). (K,(1. (K,n))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (K,(1. (K,n)))
the carrier of ((0). (K,(1. (K,n)))) is non empty set
BA is set
len (1. (K,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
(K,(1. (K,n)),(n |-> (0. K))) is Element of bool the carrier of ((width (1. (K,n))) -VectSp_over K)
the carrier of ((width (1. (K,n))) -VectSp_over K) is non empty set
bool the carrier of ((width (1. (K,n))) -VectSp_over K) is set
( the carrier of K,(n |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len (n |-> (0. K)),1, the carrier of K
len (n |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*(n |-> (0. K))*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len (n |-> (0. K)), the carrier of K
[1,(n |-> (0. K))] is set
{1,(n |-> (0. K))} is non empty finite V37() set
{{1,(n |-> (0. K))},{1}} is non empty finite V37() without_zero V103() set
{[1,(n |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(n |-> (0. K))*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,(1. (K,n)),( the carrier of K,(n |-> (0. K)))) is set
width ( the carrier of K,(n |-> (0. K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width (1. (K,n)) & width b₁ = width ( the carrier of K,(n |-> (0. K))) & (1. (K,n)) * b₁ = ( the carrier of K,(n |-> (0. K))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K : ( the carrier of K,b₁) in (K,(1. (K,n)),( the carrier of K,(n |-> (0. K)))) } is set
AB is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
( the carrier of K,AB) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of K
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*AB*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len AB, the carrier of K
[1,AB] is set
{1,AB} is non empty finite V37() set
{{1,AB},{1}} is non empty finite V37() without_zero V103() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*AB*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(1. (K,n)) * x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
Col (x,1) is Relation-like NAT -defined the carrier of K -valued Function-like finite len x -element FinSequence-like FinSubsequence-like Element of (len x) -tuples_on the carrier of K
(len x) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len x } is set
0. (K,(1. (K,n))) is zero Element of the carrier of (K,(1. (K,n)))
0. ((width (1. (K,n))) -VectSp_over K) is Relation-like Function-like zero Element of the carrier of ((width (1. (K,n))) -VectSp_over K)
{(0. (K,(1. (K,n))))} is non empty trivial finite 1 -element Element of bool the carrier of (K,(1. (K,n)))
bool the carrier of (K,(1. (K,n))) is set
(Omega). (K,(1. (K,n))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (K,(1. (K,n)))
the addF of (K,(1. (K,n))) is Relation-like [: the carrier of (K,(1. (K,n))), the carrier of (K,(1. (K,n))):] -defined the carrier of (K,(1. (K,n))) -valued Function-like total quasi_total V223( the carrier of (K,(1. (K,n)))) V224( the carrier of (K,(1. (K,n)))) Element of bool [:[: the carrier of (K,(1. (K,n))), the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):]
[: the carrier of (K,(1. (K,n))), the carrier of (K,(1. (K,n))):] is Relation-like set
[:[: the carrier of (K,(1. (K,n))), the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):] is Relation-like set
bool [:[: the carrier of (K,(1. (K,n))), the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):] is set
the ZeroF of (K,(1. (K,n))) is Element of the carrier of (K,(1. (K,n)))
the lmult of (K,(1. (K,n))) is Relation-like [: the carrier of K, the carrier of (K,(1. (K,n))):] -defined the carrier of (K,(1. (K,n))) -valued Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):]
[: the carrier of K, the carrier of (K,(1. (K,n))):] is Relation-like set
[:[: the carrier of K, the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):] is Relation-like set
bool [:[: the carrier of K, the carrier of (K,(1. (K,n))):], the carrier of (K,(1. (K,n))):] is set
VectSpStr(# the carrier of (K,(1. (K,n))), the addF of (K,(1. (K,n))), the ZeroF of (K,(1. (K,n))), the lmult of (K,(1. (K,n))) #) is non empty strict VectSpStr over K
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,K) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
(width K) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
dim (n,K) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) - (the_rank_of K) is V105() ext-real complex set
dim ((width K) -VectSp_over n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(n,BA) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width BA) -VectSp_over n
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
1_ n is Element of the carrier of n
1. n is non zero Element of the carrier of n
- (1_ n) is Element of the carrier of n
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
dom AB is finite Element of bool NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
f is Element of the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len AB) is finite len AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len AB ) } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (AB,X) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
(width AB) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width AB } is set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (AB,BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
f * (Line (AB,BX)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
f multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
the multF of n [;] (f,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
K501( the carrier of n, the carrier of n,(Line (AB,BX)),(f multfield)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(Line (AB,X)) + (f * (Line (AB,BX))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
K498( the carrier of n, the carrier of n, the carrier of n, the addF of n,(Line (AB,X)),(f * (Line (AB,BX)))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
ReplaceLine (AB,X,((Line (AB,X)) + (f * (Line (AB,BX))))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(n,(ReplaceLine (AB,X,((Line (AB,X)) + (f * (Line (AB,BX))))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (ReplaceLine (AB,X,((Line (AB,X)) + (f * (Line (AB,BX))))))) -VectSp_over n
width (ReplaceLine (AB,X,((Line (AB,X)) + (f * (Line (AB,BX)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (ReplaceLine (AB,X,((Line (AB,X)) + (f * (Line (AB,BX))))))) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
(n,AB) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width AB) -VectSp_over n
(width AB) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
MV is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(n,MV) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width MV) -VectSp_over n
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width MV) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
(n,AB) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width AB) -VectSp_over n
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
Seg (width K) is finite width K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width K ) } is set
the_rank_of BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. n is zero Element of the carrier of n
(width K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width K } is set
(Seg (width K)) --> (0. n) is Relation-like Seg (width K) -defined Seg (width K) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width K)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (width K)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width K)),{(0. n)}:] is finite V37() set
f is finite without_zero Element of bool NAT
AB is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
the_rank_of AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width K, the carrier of n
Seg (card f) is finite card f -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card f ) } is set
Segm (AB,(Seg (card f)),f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card f, the carrier of n
card (Seg (card f)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (card f)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card f)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card f))) -tuples_on NAT
(card (Seg (card f))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card f)) } is set
Sgm f is Relation-like NAT -defined NAT -valued Function-like finite card f -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card f) -tuples_on NAT
(card f) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card f } is set
Segm (AB,(Sgm (Seg (card f))),(Sgm f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card f, the carrier of n
1. (n,(card f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card f, card f, the carrier of n
dom AB is finite Element of bool NAT
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width AB) is finite width AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width AB ) } is set
card (Seg (width AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Seg (width K)) \ f is finite without_zero Element of bool NAT
