:: MATRLIN2 semantic presentation

REAL is set
NAT is non empty non trivial V21() V22() V23() non finite cardinal limit_cardinal Element of bool REAL
bool REAL is set
K583() is strict doubleLoopStr
the carrier of K583() is set
NAT is non empty non trivial V21() V22() V23() non finite cardinal limit_cardinal set
bool NAT is non trivial non finite set
bool NAT is non trivial non finite set
COMPLEX is set
RAT is set
INT is set
{} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V21() V22() V23() V25() V26() V27() finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V93() V94() set
2 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V21() V22() V23() V25() V26() V27() finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V93() V94() Element of NAT
K521(0,1,2) is finite set
[:K521(0,1,2),K521(0,1,2):] is Relation-like finite set
[:[:K521(0,1,2),K521(0,1,2):],K521(0,1,2):] is Relation-like finite set
bool [:[:K521(0,1,2),K521(0,1,2):],K521(0,1,2):] is finite V32() set
bool [:K521(0,1,2),K521(0,1,2):] is finite V32() set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is set
{{},1} is non empty finite V32() set
K670() is set
bool K670() is set
K671() is Element of bool K670()
3 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
card {} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V21() V22() V23() V25() V26() V27() finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V93() V94() set
Seg 1 is non empty trivial finite 1 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V32() 1 -element set
Seg 2 is non empty finite 2 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V32() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
f is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 /\ V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 + V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
g is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of f
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of f
g /\ A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of f
g + A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of f
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
the carrier of A is non empty set
the carrier of (V2 /\ V3) is non empty set
AI is set
AI is set
the carrier of (V2 + V3) is non empty set
AI is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
the carrier of AI is non empty set
S is set
the carrier of V1 is non empty set
KER is Element of the carrier of V1
MAI is Element of the carrier of V1
KER + MAI is Element of the carrier of V1
MK is Element of the carrier of AI
x is Element of the carrier of AI
MK + x is Element of the carrier of AI
S is set
the carrier of f is non empty set
KER is Element of the carrier of f
MAI is Element of the carrier of f
KER + MAI is Element of the carrier of f
the carrier of V1 is non empty set
MK is Element of the carrier of V1
x is Element of the carrier of V1
MK + x is Element of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 /\ V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
the carrier of V2 is non empty set
bool the carrier of V2 is set
the carrier of V3 is non empty set
bool the carrier of V3 is set
V2 + V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
the carrier of (V2 + V3) is non empty set
bool the carrier of (V2 + V3) is set
A is linearly-independent Element of bool the carrier of V2
A is linearly-independent Element of bool the carrier of V3
A \/ A is set
AI is Element of bool the carrier of (V2 + V3)
the carrier of K is non empty non trivial set
MAI is Relation-like Function-like total quasi_total Linear_Combination of AI
Sum MAI is Element of the carrier of (V2 + V3)
0. (V2 + V3) is V47(V2 + V3) Element of the carrier of (V2 + V3)
the ZeroF of (V2 + V3) is Element of the carrier of (V2 + V3)
Carrier MAI is finite Element of bool the carrier of (V2 + V3)
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not MAI . b₁ = 0. K } is set
(0. (V2 + V3)) + (0. (V2 + V3)) is Element of the carrier of (V2 + V3)
(Carrier MAI) /\ A is finite Element of bool the carrier of V2
(Carrier MAI) \ A is finite Element of bool the carrier of (V2 + V3)
w is finite Element of bool the carrier of (V2 + V3)
KER is Element of bool the carrier of (V2 + V3)
W is set
S is Element of bool the carrier of (V2 + V3)
(0). (V2 + V3) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
f is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
g is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
f /\ g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
f + g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
S is Element of bool the carrier of (V2 + V3)
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
0. V3 is V47(V3) Element of the carrier of V3
the ZeroF of V3 is Element of the carrier of V3
W is set
LLw is Element of the carrier of (V2 + V3)
MAI . LLw is Element of the carrier of K
MAI . W is set
[: the carrier of (V2 + V3), the carrier of K:] is Relation-like set
bool [: the carrier of (V2 + V3), the carrier of K:] is set
W is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of (V2 + V3), the carrier of K:]
LLw is set
i is Element of the carrier of (V2 + V3)
MAI . i is Element of the carrier of K
MAI . LLw is set
LLw is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of (V2 + V3), the carrier of K:]
Funcs ( the carrier of (V2 + V3), the carrier of K) is functional non empty FUNCTION_DOMAIN of the carrier of (V2 + V3), the carrier of K
i is Relation-like Function-like total quasi_total Element of Funcs ( the carrier of (V2 + V3), the carrier of K)
A12 is Element of the carrier of (V2 + V3)
i . A12 is Element of the carrier of K
A12 is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
Carrier A12 is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not A12 . b₁ = 0. K } is set
i is set
fb is Element of the carrier of (V2 + V3)
A12 . fb is Element of the carrier of K
x is finite Element of bool the carrier of (V2 + V3)
fb is Relation-like Function-like total quasi_total Element of Funcs ( the carrier of (V2 + V3), the carrier of K)
fbi is Element of the carrier of (V2 + V3)
fb . fbi is Element of the carrier of K
fbi is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
Carrier fbi is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not fbi . b₁ = 0. K } is set
b2n is set
v is Element of the carrier of (V2 + V3)
fbi . v is Element of the carrier of K
the carrier of f is non empty set
b2n is Relation-like Function-like total quasi_total Linear_Combination of S
Carrier b2n is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not b2n . b₁ = 0. K } is set
Sum b2n is Element of the carrier of (V2 + V3)
v is Relation-like Function-like total quasi_total Linear_Combination of f
Carrier v is finite Element of bool the carrier of f
bool the carrier of f is set
{ b₁ where b₁ is Element of the carrier of f : not v . b₁ = 0. K } is set
Sum v is Element of the carrier of f
i is Relation-like Function-like total quasi_total Linear_Combination of KER
b2n + i is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
v is Element of the carrier of (V2 + V3)
MAI . v is Element of the carrier of K
(b2n + i) . v is Element of the carrier of K
b2n . v is Element of the carrier of K
i . v is Element of the carrier of K
(b2n . v) + (i . v) is Element of the carrier of K
(MAI . v) + (i . v) is Element of the carrier of K
(MAI . v) + (0. K) is Element of the carrier of K
b2n . v is Element of the carrier of K
i . v is Element of the carrier of K
(b2n . v) + (i . v) is Element of the carrier of K
(i . v) + (0. K) is Element of the carrier of K
b2n . v is Element of the carrier of K
i . v is Element of the carrier of K
(b2n . v) + (i . v) is Element of the carrier of K
(0. K) + (i . v) is Element of the carrier of K
(0. K) + (0. K) is Element of the carrier of K
Sum i is Element of the carrier of (V2 + V3)
(Sum b2n) + (Sum i) is Element of the carrier of (V2 + V3)
Carrier i is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not i . b₁ = 0. K } is set
the carrier of g is non empty set
v is Relation-like Function-like total quasi_total Linear_Combination of g
Carrier v is finite Element of bool the carrier of g
bool the carrier of g is set
{ b₁ where b₁ is Element of the carrier of g : not v . b₁ = 0. K } is set
Sum v is Element of the carrier of g
0. f is V47(f) Element of the carrier of f
the ZeroF of f is Element of the carrier of f
0. g is V47(g) Element of the carrier of g
the ZeroF of g is Element of the carrier of g
{} \/ {} is Relation-like finite V32() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 /\ V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V2 + V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
f is Basis of V2
g is Basis of V3
f \/ g is set
the carrier of V3 is non empty set
the carrier of (V2 + V3) is non empty set
the carrier of V2 is non empty set
bool the carrier of (V2 + V3) is set
(Omega). V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V3
the addF of V3 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V3, the carrier of V3:], the carrier of V3:]
[: the carrier of V3, the carrier of V3:] is Relation-like set
[:[: the carrier of V3, the carrier of V3:], the carrier of V3:] is Relation-like set
bool [:[: the carrier of V3, the carrier of V3:], the carrier of V3:] is set
the ZeroF of V3 is Element of the carrier of V3
the lmult of V3 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:]
the carrier of K is non empty non trivial set
[: the carrier of K, the carrier of V3:] is Relation-like set
[:[: the carrier of K, the carrier of V3:], the carrier of V3:] is Relation-like set
bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:] is set
VectSpStr(# the carrier of V3, the addF of V3, the ZeroF of V3, the lmult of V3 #) is non empty strict VectSpStr over K
Lin g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V3
AI is Element of bool the carrier of (V2 + V3)
Lin AI is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
A is Element of bool the carrier of (V2 + V3)
Lin A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
A is Element of bool the carrier of (V2 + V3)
Lin A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
(Lin A) + (Lin AI) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2 + V3
(Omega). V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2
the addF of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:]
[: the carrier of V2, the carrier of V2:] is Relation-like set
[:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is set
the ZeroF of V2 is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
VectSpStr(# the carrier of V2, the addF of V2, the ZeroF of V2, the lmult of V2 #) is non empty strict VectSpStr over K
Lin f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V2
the carrier of (Lin A) is non empty set
S is set
KER is Element of the carrier of (V2 + V3)
the carrier of V1 is non empty set
MAI is Element of the carrier of V1
MK is Element of the carrier of V1
MAI + MK is Element of the carrier of V1
x is Element of the carrier of (V2 + V3)
v is Element of the carrier of (V2 + V3)
x + v is Element of the carrier of (V2 + V3)
the carrier of (Lin AI) is non empty set
the carrier of (Lin A) is non empty set
the addF of (V2 + V3) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of (V2 + V3), the carrier of (V2 + V3):], the carrier of (V2 + V3):]
[: the carrier of (V2 + V3), the carrier of (V2 + V3):] is Relation-like set
[:[: the carrier of (V2 + V3), the carrier of (V2 + V3):], the carrier of (V2 + V3):] is Relation-like set
bool [:[: the carrier of (V2 + V3), the carrier of (V2 + V3):], the carrier of (V2 + V3):] is set