card ((Seg (width K)) \ f) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width K) - (card f) is V105() ext-real complex set
Segm (AB,(Seg (card f)),(Seg (width AB))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card (Seg (width AB)), the carrier of n
Sgm (Seg (width AB)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (width AB)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (width AB))) -tuples_on NAT
(card (Seg (width AB))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (width AB)) } is set
Segm (AB,(Sgm (Seg (card f))),(Sgm (Seg (width AB)))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card (Seg (width AB)), the carrier of n
(width K) -' (card f) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
idseq (card f) is Relation-like NAT -defined Function-like finite card f -element FinSequence-like FinSubsequence-like set
id (Seg (card f)) is Relation-like Seg (card f) -defined Seg (card f) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (card f)),(Seg (card f)):]
[:(Seg (card f)),(Seg (card f)):] is Relation-like finite set
bool [:(Seg (card f)),(Seg (card f)):] is finite V37() set
(Sgm (Seg (card f))) " (Seg (card f)) is finite set
idseq (width AB) is Relation-like NAT -defined Function-like finite width AB -element FinSequence-like FinSubsequence-like set
id (Seg (width AB)) is Relation-like Seg (width AB) -defined Seg (width AB) -valued Function-like one-to-one total quasi_total finite Element of bool [:(Seg (width AB)),(Seg (width AB)):]
[:(Seg (width AB)),(Seg (width AB)):] is Relation-like finite set
bool [:(Seg (width AB)),(Seg (width AB)):] is finite V37() set
(Sgm (Seg (width AB))) " f is finite set
Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Seg (card f)),f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card f, the carrier of n
Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Sgm (Seg (card f))),(Sgm f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card f, the carrier of n
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len AB) is finite len AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len AB ) } is set
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(dom AB) \ (Seg (card f)) is finite Element of bool NAT
Line (AB,BX) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
(width AB) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = width AB } is set
(width AB) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite width AB -element FinSequence-like FinSubsequence-like Element of (width AB) -tuples_on the carrier of n
(Seg (width AB)) --> (0. n) is Relation-like Seg (width AB) -defined Seg (width AB) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (width AB)),{(0. n)}:]
[:(Seg (width AB)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (width AB)),{(0. n)}:] is finite V37() set
(n,(Segm (AB,(Seg (card f)),(Seg (width AB))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (Segm (AB,(Seg (card f)),(Seg (width AB))))) -VectSp_over n
width (Segm (AB,(Seg (card f)),(Seg (width AB)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (Segm (AB,(Seg (card f)),(Seg (width AB))))) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
(n,AB) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width AB) -VectSp_over n
(width AB) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
((width K) -' (card f)) + {} is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
1. (n,((width K) -' (card f))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of (width K) -' (card f),(width K) -' (card f), the carrier of n
Det (1. (n,((width K) -' (card f)))) is Element of the carrier of n
Permutations ((width K) -' (card f)) is set
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total quasi_total V223( the carrier of n) V224( 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
FinOmega (Permutations ((width K) -' (card f))) is Element of K96((Permutations ((width K) -' (card f))))
K96((Permutations ((width K) -' (card f)))) is V24() set
Path_product (1. (n,((width K) -' (card f)))) is Relation-like Permutations ((width K) -' (card f)) -defined the carrier of n -valued Function-like total quasi_total Element of bool [:(Permutations ((width K) -' (card f))), the carrier of n:]
[:(Permutations ((width K) -' (card f))), the carrier of n:] is Relation-like set
bool [:(Permutations ((width K) -' (card f))), the carrier of n:] is set
K103((Permutations ((width K) -' (card f))), the carrier of n, the addF of n,(FinOmega (Permutations ((width K) -' (card f)))),(Path_product (1. (n,((width K) -' (card f)))))) is Element of the carrier of n
Seg ((width K) -' (card f)) is finite (width K) -' (card f) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= (width K) -' (card f) ) } is set
card (Seg ((width K) -' (card f))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Seg (card f)),((Seg (width K)) \ f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card ((Seg (width K)) \ f), the carrier of n
Sgm ((Seg (width K)) \ f) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg (width K)) \ f) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg (width K)) \ f)) -tuples_on NAT
(card ((Seg (width K)) \ f)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg (width K)) \ f) } is set
Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Sgm (Seg (card f))),(Sgm ((Seg (width K)) \ f))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card f)), card ((Seg (width K)) \ f), the carrier of n
(Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Seg (card f)),((Seg (width K)) \ f))) @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
- ((Segm ((Segm (AB,(Seg (card f)),(Seg (width AB)))),(Seg (card f)),((Seg (width K)) \ f))) @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BX is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of (width K) -' (card f), width K, the carrier of n
Segm (BX,(Seg ((width K) -' (card f))),((Seg (width K)) \ f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg ((width K) -' (card f))), card ((Seg (width K)) \ f), the carrier of n
Sgm (Seg ((width K) -' (card f))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg ((width K) -' (card f))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg ((width K) -' (card f)))) -tuples_on NAT
(card (Seg ((width K) -' (card f)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg ((width K) -' (card f))) } is set
Segm (BX,(Sgm (Seg ((width K) -' (card f)))),(Sgm ((Seg (width K)) \ f))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg ((width K) -' (card f))), card ((Seg (width K)) \ f), the carrier of n
Segm (BX,(Seg ((width K) -' (card f))),f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg ((width K) -' (card f))), card f, the carrier of n
Segm (BX,(Sgm (Seg ((width K) -' (card f)))),(Sgm f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg ((width K) -' (card f))), card f, the carrier of n
lines BX is finite Element of bool the carrier of ((width K) -VectSp_over n)
the carrier of ((width K) -VectSp_over n) is non empty set
bool the carrier of ((width K) -VectSp_over n) is set
Lin (lines BX) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width K) -VectSp_over n
len BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Indices BX is set
dom BX is finite Element of bool NAT
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
[:(dom BX),(Seg (width BX)):] is Relation-like finite set
[:(Seg ((width K) -' (card f))),(Seg (width K)):] is Relation-like finite set
[:(Seg ((width K) -' (card f))),((Seg (width K)) \ f):] is Relation-like finite set
Seg (len BX) is finite len BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BX ) } is set
EqSegm (BX,(Seg ((width K) -' (card f))),((Seg (width K)) \ f)) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg ((width K) -' (card f))), card (Seg ((width K) -' (card f))), the carrier of n
MV is Element of bool the carrier of ((width K) -VectSp_over n)
card MV is V26() V27() V28() cardinal set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of A is non empty non trivial V103() set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
1_ A is Element of the carrier of A
1. A is non zero Element of the carrier of A
- (1_ A) is Element of the carrier of A
K -VectSp_over A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over A
B is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
dom B is finite Element of bool NAT
lines B is finite Element of bool the carrier of (K -VectSp_over A)
the carrier of (K -VectSp_over A) is non empty set
bool the carrier of (K -VectSp_over A) is set
Lin (lines B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of K -VectSp_over A
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (B,BA) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
(width B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b₁ where b₁ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b₁ = width B } is set
AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (B,AB) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
x is Element of the carrier of A
x * (Line (B,AB)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
x multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
bool [: the carrier of A, the carrier of A:] is set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
the multF of A [;] (x,(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K501( the carrier of A, the carrier of A,(Line (B,AB)),(x multfield)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Line (B,BA)) + (x * (Line (B,AB))) is Relation-like NAT -defined the carrier of A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
K498( the carrier of A, the carrier of A, the carrier of A, the addF of A,(Line (B,BA)),(x * (Line (B,AB)))) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
ReplaceLine (B,BA,((Line (B,BA)) + (x * (Line (B,AB))))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,K, the carrier of A
lines (ReplaceLine (B,BA,((Line (B,BA)) + (x * (Line (B,AB)))))) is finite Element of bool the carrier of (K -VectSp_over A)
Lin (lines (ReplaceLine (B,BA,((Line (B,BA)) + (x * (Line (B,AB))))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of K -VectSp_over A
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
MV is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
x * MV is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
BX is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
BX + (x * MV) is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
the addF of (K -VectSp_over A) is Relation-like [: the carrier of (K -VectSp_over A), the carrier of (K -VectSp_over A):] -defined the carrier of (K -VectSp_over A) -valued Function-like total quasi_total V223( the carrier of (K -VectSp_over A)) V224( the carrier of (K -VectSp_over A)) Element of bool [:[: the carrier of (K -VectSp_over A), the carrier of (K -VectSp_over A):], the carrier of (K -VectSp_over A):]
[: the carrier of (K -VectSp_over A), the carrier of (K -VectSp_over A):] is Relation-like set
[:[: the carrier of (K -VectSp_over A), the carrier of (K -VectSp_over A):], the carrier of (K -VectSp_over A):] is Relation-like set
bool [:[: the carrier of (K -VectSp_over A), the carrier of (K -VectSp_over A):], the carrier of (K -VectSp_over A):] is set
K560( the carrier of (K -VectSp_over A), the addF of (K -VectSp_over A),BX,(x * MV)) is Relation-like Function-like Element of the carrier of (K -VectSp_over A)
0. (K -VectSp_over A) is Relation-like Function-like zero Element of the carrier of (K -VectSp_over A)
B . BA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B . AB is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
lA is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
{BA} is non empty trivial finite V37() 1 -element Element of bool NAT
{BA} --> lA is Relation-like {BA} -defined the carrier of A * -valued Function-like constant non empty total quasi_total finite Function-yielding V147() Element of bool [:{BA},( the carrier of A *):]
[:{BA},( the carrier of A *):] is Relation-like set
bool [:{BA},( the carrier of A *):] is set
len ((Line (B,BA)) + (x * (Line (B,AB)))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ is set
Replace (BA,c₁₃,) is Relation-like Function-like set
BA .--> c₁₃ is Relation-like {BA} -defined Function-like one-to-one finite set
{BA} is non empty trivial finite V37() 1 -element set
B +* (BA .--> c₁₃) is Relation-like Function-like finite set
B +* ({BA} --> lA) is Relation-like Function-like finite set
dom ({BA} --> lA) is non empty finite set
(dom B) \ (dom ({BA} --> lA)) is finite Element of bool NAT
B .: ((dom B) \ (dom ({BA} --> lA))) is finite set
B .: (dom B) is finite set
B .: (dom ({BA} --> lA)) is finite set
(B .: (dom B)) \ (B .: (dom ({BA} --> lA))) is finite Element of bool (B .: (dom B))
bool (B .: (dom B)) is finite V37() set
rng B is finite set
(rng B) \ (B .: (dom ({BA} --> lA))) is finite Element of bool (rng B)
bool (rng B) is finite V37() set
Im (B,BA) is set
B .: {BA} is finite set
(rng B) \ (Im (B,BA)) is finite Element of bool (rng B)
B . BA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{(B . BA)} is functional non empty trivial finite V37() 1 -element set
(rng B) \ {(B . BA)} is finite Element of bool (rng B)
{(Line (B,BA))} is functional non empty trivial finite V37() 1 -element FinSequence-membered Element of bool ((width B) -tuples_on the carrier of A)
bool ((width B) -tuples_on the carrier of A) is set
(rng B) \ {(Line (B,BA))} is finite Element of bool (rng B)
(lines B) \ {(Line (B,BA))} is finite Element of bool the carrier of (K -VectSp_over A)
rng ({BA} --> lA) is non empty trivial finite 1 -element set
((lines B) \ {(Line (B,BA))}) \/ (rng ({BA} --> lA)) is non empty finite set
{((Line (B,BA)) + (x * (Line (B,AB))))} is functional non empty trivial finite V37() 1 -element FinSequence-membered Element of bool ((width B) -tuples_on the carrier of A)
((lines B) \ {(Line (B,BA))}) \/ {((Line (B,BA)) + (x * (Line (B,AB))))} is non empty finite set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
n -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is non empty right_complementable V95() vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
dim A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg (dim A) is finite dim A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= dim A ) } is set
1. (K,(dim A)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of dim A, dim A, the carrier of K
the carrier of A is non empty set
bool the carrier of A is set
B is finite Element of bool the carrier of A
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is set
card B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Omega). A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
the ZeroF of A is Element of the carrier of A
the lmult of A is Relation-like [: the carrier of K, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of A:], the carrier of A:]
[: the carrier of K, the carrier of A:] is Relation-like set
[:[: the carrier of K, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of K, the carrier of A:], the carrier of A:] is set
VectSpStr(# the carrier of A, the addF of A, the ZeroF of A, the lmult of A #) is non empty strict VectSpStr over K
BA is linearly-independent Element of bool the carrier of (n -VectSp_over K)
AB is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
lines AB is finite Element of bool the carrier of (n -VectSp_over K)
1_ K is Element of the carrier of K
1. K is non zero Element of the carrier of K
- (1_ K) is Element of the carrier of K
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
dom x is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (len x) is finite len x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
X is Element of the carrier of K
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (x,BX) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of K
(width x) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width x } is set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Line (x,MV) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of K
X * (Line (x,MV)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -tuples_on the carrier of K
X multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like 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 quasi_total V223( 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 Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
the multF of K [;] (X,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K501( the carrier of K, the carrier of K,(Line (x,MV)),(X multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line (x,BX)) + (X * (Line (x,MV))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width x -element FinSequence-like FinSubsequence-like Element of (width x) -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 quasi_total V223( the carrier of K) V224( the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
K498( the carrier of K, the carrier of K, the carrier of K, the addF of K,(Line (x,BX)),(X * (Line (x,MV)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine (x,BX,((Line (x,BX)) + (X * (Line (x,MV))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
lines x is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines x) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
lines (ReplaceLine (x,BX,((Line (x,BX)) + (X * (Line (x,MV)))))) is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines (ReplaceLine (x,BX,((Line (x,BX)) + (X * (Line (x,MV))))))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
the_rank_of x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of (ReplaceLine (x,BX,((Line (x,BX)) + (X * (Line (x,MV)))))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
c₁₃ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
lines c₁₃ is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines c₁₃) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
Lin B is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
lines x is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines x) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
the_rank_of AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