the ZeroF of (V2 + V3) is Element of the carrier of (V2 + V3)
the lmult of (V2 + V3) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of (V2 + V3):], the carrier of (V2 + V3):]
[: the carrier of K, the carrier of (V2 + V3):] is Relation-like set
[:[: the carrier of K, the carrier of (V2 + V3):], the carrier of (V2 + V3):] is Relation-like set
bool [:[: the carrier of K, the carrier of (V2 + V3):], the carrier of (V2 + V3):] is set
VectSpStr(# the carrier of (V2 + V3), the addF of (V2 + V3), the ZeroF of (V2 + V3), the lmult of (V2 + V3) #) is non empty strict VectSpStr over K
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(Omega). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of V1 is non empty set
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the ZeroF of V1 is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
the carrier of K is non empty non trivial set
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) is non empty strict VectSpStr over K
the carrier of ((Omega). V1) is non empty set
V2 is Relation-like NAT -defined the carrier of ((Omega). V1) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of (Omega). V1
rng V2 is finite set
V3 is Basis of (Omega). V1
bool the carrier of V1 is set
f is linearly-independent Element of bool the carrier of V1
Lin f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
Lin V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (Omega). V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
bool the carrier of V1 is set
V2 is finite Element of bool the carrier of V1
Lin V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
dim (Lin V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
card V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
the carrier of (Lin V2) is non empty set
f is set
bool the carrier of (Lin V2) is set
f is Element of bool the carrier of (Lin V2)
Lin f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of Lin V2
g is Element of bool the carrier of (Lin V2)
Lin g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of Lin V2
A is finite Element of bool the carrier of (Lin V2)
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
card A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
bool the carrier of V1 is set
V2 is finite Element of bool the carrier of V1
Lin V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
dim (Lin V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
card V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
the carrier of (Lin V2) is non empty set
f is set
bool the carrier of (Lin V2) is set
f is Element of bool the carrier of (Lin V2)
Lin f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of Lin V2
g is Element of bool the carrier of (Lin V2)
Lin g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of Lin V2
A is finite Element of bool the carrier of (Lin V2)
card A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom V2 is finite Element of bool NAT
V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt (V3,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (V3,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom (lmlt (V3,V2)) is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom V3) /\ (dom V2) is finite Element of bool NAT
rng V3 is finite set
rng V2 is finite set
[:(rng V3),(rng V2):] is Relation-like finite set
dom the lmult of V1 is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom V1 is finite Element of bool NAT
V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V1 + V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: (V1,V2) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom (V1 + V2) is finite Element of bool NAT
dom V2 is finite Element of bool NAT
(dom V1) /\ (dom V2) is finite Element of bool NAT
rng V1 is finite set
rng V2 is finite set
[:(rng V1),(rng V2):] is Relation-like finite set
dom the addF of K is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom V2 is finite Element of bool NAT
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
V2 + V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the addF of V1 .: (V2,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom (V2 + V3) is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom V2) /\ (dom V3) is finite Element of bool NAT
rng V2 is finite set
rng V3 is finite set
[:(rng V2),(rng V3):] is Relation-like finite set
dom the addF of V1 is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt (V3,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (V3,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V3 + 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 addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: (V3,f) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt ((V3 + f),V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: ((V3 + f),V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
lmlt (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
(lmlt (V3,V2)) + (lmlt (f,V2)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the addF of V1 .: ((lmlt (V3,V2)),(lmlt (f,V2))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom ((lmlt (V3,V2)) + (lmlt (f,V2))) is finite Element of bool NAT
dom (lmlt (V3,V2)) is finite Element of bool NAT
dom (lmlt (f,V2)) is finite Element of bool NAT
(dom (lmlt (V3,V2))) /\ (dom (lmlt (f,V2))) is finite Element of bool NAT
dom (lmlt ((V3 + f),V2)) is finite Element of bool NAT
dom (V3 + f) is finite Element of bool NAT
dom V2 is finite Element of bool NAT
(dom (V3 + f)) /\ (dom V2) is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom V3) /\ (dom V2) is finite Element of bool NAT
dom f is finite Element of bool NAT
(dom f) /\ (dom V2) is finite Element of bool NAT
((dom V3) /\ (dom V2)) /\ (dom f) is finite Element of bool NAT
(((dom V3) /\ (dom V2)) /\ (dom f)) /\ (dom V2) is finite Element of bool NAT