X is finite without_zero Element of bool NAT
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
the_rank_of x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
f is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card B,n, the carrier of K
Seg (card X) is finite card X -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= card X ) } is set
Segm (x,(Seg (card X)),X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card X)), card X, the carrier of K
card (Seg (card X)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (card X)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (card X)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (card X))) -tuples_on NAT
(card (Seg (card X))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (card X)) } is set
Sgm X is Relation-like NAT -defined NAT -valued Function-like finite card X -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card X) -tuples_on NAT
(card X) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card X } is set
Segm (x,(Sgm (Seg (card X))),(Sgm X)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (card X)), card X, the carrier of K
1. (K,(card X)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card X, card X, the carrier of K
dom x is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width x) is finite width x -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
BX is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of dim A,n, the carrier of K
lines BX is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines BX) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
the carrier of (Lin (lines BX)) is non empty set
MV is set
the_rank_of BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (BX,(Seg (dim A)),X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card X, the carrier of K
card (Seg (dim A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (dim A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (dim A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (dim A))) -tuples_on NAT
(card (Seg (dim A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (dim A)) } is set
Segm (BX,(Sgm (Seg (dim A))),(Sgm X)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card X, the carrier of K
MV is linearly-independent Element of bool the carrier of A
Lin MV is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
Seg n is finite n -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
n -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
dim A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
n -' (dim A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (n -' (dim A)) is finite n -' (dim A) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= n -' (dim A) ) } is set
1. (K,(n -' (dim A))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n -' (dim A),n -' (dim A), the carrier of K
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding 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 V147() tabular FinSequence-yielding Matrix of n -' (dim A),n, the carrier of K
(K,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width B) -VectSp_over K
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width B) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Segm (B,(Seg (n -' (dim A))),(Seg n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card (Seg n), the carrier of K
card (Seg (n -' (dim A))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (n -' (dim A))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (n -' (dim A))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (n -' (dim A)))) -tuples_on NAT
(card (Seg (n -' (dim A)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (n -' (dim A))) } is set
Sgm (Seg n) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg n) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg n)) -tuples_on NAT
(card (Seg n)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg n) } is set
Segm (B,(Sgm (Seg (n -' (dim A)))),(Sgm (Seg n))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card (Seg n), the carrier of K
len (1. (K,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
dim (K,B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
width (1. (K,n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Omega). (K,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (K,B)
the carrier of (K,B) is non empty set
the addF of (K,B) is Relation-like [: the carrier of (K,B), the carrier of (K,B):] -defined the carrier of (K,B) -valued Function-like total quasi_total V223( the carrier of (K,B)) V224( the carrier of (K,B)) Element of bool [:[: the carrier of (K,B), the carrier of (K,B):], the carrier of (K,B):]
[: the carrier of (K,B), the carrier of (K,B):] is Relation-like set
[:[: the carrier of (K,B), the carrier of (K,B):], the carrier of (K,B):] is Relation-like set
bool [:[: the carrier of (K,B), the carrier of (K,B):], the carrier of (K,B):] is set
the ZeroF of (K,B) is Element of the carrier of (K,B)
the lmult of (K,B) is Relation-like [: the carrier of K, the carrier of (K,B):] -defined the carrier of (K,B) -valued Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of (K,B):], the carrier of (K,B):]
[: the carrier of K, the carrier of (K,B):] is Relation-like set
[:[: the carrier of K, the carrier of (K,B):], the carrier of (K,B):] is Relation-like set
bool [:[: the carrier of K, the carrier of (K,B):], the carrier of (K,B):] is set
VectSpStr(# the carrier of (K,B), the addF of (K,B), the ZeroF of (K,B), the lmult of (K,B) #) is non empty strict VectSpStr over K
(0). (K,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (K,B)
(0). A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
(Omega). A is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
the carrier of A is non empty set
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
the ZeroF of A is Element of the carrier of A
the lmult of A is Relation-like [: the carrier of K, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of A:], the carrier of A:]
[: the carrier of K, the carrier of A:] is Relation-like set
[:[: the carrier of K, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of K, the carrier of A:], the carrier of A:] is set
VectSpStr(# the carrier of A, the addF of A, the ZeroF of A, the lmult of A #) is non empty strict VectSpStr over K
0. (K,(n -' (dim A)),n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n -' (dim A),n, the carrier of K
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = n } is set
0. K is zero Element of the carrier of K
n |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
(Seg n) --> (0. K) is Relation-like Seg n -defined Seg n -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg n),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg n),{(0. K)}:] is Relation-like finite set
bool [:(Seg n),{(0. K)}:] is finite V37() set
(n -' (dim A)) |-> (n |-> (0. K)) is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like finite n -' (dim A) -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (n -' (dim A)) -tuples_on (n -tuples_on the carrier of K)
(n -' (dim A)) -tuples_on (n -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
(n -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (n -tuples_on the carrier of K) * : len b₁ = n -' (dim A) } is set
(Seg (n -' (dim A))) --> (n |-> (0. K)) is Relation-like Seg (n -' (dim A)) -defined Seg (n -' (dim A)) -defined n -tuples_on the carrier of K -valued {(n |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (n -' (dim A))),{(n |-> (0. K))}:]
{(n |-> (0. K))} is functional non empty trivial finite V37() 1 -element set
[:(Seg (n -' (dim A))),{(n |-> (0. K))}:] is Relation-like finite set
bool [:(Seg (n -' (dim A))),{(n |-> (0. K))}:] is finite V37() set
n - (dim A) is V105() ext-real complex set
(dim A) - (dim A) is V105() ext-real complex set
len (0. (K,(n -' (dim A)),n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (0. (K,(n -' (dim A)),n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg n) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (dim A) is finite dim A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= dim A ) } is set
1. (K,(dim A)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of dim A, dim A, the carrier of K
AB is finite without_zero Element of bool NAT
card AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
BA is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of dim A,n, the carrier of K
Segm (BA,(Seg (dim A)),AB) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card AB, the carrier of K
card (Seg (dim A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm (Seg (dim A)) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (dim A)) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (dim A))) -tuples_on NAT
(card (Seg (dim A))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (dim A)) } is set
Sgm AB is Relation-like NAT -defined NAT -valued Function-like finite card AB -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card AB) -tuples_on NAT
(card AB) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card AB } is set