(dom V3) /\ (dom f) is finite Element of bool NAT
((dom V3) /\ (dom f)) /\ (dom V2) is finite Element of bool NAT
(((dom V3) /\ (dom f)) /\ (dom V2)) /\ (dom V2) is finite Element of bool NAT
(dom V2) /\ (dom V2) is finite Element of bool NAT
((dom V3) /\ (dom f)) /\ ((dom V2) /\ (dom V2)) is finite Element of bool NAT
AI is set
(lmlt (f,V2)) /. AI is Element of the carrier of V1
(lmlt (f,V2)) . AI is set
f /. AI is Element of the carrier of K
f . AI is set
(V3 + f) . AI is set
(V3 + f) /. AI is Element of the carrier of K
V3 /. AI is Element of the carrier of K
V3 . AI is set
V2 /. AI is Element of the carrier of V1
V2 . AI is set
(lmlt (V3,V2)) /. AI is Element of the carrier of V1
(lmlt (V3,V2)) . AI is set
((lmlt (V3,V2)) + (lmlt (f,V2))) . AI is set
((lmlt (V3,V2)) /. AI) + ((lmlt (f,V2)) /. AI) is Element of the carrier of V1
the lmult of V1 . ((V3 /. AI),(V2 /. AI)) is Element of the carrier of V1
( the lmult of V1 . ((V3 /. AI),(V2 /. AI))) + ((lmlt (f,V2)) /. AI) is Element of the carrier of V1
(V3 /. AI) * (V2 /. AI) is Element of the carrier of V1
(f /. AI) * (V2 /. AI) is Element of the carrier of V1
the lmult of V1 . ((f /. AI),(V2 /. AI)) is Element of the carrier of V1
((V3 /. AI) * (V2 /. AI)) + ((f /. AI) * (V2 /. AI)) is Element of the carrier of V1
(V3 /. AI) + (f /. AI) is Element of the carrier of K
((V3 /. AI) + (f /. AI)) * (V2 /. AI) is Element of the carrier of V1
the lmult of V1 . (((V3 /. AI) + (f /. AI)),(V2 /. AI)) is Element of the carrier of V1
((V3 + f) /. AI) * (V2 /. AI) is Element of the carrier of V1
the lmult of V1 . (((V3 + f) /. AI),(V2 /. AI)) is Element of the carrier of V1
(lmlt ((V3 + f),V2)) . AI is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
V2 + V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the addF of V1 .: (V2,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt (f,(V2 + V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (f,(V2 + V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
lmlt (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
lmlt (f,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: (f,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
(lmlt (f,V2)) + (lmlt (f,V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the addF of V1 .: ((lmlt (f,V2)),(lmlt (f,V3))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
dom ((lmlt (f,V2)) + (lmlt (f,V3))) is finite Element of bool NAT
dom (lmlt (f,V2)) is finite Element of bool NAT
dom (lmlt (f,V3)) is finite Element of bool NAT
(dom (lmlt (f,V2))) /\ (dom (lmlt (f,V3))) is finite Element of bool NAT
dom (lmlt (f,(V2 + V3))) is finite Element of bool NAT
dom f is finite Element of bool NAT
dom (V2 + V3) is finite Element of bool NAT
(dom f) /\ (dom (V2 + V3)) is finite Element of bool NAT
dom V2 is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom V2) /\ (dom V3) is finite Element of bool NAT
(dom f) /\ (dom V2) is finite Element of bool NAT
(dom f) /\ (dom V3) is finite Element of bool NAT
((dom f) /\ (dom V2)) /\ (dom f) is finite Element of bool NAT
(((dom f) /\ (dom V2)) /\ (dom f)) /\ (dom V3) is finite Element of bool NAT
(dom f) /\ (dom f) is finite Element of bool NAT
((dom f) /\ (dom f)) /\ (dom V2) is finite Element of bool NAT
(((dom f) /\ (dom f)) /\ (dom V2)) /\ (dom V3) is finite Element of bool NAT
AI is set
(lmlt (f,V3)) /. AI is Element of the carrier of V1
(lmlt (f,V3)) . AI is set
V3 /. AI is Element of the carrier of V1
V3 . AI is set
(V2 + V3) . AI is set
(V2 + V3) /. AI is Element of the carrier of V1
f /. AI is Element of the carrier of K
f . AI is set
V2 /. AI is Element of the carrier of V1
V2 . AI is set
(lmlt (f,V2)) /. AI is Element of the carrier of V1
(lmlt (f,V2)) . AI is set
((lmlt (f,V2)) + (lmlt (f,V3))) . AI is set
((lmlt (f,V2)) /. AI) + ((lmlt (f,V3)) /. AI) is Element of the carrier of V1
the lmult of V1 . ((f /. AI),(V2 /. AI)) is Element of the carrier of V1
( the lmult of V1 . ((f /. AI),(V2 /. AI))) + ((lmlt (f,V3)) /. AI) is Element of the carrier of V1
(f /. AI) * (V2 /. AI) is Element of the carrier of V1
(f /. AI) * (V3 /. AI) is Element of the carrier of V1
the lmult of V1 . ((f /. AI),(V3 /. AI)) is Element of the carrier of V1
((f /. AI) * (V2 /. AI)) + ((f /. AI) * (V3 /. AI)) is Element of the carrier of V1
(V2 /. AI) + (V3 /. AI) is Element of the carrier of V1
(f /. AI) * ((V2 /. AI) + (V3 /. AI)) is Element of the carrier of V1
the lmult of V1 . ((f /. AI),((V2 /. AI) + (V3 /. AI))) is Element of the carrier of V1
(f /. AI) * ((V2 + V3) /. AI) is Element of the carrier of V1
the lmult of V1 . ((f /. AI),((V2 + V3) /. AI)) is Element of the carrier of V1
(lmlt (f,(V2 + V3))) . AI is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V2 ^ V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
lmlt (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (f,V2) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f ^ g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt ((f ^ g),(V2 ^ V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: ((f ^ g),(V2 ^ V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
lmlt (g,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 .: (g,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
(lmlt (f,V2)) ^ (lmlt (g,V3)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
(len g) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
(len f) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
(len f) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(len g) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
AI is Relation-like NAT -defined the carrier of K -valued Function-like finite len f -element FinSequence-like FinSubsequence-like Element of (len f) -tuples_on the carrier of K