Segm (BA,(Sgm (Seg (dim A))),(Sgm AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card AB, the carrier of K
the_rank_of BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
lines BA is finite Element of bool the carrier of (n -VectSp_over K)
the carrier of (n -VectSp_over K) is non empty set
bool the carrier of (n -VectSp_over K) is set
(Seg n) \ AB is finite without_zero Element of bool NAT
Segm (BA,(Seg (dim A)),((Seg n) \ AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card ((Seg n) \ AB), the carrier of K
card ((Seg n) \ AB) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Sgm ((Seg n) \ AB) is Relation-like NAT -defined NAT -valued Function-like finite card ((Seg n) \ AB) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card ((Seg n) \ AB)) -tuples_on NAT
(card ((Seg n) \ AB)) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card ((Seg n) \ AB) } is set
Segm (BA,(Sgm (Seg (dim A))),(Sgm ((Seg n) \ AB))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card ((Seg n) \ AB), the carrier of K
(card (Seg n)) - (card AB) is V105() ext-real complex set
width (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n - (card AB) is V105() ext-real complex set
len (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
- ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width (- ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
card (Seg (n -' (dim A))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(Seg n) \ ((Seg n) \ AB) is finite without_zero Element of bool NAT
(Seg n) /\ AB is finite Element of bool NAT
n -' (n -' (dim A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
n - (n -' (dim A)) is V105() ext-real complex set
Indices (0. (K,(n -' (dim A)),n)) is set
dom (0. (K,(n -' (dim A)),n)) is finite Element of bool NAT
Seg (width (0. (K,(n -' (dim A)),n))) is finite width (0. (K,(n -' (dim A)),n)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (0. (K,(n -' (dim A)),n)) ) } is set
[:(dom (0. (K,(n -' (dim A)),n))),(Seg (width (0. (K,(n -' (dim A)),n)))):] is Relation-like finite set
[:(Seg (n -' (dim A))),(Seg n):] is Relation-like finite set
[:(Seg (n -' (dim A))),AB:] is Relation-like finite set
len ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (- ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[:(Seg (n -' (dim A))),((Seg n) \ AB):] is Relation-like finite set
[:(Seg (n -' (dim A))),AB:] /\ [:(Seg (n -' (dim A))),((Seg n) \ AB):] is Relation-like finite set
AB /\ ((Seg n) \ AB) is finite Element of bool NAT
[:(Seg (n -' (dim A))),(AB /\ ((Seg n) \ AB)):] is Relation-like finite set
[:(Seg (n -' (dim A))),{}:] is Relation-like finite set
Sgm (Seg (n -' (dim A))) is Relation-like NAT -defined NAT -valued Function-like finite card (Seg (n -' (dim A))) -element FinSequence-like FinSubsequence-like V185() V186() V187() V188() Element of (card (Seg (n -' (dim A)))) -tuples_on NAT
(card (Seg (n -' (dim A)))) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = card (Seg (n -' (dim A))) } is set
(Sgm (Seg (n -' (dim A)))) " is Relation-like Function-like set
(Sgm AB) " is Relation-like Function-like set
(Sgm ((Seg n) \ AB)) " is Relation-like Function-like set
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card AB, the carrier of K
f is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card ((Seg n) \ AB), the carrier of K
BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[BX,MV] is set
{BX,MV} is non empty finite V37() set
{BX} is non empty trivial finite V37() 1 -element set
{{BX,MV},{BX}} is non empty finite V37() without_zero V103() set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm (Seg (n -' (dim A)))) ") . BX is set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm AB) ") . MV is set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
((Sgm ((Seg n) \ AB)) ") . MV is set
X * (lA,c₁₃) is Element of the carrier of K
f * (x,x) is Element of the carrier of K
[:(Seg (n -' (dim A))),AB:] \/ [:(Seg (n -' (dim A))),((Seg n) \ AB):] is Relation-like finite set
BX is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n -' (dim A),n, the carrier of K
Segm (BX,(Seg (n -' (dim A))),AB) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card AB, the carrier of K
Segm (BX,(Sgm (Seg (n -' (dim A)))),(Sgm AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card AB, the carrier of K
Segm (BX,(Seg (n -' (dim A))),((Seg n) \ AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card ((Seg n) \ AB), the carrier of K
Segm (BX,(Sgm (Seg (n -' (dim A)))),(Sgm ((Seg n) \ AB))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (n -' (dim A))), card ((Seg n) \ AB), the carrier of K
Indices BX is set
dom BX is finite Element of bool NAT
width BX is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width BX) is finite width BX -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BX ) } is set
[:(dom BX),(Seg (width BX)):] is Relation-like finite set
(Indices BX) \ ([:(Seg (n -' (dim A))),AB:] \/ [:(Seg (n -' (dim A))),((Seg n) \ AB):]) is Element of bool (Indices BX)
bool (Indices BX) is set
(Segm (BX,(Seg (n -' (dim A))),AB)) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
- ((Segm (BX,(Seg (n -' (dim A))),AB)) @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,BX) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width BX) -VectSp_over K
(width BX) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
MV is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of dim A,n, the carrier of K
Segm (MV,(Seg (dim A)),AB) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card AB, the carrier of K
Segm (MV,(Sgm (Seg (dim A))),(Sgm AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card AB, the carrier of K
Segm (MV,(Seg (dim A)),((Seg n) \ AB)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card ((Seg n) \ AB), the carrier of K
Segm (MV,(Sgm (Seg (dim A))),(Sgm ((Seg n) \ AB))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of card (Seg (dim A)), card ((Seg n) \ AB), the carrier of K
lines MV is finite Element of bool the carrier of (n -VectSp_over K)
Lin (lines MV) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of n -VectSp_over K
Indices BA is set
dom BA is finite Element of bool NAT
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width BA) is finite width BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width BA ) } is set
[:(dom BA),(Seg (width BA)):] is Relation-like finite set
[:(Seg (dim A)),(Seg n):] is Relation-like finite set
lA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
c₁₃ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[lA,c₁₃] is set
{lA,c₁₃} is non empty finite V37() set
{lA} is non empty trivial finite V37() 1 -element set
{{lA,c₁₃},{lA}} is non empty finite V37() without_zero V103() set
Indices MV is set
dom MV is finite Element of bool NAT
width MV is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width MV) is finite width MV -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width MV ) } is set
[:(dom MV),(Seg (width MV)):] is Relation-like finite set
rng (Sgm (Seg (dim A))) is finite V195() V196() V197() V200() set
dom (Sgm (Seg (dim A))) is finite card (Seg (dim A)) -element Element of bool NAT
x is set
(Sgm (Seg (dim A))) . x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
[:(Seg (dim A)),AB:] is Relation-like finite set
rng (Sgm AB) is finite V195() V196() V197() V200() set
dom (Sgm AB) is finite card AB -element Element of bool NAT
y is set
(Sgm AB) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[x,y] is Element of [:NAT,NAT:]
{x,y} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,y},{x}} is non empty finite V37() without_zero V103() set
Indices (Segm (MV,(Seg (dim A)),AB)) is set
dom (Segm (MV,(Seg (dim A)),AB)) is finite Element of bool NAT
width (Segm (MV,(Seg (dim A)),AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (MV,(Seg (dim A)),AB))) is finite width (Segm (MV,(Seg (dim A)),AB)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (MV,(Seg (dim A)),AB)) ) } is set
[:(dom (Segm (MV,(Seg (dim A)),AB))),(Seg (width (Segm (MV,(Seg (dim A)),AB)))):] is Relation-like finite set
Indices (Segm (BA,(Seg (dim A)),AB)) is set
dom (Segm (BA,(Seg (dim A)),AB)) is finite Element of bool NAT
width (Segm (BA,(Seg (dim A)),AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (BA,(Seg (dim A)),AB))) is finite width (Segm (BA,(Seg (dim A)),AB)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (BA,(Seg (dim A)),AB)) ) } is set
[:(dom (Segm (BA,(Seg (dim A)),AB))),(Seg (width (Segm (BA,(Seg (dim A)),AB)))):] is Relation-like finite set
BA * (lA,c₁₃) is Element of the carrier of K
(Segm (MV,(Seg (dim A)),AB)) * (x,y) is Element of the carrier of K
MV * (lA,c₁₃) is Element of the carrier of K
[:(Seg (dim A)),((Seg n) \ AB):] is Relation-like finite set
rng (Sgm ((Seg n) \ AB)) is finite V195() V196() V197() V200() set
dom (Sgm ((Seg n) \ AB)) is finite card ((Seg n) \ AB) -element Element of bool NAT
y is set
(Sgm ((Seg n) \ AB)) . y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
y is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
[x,y] is Element of [:NAT,NAT:]