A is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len f -element FinSequence-like FinSubsequence-like Element of (len f) -tuples_on the carrier of V1
the lmult of V1 .: (AI,A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
S is Relation-like NAT -defined the carrier of K -valued Function-like finite len g -element FinSequence-like FinSubsequence-like Element of (len g) -tuples_on the carrier of K
A is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len g -element FinSequence-like FinSubsequence-like Element of (len g) -tuples_on the carrier of V1
the lmult of V1 .: (S,A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
( the lmult of V1 .: (AI,A)) ^ ( the lmult of V1 .: (S,A)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum V2 is Element of the carrier of V1
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V2 + V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the addF of V1 .: (V2,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (V2 + V3) is Element of the carrier of V1
Sum V3 is Element of the carrier of V1
(Sum V2) + (Sum V3) is Element of the carrier of V1
(len V2) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of V1
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of V1
f + g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of V1
the addF of V1 .: (f,g) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (f + g) is Element of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is Element of the carrier of K
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) |-> V1 is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
(len V3) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt (((len V3) |-> V1),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: (((len V3) |-> V1),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (((len V3) |-> V1),V3)) is Element of the carrier of V2
Sum V3 is Element of the carrier of V2
V1 * (Sum V3) is Element of the carrier of V2
the lmult of V2 . (V1,(Sum V3)) is Element of the carrier of V2
f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
f + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum A is Element of the carrier of V2
A is Element of the carrier of K
(len A) |-> A 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
lmlt (((len A) |-> A),A) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (((len A) |-> A),A) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (((len A) |-> A),A)) is Element of the carrier of V2
A * (Sum A) is Element of the carrier of V2
the lmult of V2 . (A,(Sum A)) is Element of the carrier of V2
A | f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Seg f is finite f -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
A | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of V2 -valued Function-like finite FinSubsequence-like set
len (A | f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (A | f) is finite Element of bool NAT
dom A is finite Element of bool NAT
A /. (f + 1) is Element of the carrier of V2
A . (f + 1) is set
<*A*> is Relation-like NAT -defined the carrier of K -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
[1,A] is set
{1,A} is non empty finite set
{{1,A},{1}} is non empty finite V32() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(A /. (f + 1))*> is Relation-like NAT -defined the carrier of V2 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
[1,(A /. (f + 1))] is set
{1,(A /. (f + 1))} is non empty finite set
{{1,(A /. (f + 1))},{1}} is non empty finite V32() set
{[1,(A /. (f + 1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
lmlt (<*A*>,<*(A /. (f + 1))*>) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (<*A*>,<*(A /. (f + 1))*>) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
A * (A /. (f + 1)) is Element of the carrier of V2
the lmult of V2 . (A,(A /. (f + 1))) is Element of the carrier of V2
<*(A * (A /. (f + 1)))*> is Relation-like NAT -defined the carrier of V2 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
[1,(A * (A /. (f + 1)))] is set
{1,(A * (A /. (f + 1)))} is non empty finite set
{{1,(A * (A /. (f + 1)))},{1}} is non empty finite V32() set
{[1,(A * (A /. (f + 1)))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len <*A*> is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
<*(A . (f + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A . (f + 1))] is set
{1,(A . (f + 1))} is non empty finite set
{{1,(A . (f + 1))},{1}} is non empty finite V32() set
{[1,(A . (f + 1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len <*(A . (f + 1))*> is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
(f + 1) |-> A is Relation-like NAT -defined the carrier of K -valued Function-like finite f + 1 -element FinSequence-like FinSubsequence-like Element of (f + 1) -tuples_on the carrier of K
(f + 1) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
f |-> A is Relation-like NAT -defined the carrier of K -valued Function-like finite f -element FinSequence-like FinSubsequence-like Element of f -tuples_on the carrier of K
f -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(f |-> A) ^ <*A*> is Relation-like NAT -defined the carrier of K -valued Function-like non empty finite f + 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (f |-> A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(A | f) ^ <*(A . (f + 1))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
lmlt ((f |-> A),(A | f)) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((f |-> A),(A | f)) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
(lmlt ((f |-> A),(A | f))) ^ (lmlt (<*A*>,<*(A /. (f + 1))*>)) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum ((lmlt ((f |-> A),(A | f))) ^ (lmlt (<*A*>,<*(A /. (f + 1))*>))) is Element of the carrier of V2
Sum (lmlt ((f |-> A),(A | f))) is Element of the carrier of V2
Sum (lmlt (<*A*>,<*(A /. (f + 1))*>)) is Element of the carrier of V2
(Sum (lmlt ((f |-> A),(A | f)))) + (Sum (lmlt (<*A*>,<*(A /. (f + 1))*>))) is Element of the carrier of V2
Sum (A | f) is Element of the carrier of V2
A * (Sum (A | f)) is Element of the carrier of V2
the lmult of V2 . (A,(Sum (A | f))) is Element of the carrier of V2
Sum <*(A * (A /. (f + 1)))*> is Element of the carrier of V2
(A * (Sum (A | f))) + (Sum <*(A * (A /. (f + 1)))*>) is Element of the carrier of V2
(A * (Sum (A | f))) + (A * (A /. (f + 1))) is Element of the carrier of V2
(Sum (A | f)) + (A /. (f + 1)) is Element of the carrier of V2
A * ((Sum (A | f)) + (A /. (f + 1))) is Element of the carrier of V2
the lmult of V2 . (A,((Sum (A | f)) + (A /. (f + 1)))) is Element of the carrier of V2
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum f is Element of the carrier of V2
g is Element of the carrier of K
(len f) |-> g is Relation-like NAT -defined the carrier of K -valued Function-like finite len f -element FinSequence-like FinSubsequence-like Element of (len f) -tuples_on the carrier of K
(len f) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt (((len f) |-> g),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (((len f) |-> g),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (((len f) |-> g),f)) is Element of the carrier of V2
g * (Sum f) is Element of the carrier of V2
the lmult of V2 . (g,(Sum f)) is Element of the carrier of V2
len ((len f) |-> g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((len f) |-> g) is finite len f -element Element of bool NAT
dom f is finite Element of bool NAT
dom (lmlt (((len f) |-> g),f)) is finite Element of bool NAT
len (lmlt (((len f) |-> g),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
<*> the carrier of V2 is Relation-like non-empty empty-yielding NAT -defined the carrier of V2 -valued Function-like one-to-one constant functional empty V21() V22() V23() V25() V26() V27() finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V93() V94() FinSequence of the carrier of V2
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Element of the carrier of V1
V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) |-> V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of V1
(len V3) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
lmlt (V3,((len V3) |-> V2)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (V3,((len V3) |-> V2)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (lmlt (V3,((len V3) |-> V2))) is Element of the carrier of V1
Sum V3 is Element of the carrier of K
(Sum V3) * V2 is Element of the carrier of V1
the lmult of V1 . ((Sum V3),V2) is Element of the carrier of V1
len ((len V3) |-> V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((len V3) |-> V2) is finite len V3 -element Element of bool NAT
dom V3 is finite Element of bool NAT
dom (lmlt (V3,((len V3) |-> V2))) is finite Element of bool NAT
A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
A is Element of the carrier of K
V3 . A is set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
((len V3) |-> V2) . A is set
(lmlt (V3,((len V3) |-> V2))) . A is set
A * V2 is Element of the carrier of V1
the lmult of V1 . (A,V2) is Element of the carrier of V1
len (lmlt (V3,((len V3) |-> V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is Element of the carrier of K
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V1 * f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V1 multfield is Relation-like 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 Function-like total quasi_total 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:]
K378( the carrier of K, the carrier of K, the multF of K,V1,(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(V1 multfield) * f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt ((V1 * f),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((V1 * f),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((V1 * f),V3)) is Element of the carrier of V2
lmlt (f,V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (f,V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (f,V3)) is Element of the carrier of V2
V1 * (Sum (lmlt (f,V3))) is Element of the carrier of V2
the lmult of V2 . (V1,(Sum (lmlt (f,V3)))) is Element of the carrier of V2
len (V1 * f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (V1 * f) is finite Element of bool NAT
dom f is finite Element of bool NAT
dom (lmlt ((V1 * f),V3)) is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom (V1 * f)) /\ (dom V3) is finite Element of bool NAT
dom (lmlt (f,V3)) is finite Element of bool NAT
(dom f) /\ (dom V3) is finite Element of bool NAT
A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt (f,V3)) . A is set
(lmlt ((V1 * f),V3)) . A is set
AI is Element of the carrier of V2
V1 * AI is Element of the carrier of V2
the lmult of V2 . (V1,AI) is Element of the carrier of V2
V3 /. A is Element of the carrier of V2
V3 . A is set
f /. A is Element of the carrier of K
f . A is set
(V1 * f) . A is set
V1 * (f /. A) is Element of the carrier of K
(V1 * (f /. A)) * (V3 /. A) is Element of the carrier of V2
the lmult of V2 . ((V1 * (f /. A)),(V3 /. A)) is Element of the carrier of V2
(f /. A) * (V3 /. A) is Element of the carrier of V2
the lmult of V2 . ((f /. A),(V3 /. A)) is Element of the carrier of V2
V1 * ((f /. A) * (V3 /. A)) is Element of the carrier of V2
the lmult of V2 . (V1,((f /. A) * (V3 /. A))) is Element of the carrier of V2
len (lmlt (f,V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (lmlt ((V1 * f),V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
lmlt (V2,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: (V2,V3) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
f is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of f is non empty set