{x,y} is non empty finite V37() set
{x} is non empty trivial finite V37() 1 -element set
{{x,y},{x}} is non empty finite V37() without_zero V103() set
Indices (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) is set
dom (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) is finite Element of bool NAT
Seg (width (Segm (BA,(Seg (dim A)),((Seg n) \ AB)))) is finite width (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (BA,(Seg (dim A)),((Seg n) \ AB))) ) } is set
[:(dom (Segm (BA,(Seg (dim A)),((Seg n) \ AB)))),(Seg (width (Segm (BA,(Seg (dim A)),((Seg n) \ AB))))):] is Relation-like finite set
Indices (Segm (MV,(Seg (dim A)),((Seg n) \ AB))) is set
dom (Segm (MV,(Seg (dim A)),((Seg n) \ AB))) is finite Element of bool NAT
width (Segm (MV,(Seg (dim A)),((Seg n) \ AB))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (MV,(Seg (dim A)),((Seg n) \ AB)))) is finite width (Segm (MV,(Seg (dim A)),((Seg n) \ AB))) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (MV,(Seg (dim A)),((Seg n) \ AB))) ) } is set
[:(dom (Segm (MV,(Seg (dim A)),((Seg n) \ AB)))),(Seg (width (Segm (MV,(Seg (dim A)),((Seg n) \ AB))))):] is Relation-like finite set
Indices ((Segm (BX,(Seg (n -' (dim A))),AB)) @) is set
dom ((Segm (BX,(Seg (n -' (dim A))),AB)) @) is finite Element of bool NAT
width ((Segm (BX,(Seg (n -' (dim A))),AB)) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width ((Segm (BX,(Seg (n -' (dim A))),AB)) @)) is finite width ((Segm (BX,(Seg (n -' (dim A))),AB)) @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((Segm (BX,(Seg (n -' (dim A))),AB)) @) ) } is set
[:(dom ((Segm (BX,(Seg (n -' (dim A))),AB)) @)),(Seg (width ((Segm (BX,(Seg (n -' (dim A))),AB)) @))):] is Relation-like finite set
[y,x] is Element of [:NAT,NAT:]
{y,x} is non empty finite V37() set
{y} is non empty trivial finite V37() 1 -element set
{{y,x},{y}} is non empty finite V37() without_zero V103() set
Indices (Segm (BX,(Seg (n -' (dim A))),AB)) is set
dom (Segm (BX,(Seg (n -' (dim A))),AB)) is finite Element of bool NAT
width (Segm (BX,(Seg (n -' (dim A))),AB)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Seg (width (Segm (BX,(Seg (n -' (dim A))),AB))) is finite width (Segm (BX,(Seg (n -' (dim A))),AB)) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width (Segm (BX,(Seg (n -' (dim A))),AB)) ) } is set
[:(dom (Segm (BX,(Seg (n -' (dim A))),AB))),(Seg (width (Segm (BX,(Seg (n -' (dim A))),AB)))):] is Relation-like finite set
Indices ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) is set
dom ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) is finite Element of bool NAT
Seg (width ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)) is finite width ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= width ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) ) } is set
[:(dom ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)),(Seg (width ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @))):] is Relation-like finite set
MV * (lA,c₁₃) is Element of the carrier of K
(- ((Segm (BX,(Seg (n -' (dim A))),AB)) @)) * (x,y) is Element of the carrier of K
((Segm (BX,(Seg (n -' (dim A))),AB)) @) * (x,y) is Element of the carrier of K
- (((Segm (BX,(Seg (n -' (dim A))),AB)) @) * (x,y)) is Element of the carrier of K
(- ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)) * (y,x) is Element of the carrier of K
- ((- ((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @)) * (y,x)) is Element of the carrier of K
((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) * (y,x) is Element of the carrier of K
- (((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) * (y,x)) is Element of the carrier of K
- (- (((Segm (BA,(Seg (dim A)),((Seg n) \ AB))) @) * (y,x))) is Element of the carrier of K
(Segm (BA,(Seg (dim A)),((Seg n) \ AB))) * (x,y) is Element of the carrier of K
BA * (lA,c₁₃) is Element of the carrier of K
BA * (lA,c₁₃) is Element of the carrier of K
MV * (lA,c₁₃) is Element of the carrier of K
BA * (lA,c₁₃) is Element of the carrier of K
MV * (lA,c₁₃) is Element of the carrier of K
the carrier of A is non empty set
bool the carrier of A is set
lA is Element of bool the carrier of A
Lin lA is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total V223( the carrier of A) V224( the carrier of A) Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is set
the ZeroF of A is Element of the carrier of A
the lmult of A is Relation-like [: the carrier of K, the carrier of A:] -defined the carrier of A -valued Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of A:], the carrier of A:]
[: the carrier of K, the carrier of A:] is Relation-like set
[:[: the carrier of K, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of K, the carrier of A:], the carrier of A:] is set
VectSpStr(# the carrier of A, the addF of A, the ZeroF of A, the lmult of A #) is non empty strict VectSpStr over K
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,A) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width A) -VectSp_over n
(width A) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,(K * A)) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (K * A)) -VectSp_over n
width (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (K * A)) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
len (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the carrier of (n,A) is non empty set
BA is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K, width A, the carrier of n
(n,BA) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width BA) -VectSp_over n
width BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width BA) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
the carrier of (n,BA) is non empty set
AB is set
0. n is zero Element of the carrier of n
(len A) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len A } is set
Seg (len A) is finite len A -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len A ) } is set
(Seg (len A)) --> (0. n) is Relation-like Seg (len A) -defined Seg (len A) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len A)),{(0. n)}:]
{(0. n)} is non empty trivial finite 1 -element set
[:(Seg (len A)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len A)),{(0. n)}:] is finite V37() set
(n,A,((len A) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width A) -VectSp_over n)
the carrier of ((width A) -VectSp_over n) is non empty set
bool the carrier of ((width A) -VectSp_over n) is set
( the carrier of n,((len A) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len A) |-> (0. n)),1, the carrier of n
len ((len A) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len A) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len A) |-> (0. n)), the carrier of n
[1,((len A) |-> (0. n))] is set
{1,((len A) |-> (0. n))} is non empty finite V37() set
{{1,((len A) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len A) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len A) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,A,( the carrier of n,((len A) |-> (0. n)))) is set
width ( the carrier of n,((len A) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width A & width b₁ = width ( the carrier of n,((len A) |-> (0. n))) & A * b₁ = ( the carrier of n,((len A) |-> (0. n))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,A,( the carrier of n,((len A) |-> (0. n)))) } is set
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
( the carrier of n,x) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len x,1, the carrier of n
len x is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*x*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len x, the carrier of n
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() without_zero V103() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*x*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len BA is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len BA) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like finite len BA -element FinSequence-like FinSubsequence-like Element of (len BA) -tuples_on the carrier of n
(len BA) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = len BA } is set
Seg (len BA) is finite len BA -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len BA ) } is set
(Seg (len BA)) --> (0. n) is Relation-like Seg (len BA) -defined Seg (len BA) -defined the carrier of n -valued {(0. n)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len BA)),{(0. n)}:]
[:(Seg (len BA)),{(0. n)}:] is Relation-like finite set
bool [:(Seg (len BA)),{(0. n)}:] is finite V37() set
( the carrier of n,((len BA) |-> (0. n))) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len BA) |-> (0. n)),1, the carrier of n
len ((len BA) |-> (0. n)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len BA) |-> (0. n))*> is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len BA) |-> (0. n)), the carrier of n
[1,((len BA) |-> (0. n))] is set
{1,((len BA) |-> (0. n))} is non empty finite V37() set