g is Relation-like NAT -defined the carrier of f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f
lmlt (V2,g) is Relation-like NAT -defined the carrier of f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f
the lmult of f is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of f:], the carrier of f:]
[: the carrier of K, the carrier of f:] is Relation-like set
[:[: the carrier of K, the carrier of f:], the carrier of f:] is Relation-like set
bool [:[: the carrier of K, the carrier of f:], the carrier of f:] is set
the lmult of f .: (V2,g) is Relation-like NAT -defined the carrier of f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f
dom (lmlt (V2,V3)) is finite Element of bool NAT
dom V2 is finite Element of bool NAT
dom V3 is finite Element of bool NAT
(dom V2) /\ (dom V3) is finite Element of bool NAT
dom (lmlt (V2,g)) is finite Element of bool NAT
dom g is finite Element of bool NAT
(dom V2) /\ (dom g) is finite Element of bool NAT
AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V2 . AI is set
V2 /. AI is Element of the carrier of K
g . AI is set
g /. AI is Element of the carrier of f
V3 . AI is set
V3 /. AI is Element of the carrier of V1
(lmlt (V2,V3)) . AI is set
(V2 /. AI) * (V3 /. AI) is Element of the carrier of V1
the lmult of V1 . ((V2 /. AI),(V3 /. AI)) is Element of the carrier of V1
(V2 /. AI) * (g /. AI) is Element of the carrier of f
the lmult of f . ((V2 /. AI),(g /. AI)) is Element of the carrier of f
(lmlt (V2,g)) . AI is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum V2 is Element of the carrier of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of V3 is non empty set
f is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum f is Element of the carrier of V3
g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
g + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum A is Element of the carrier of V1
AI is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of AI is non empty set
S is Relation-like NAT -defined the carrier of AI -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AI
Sum S is Element of the carrier of AI
A | g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Seg g is finite g -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= g ) } is set
A | (Seg g) is Relation-like NAT -defined Seg g -defined NAT -defined the carrier of V1 -valued Function-like finite FinSubsequence-like set
len (A | g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum (A | g) is Element of the carrier of V1
S | g is Relation-like NAT -defined the carrier of AI -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of AI
S | (Seg g) is Relation-like NAT -defined Seg g -defined NAT -defined the carrier of AI -valued Function-like finite FinSubsequence-like set
Sum (S | g) is Element of the carrier of AI
dom A is finite Element of bool NAT
S . (g + 1) is set
S /. (g + 1) is Element of the carrier of AI
A . (g + 1) is set
A /. (g + 1) is Element of the carrier of V1
dom (A | g) is finite Element of bool NAT
(Sum (A | g)) + (A /. (g + 1)) is Element of the carrier of V1
(Sum (S | g)) + (S /. (g + 1)) is Element of the carrier of AI
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum g is Element of the carrier of V1
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of A is non empty set
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 A is Element of the carrier of A
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
0. A is V47(A) Element of the carrier of A
the ZeroF of A is Element of the carrier of A
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of A is non empty set
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
A is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum g is Element of the carrier of V1
Sum A is Element of the carrier of A
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of V1 is non empty non trivial set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom V3 is finite Element of bool NAT
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
1. (V1,(len V3)) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of V1
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
Line ((1. (V1,(len V3))),K) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width (1. (V1,(len V3))) -element FinSequence-like FinSubsequence-like Element of (width (1. (V1,(len V3)))) -tuples_on the carrier of V1
width (1. (V1,(len V3))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (1. (V1,(len V3)))) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
lmlt ((Line ((1. (V1,(len V3))),K)),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
[:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line ((1. (V1,(len V3))),K)),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line ((1. (V1,(len V3))),K)),V3)) is Element of the carrier of V2
V3 /. K is Element of the carrier of V2
len (Line ((1. (V1,(len V3))),K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line ((1. (V1,(len V3))),K)) is finite width (1. (V1,(len V3))) -element Element of bool NAT
dom (lmlt ((Line ((1. (V1,(len V3))),K)),V3)) is finite Element of bool NAT
len (lmlt ((Line ((1. (V1,(len V3))),K)),V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
[:NAT, the carrier of V2:] is Relation-like non trivial non finite set
bool [:NAT, the carrier of V2:] is non trivial non finite set
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
A is Relation-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of V2:]
A . (len (lmlt ((Line ((1. (V1,(len V3))),K)),V3))) is Element of the carrier of V2
A . {} is set
len (1. (V1,(len V3))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (1. (V1,(len V3))) is finite Element of bool NAT
AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
A . AI is set
AI + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