{{1,((len BA) |-> (0. n))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len BA) |-> (0. n))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len BA) |-> (0. n))*> @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
0. (n,(len K),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len K,1, the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b₁ = 1 } is set
1 |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
(Seg 1) --> (0. n) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of n -valued {(0. n)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. n)}:]
[:(Seg 1),{(0. n)}:] is Relation-like finite set
bool [:(Seg 1),{(0. n)}:] is finite V37() set
(len K) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len K -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len K) -tuples_on (1 -tuples_on the carrier of n)
(len K) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
(1 -tuples_on the carrier of n) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len K } is set
Seg (len K) is finite len K -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len K ) } is set
(Seg (len K)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len K) -defined Seg (len K) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len K)),{(1 |-> (0. n))}:]
{(1 |-> (0. n))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len K)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len K)),{(1 |-> (0. n))}:] is finite V37() set
0. (n,(len A),1) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len A,1, the carrier of n
(len A) |-> (1 |-> (0. n)) is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len A) -tuples_on (1 -tuples_on the carrier of n)
(len A) -tuples_on (1 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of n
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of n) * : len b₁ = len A } is set
(Seg (len A)) --> (1 |-> (0. n)) is Relation-like non-empty Seg (len A) -defined Seg (len A) -defined 1 -tuples_on the carrier of n -valued {(1 |-> (0. n))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len A)),{(1 |-> (0. n))}:]
[:(Seg (len A)),{(1 |-> (0. n))}:] is Relation-like finite set
bool [:(Seg (len A)),{(1 |-> (0. n))}:] is finite V37() set
(n,BA,( the carrier of n,((len BA) |-> (0. n)))) is set
width ( the carrier of n,((len BA) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width BA & width b₁ = width ( the carrier of n,((len BA) |-> (0. n))) & BA * b₁ = ( the carrier of n,((len BA) |-> (0. n))) ) } is set
width ( the carrier of n,((len BA) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * (A * f) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
BA * f is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(n,BA,( the carrier of n,((len BA) |-> (0. n)))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width BA & width b₁ = width ( the carrier of n,((len BA) |-> (0. n))) & BA * b₁ = ( the carrier of n,((len BA) |-> (0. n))) ) } is set
(n,BA,( the carrier of n,((len BA) |-> (0. n)))) is set
width ( the carrier of n,((len BA) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width BA & width b₁ = width ( the carrier of n,((len BA) |-> (0. n))) & BA * b₁ = ( the carrier of n,((len BA) |-> (0. n))) ) } is set
(n,BA,( the carrier of n,((len BA) |-> (0. n)))) is set
width ( the carrier of n,((len BA) |-> (0. n))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n * : ( len b₁ = width BA & width b₁ = width ( the carrier of n,((len BA) |-> (0. n))) & BA * b₁ = ( the carrier of n,((len BA) |-> (0. n))) ) } is set
(n,BA,((len BA) |-> (0. n))) is linearly-closed Element of bool the carrier of ((width BA) -VectSp_over n)
the carrier of ((width BA) -VectSp_over n) is non empty set
bool the carrier of ((width BA) -VectSp_over n) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n : ( the carrier of n,b₁) in (n,BA,( the carrier of n,((len BA) |-> (0. n)))) } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of n is non empty non trivial V103() set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
the_rank_of (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (K * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
K @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
len (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
(A @) * (K @) is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
width (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len K is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,(K @)) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (K @)) -VectSp_over n
(width (K @)) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
dim (n,(K @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
the_rank_of (K @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (K @)) - (the_rank_of (K @)) is V105() ext-real complex set
width (A @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(n,((A @) * (K @))) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width ((A @) * (K @))) -VectSp_over n
width ((A @) * (K @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width ((A @) * (K @))) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
dim (n,((A @) * (K @))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
the_rank_of ((A @) * (K @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width ((A @) * (K @))) - (the_rank_of ((A @) * (K @))) is V105() ext-real complex set
(width (K @)) - (the_rank_of ((A @) * (K @))) is V105() ext-real complex set
(n,A) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width A) -VectSp_over n
(width A) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
(n,(K * A)) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (K * A)) -VectSp_over n
(width (K * A)) -VectSp_over n is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over n
dim (n,A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width A) - (the_rank_of A) is V105() ext-real complex set
dim (n,(K * A)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width (K * A)) - (the_rank_of (K * A)) is V105() ext-real complex set
(width A) - (the_rank_of (K * A)) is V105() ext-real complex set
(K * A) @ is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of n *
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
Det A is Element of the carrier of K
Permutations n is set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total quasi_total V223( the carrier of K) V224( 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
FinOmega (Permutations n) is Element of K96((Permutations n))
K96((Permutations n)) is V24() set
Path_product A is Relation-like Permutations n -defined the carrier of K -valued Function-like total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is set
K103((Permutations n), the carrier of K, the addF of K,(FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(K,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width B) -VectSp_over K
(width B) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,(A * B)) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (A * B)) -VectSp_over K
width (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width (A * B)) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
len (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n, width B, the carrier of K
(K,AB) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width AB) -VectSp_over K
width AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(width AB) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of (K,AB) is non empty set
the carrier of (K,B) is non empty set
A ~ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
(A ~) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
len (A ~) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
x is set
len AB is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(len AB) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len AB -element FinSequence-like FinSubsequence-like Element of (len AB) -tuples_on the carrier of K
(len AB) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len AB } is set
Seg (len AB) is finite len AB -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len AB ) } is set
(Seg (len AB)) --> (0. K) is Relation-like Seg (len AB) -defined Seg (len AB) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len AB)),{(0. K)}:]
{(0. K)} is non empty trivial finite 1 -element set
[:(Seg (len AB)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (len AB)),{(0. K)}:] is finite V37() set
(K,AB,((len AB) |-> (0. K))) is linearly-closed Element of bool the carrier of ((width AB) -VectSp_over K)
the carrier of ((width AB) -VectSp_over K) is non empty set
bool the carrier of ((width AB) -VectSp_over K) is set
( the carrier of K,((len AB) |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len AB) |-> (0. K)),1, the carrier of K
len ((len AB) |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len AB) |-> (0. K))*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len AB) |-> (0. K)), the carrier of K
[1,((len AB) |-> (0. K))] is set
{1,((len AB) |-> (0. K))} is non empty finite V37() set
{{1,((len AB) |-> (0. K))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len AB) |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len AB) |-> (0. K))*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,AB,( the carrier of K,((len AB) |-> (0. K)))) is set
width ( the carrier of K,((len AB) |-> (0. K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width AB & width b₁ = width ( the carrier of K,((len AB) |-> (0. K))) & AB * b₁ = ( the carrier of K,((len AB) |-> (0. K))) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K : ( the carrier of K,b₁) in (K,AB,( the carrier of K,((len AB) |-> (0. K)))) } is set
f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
( the carrier of K,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len f,1, the carrier of K
len f is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*f*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len f, the carrier of K
[1,f] is set
{1,f} is non empty finite V37() set
{{1,f},{1}} is non empty finite V37() without_zero V103() set
{[1,f]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*f*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width X is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
AB * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width (A ~) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len (B * X) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(1. (K,n)) * (B * X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
A * (B * X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(A ~) * (A * (B * X)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(A ~) * ( the carrier of K,((len AB) |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
0. (K,(len AB),1) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len AB,1, the carrier of K
1 -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = 1 } is set
1 |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of K
(Seg 1) --> (0. K) is Relation-like Seg 1 -defined Seg 1 -defined the carrier of K -valued {(0. K)} -valued Function-like constant non empty total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg 1),{(0. K)}:]
[:(Seg 1),{(0. K)}:] is Relation-like finite set
bool [:(Seg 1),{(0. K)}:] is finite V37() set
(len AB) |-> (1 |-> (0. K)) is Relation-like NAT -defined 1 -tuples_on the carrier of K -valued Function-like finite len AB -element FinSequence-like FinSubsequence-like Function-yielding V147() Element of (len AB) -tuples_on (1 -tuples_on the carrier of K)
(len AB) -tuples_on (1 -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of K
(1 -tuples_on the carrier of K) * is functional non empty FinSequence-membered FinSequenceSet of 1 -tuples_on the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined 1 -tuples_on the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of (1 -tuples_on the carrier of K) * : len b₁ = len AB } is set
(Seg (len AB)) --> (1 |-> (0. K)) is Relation-like non-empty Seg (len AB) -defined Seg (len AB) -defined 1 -tuples_on the carrier of K -valued {(1 |-> (0. K))} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Function-yielding V147() Element of bool [:(Seg (len AB)),{(1 |-> (0. K))}:]
{(1 |-> (0. K))} is functional non empty trivial finite V37() 1 -element without_zero V103() set
[:(Seg (len AB)),{(1 |-> (0. K))}:] is Relation-like finite set
bool [:(Seg (len AB)),{(1 |-> (0. K))}:] is finite V37() set
(A ~) * (0. (K,(len AB),1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(len B) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of K
(len B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len B } is set
Seg (len B) is finite len B -element without_zero Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
(Seg (len B)) --> (0. K) is Relation-like Seg (len B) -defined Seg (len B) -defined the carrier of K -valued {(0. K)} -valued Function-like constant total total quasi_total finite FinSequence-like FinSubsequence-like Element of bool [:(Seg (len B)),{(0. K)}:]
[:(Seg (len B)),{(0. K)}:] is Relation-like finite set
bool [:(Seg (len B)),{(0. K)}:] is finite V37() set
( the carrier of K,((len B) |-> (0. K))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of len ((len B) |-> (0. K)),1, the carrier of K
len ((len B) |-> (0. K)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
<*((len B) |-> (0. K))*> is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of 1, len ((len B) |-> (0. K)), the carrier of K
[1,((len B) |-> (0. K))] is set
{1,((len B) |-> (0. K))} is non empty finite V37() set
{{1,((len B) |-> (0. K))},{1}} is non empty finite V37() without_zero V103() set
{[1,((len B) |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*((len B) |-> (0. K))*> @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
(K,B,( the carrier of K,((len B) |-> (0. K)))) is set
width ( the carrier of K,((len B) |-> (0. K))) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K * : ( len b₁ = width B & width b₁ = width ( the carrier of K,((len B) |-> (0. K))) & B * b₁ = ( the carrier of K,((len B) |-> (0. K))) ) } is set
(K,B,((len B) |-> (0. K))) is linearly-closed Element of bool the carrier of ((width B) -VectSp_over K)
the carrier of ((width B) -VectSp_over K) is non empty set
bool the carrier of ((width B) -VectSp_over K) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K : ( the carrier of K,b₁) in (K,B,( the carrier of K,((len B) |-> (0. K)))) } is set
the carrier of (K,(A * B)) is non empty set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Det A is Element of the carrier of K
Permutations n is set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total quasi_total V223( the carrier of K) V224( 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
FinOmega (Permutations n) is Element of K96((Permutations n))
K96((Permutations n)) is V24() set
Path_product A is Relation-like Permutations n -defined the carrier of K -valued Function-like total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is set
K103((Permutations n), the carrier of K, the addF of K,(FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
the_rank_of (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (A * B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(K,B) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width B) -VectSp_over K
(width B) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(K,(A * B)) is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (A * B)) -VectSp_over K
(width (A * B)) -VectSp_over K is non empty right_complementable V95() strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim (K,B) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
(width B) - (the_rank_of B) is V105() ext-real complex set
(width B) - (the_rank_of (A * B)) is V105() ext-real complex set
n is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital V139() V141() right-distributive left-distributive right_unital well-unital V153() left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial V103() set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is zero Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
len A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Det A is Element of the carrier of K
Permutations n is set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like total quasi_total V223( the carrier of K) V224( 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
FinOmega (Permutations n) is Element of K96((Permutations n))
K96((Permutations n)) is V24() set
Path_product A is Relation-like Permutations n -defined the carrier of K -valued Function-like total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is set
K103((Permutations n), the carrier of K, the addF of K,(FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
width B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
the_rank_of (B * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len (B * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
len B is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width (B * A) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
width A is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
A @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding Matrix of n,n, the carrier of K
width (A @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
B @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
len (B @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
Det (A @) is Element of the carrier of K
Path_product (A @) is Relation-like Permutations n -defined the carrier of K -valued Function-like total quasi_total Element of bool [:(Permutations n), the carrier of K:]
K103((Permutations n), the carrier of K, the addF of K,(FinOmega (Permutations n)),(Path_product (A @))) is Element of the carrier of K
(B * A) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
the_rank_of ((B * A) @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
(A @) * (B @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular FinSequence-yielding FinSequence of the carrier of K *
the_rank_of ((A @) * (B @)) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT
the_rank_of (B @) is V26() V27() V28() V32() finite cardinal V105() ext-real non negative complex Element of NAT