A . (AI + 1) is set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
(Line ((1. (V1,(len V3))),K)) . (AI + 1) is set
(1. (V1,(len V3))) * (K,(AI + 1)) is Element of the carrier of V1
[K,(AI + 1)] is set
{K,(AI + 1)} is non empty finite V32() set
{K} is non empty trivial finite V32() 1 -element set
{{K,(AI + 1)},{K}} is non empty finite V32() set
Indices (1. (V1,(len V3))) is set
Seg (width (1. (V1,(len V3)))) is finite width (1. (V1,(len V3))) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (V1,(len V3))) ) } is set
[:(dom (1. (V1,(len V3)))),(Seg (width (1. (V1,(len V3))))):] is Relation-like finite set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
V3 . (AI + 1) is set
V3 /. (AI + 1) is Element of the carrier of V2
(lmlt ((Line ((1. (V1,(len V3))),K)),V3)) . (AI + 1) is set
(0. V1) * (V3 /. (AI + 1)) is Element of the carrier of V2
the lmult of V2 . ((0. V1),(V3 /. (AI + 1))) is Element of the carrier of V2
A . (AI + 1) is Element of the carrier of V2
KER is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A . KER is Element of the carrier of V2
(A . KER) + (0. V2) is Element of the carrier of V2
AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
A . AI is set
AI + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
A . (AI + 1) is set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
(Line ((1. (V1,(len V3))),K)) . (AI + 1) is set
(1. (V1,(len V3))) * (K,(AI + 1)) is Element of the carrier of V1
[K,(AI + 1)] is set
{K,(AI + 1)} is non empty finite V32() set
{K} is non empty trivial finite V32() 1 -element set
{{K,(AI + 1)},{K}} is non empty finite V32() set
Indices (1. (V1,(len V3))) is set
Seg (width (1. (V1,(len V3)))) is finite width (1. (V1,(len V3))) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (V1,(len V3))) ) } is set
[:(dom (1. (V1,(len V3)))),(Seg (width (1. (V1,(len V3))))):] is Relation-like finite set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
V3 . (AI + 1) is set
V3 /. (AI + 1) is Element of the carrier of V2
(lmlt ((Line ((1. (V1,(len V3))),K)),V3)) . (AI + 1) is set
(0. V1) * (V3 /. (AI + 1)) is Element of the carrier of V2
the lmult of V2 . ((0. V1),(V3 /. (AI + 1))) is Element of the carrier of V2
A . (AI + 1) is Element of the carrier of V2
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A . S is Element of the carrier of V2
(A . S) + (0. V2) is Element of the carrier of V2
A . K is set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
(Line ((1. (V1,(len V3))),K)) . K is set
(1. (V1,(len V3))) * (K,K) is Element of the carrier of V1
[K,K] is set
{K,K} is non empty finite V32() set
{K} is non empty trivial finite V32() 1 -element set
{{K,K},{K}} is non empty finite V32() set
Indices (1. (V1,(len V3))) is set
Seg (width (1. (V1,(len V3)))) is finite width (1. (V1,(len V3))) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (V1,(len V3))) ) } is set
[:(dom (1. (V1,(len V3)))),(Seg (width (1. (V1,(len V3))))):] is Relation-like finite set
1_ V1 is Element of the carrier of V1
1. V1 is V47(V1) Element of the carrier of V1
the OneF of V1 is Element of the carrier of V1
K - 1 is ext-real V93() V94() set
AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
AI + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
A . AI is Element of the carrier of V2
V3 . K is set
(lmlt ((Line ((1. (V1,(len V3))),K)),V3)) . K is set
(1_ V1) * (V3 /. K) is Element of the carrier of V2
the lmult of V2 . ((1_ V1),(V3 /. K)) is Element of the carrier of V2
A . (AI + 1) is Element of the carrier of V2
(A . AI) + (V3 /. K) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
V3 is Element of the carrier of V1
V3 |-- V2 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 is non empty non trivial set
f is Element of the carrier of V1
V3 + f is Element of the carrier of V1
(V3 + f) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
f |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(V3 |-- V2) + (f |-- V2) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: ((V3 |-- V2),(f |-- V2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
rng V2 is finite set
len (V3 |-- V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
AI is Relation-like Function-like total quasi_total Linear_Combination of V1
Sum AI is Element of the carrier of V1
Carrier AI is finite Element of bool the carrier of V1
bool the carrier of V1 is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of V1 : not AI . b₁ = 0. K } is set
len ((V3 + f) |-- V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
S is Relation-like Function-like total quasi_total Linear_Combination of V1
Sum S is Element of the carrier of V1
Carrier S is finite Element of bool the carrier of V1
{ b₁ where b₁ is Element of the carrier of V1 : not S . b₁ = 0. K } is set
len (f |-- V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
MAI is Relation-like Function-like total quasi_total Linear_Combination of V1
Sum MAI is Element of the carrier of V1
Carrier MAI is finite Element of bool the carrier of V1
{ b₁ where b₁ is Element of the carrier of V1 : not MAI . b₁ = 0. K } is set
(len V2) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
KER is Basis of V1
AI + MAI is Relation-like Function-like total quasi_total Linear_Combination of V1
dom V2 is finite Element of bool NAT
Seg (len V2) is finite len V2 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V2 ) } is set
w is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
x is Relation-like NAT -defined the carrier of K -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
dom x is finite len V2 -element Element of bool NAT
x . w is set
x /. w is Element of the carrier of K
MK is Relation-like NAT -defined the carrier of K -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
dom MK is finite len V2 -element Element of bool NAT
MK . w is set
MK /. w is Element of the carrier of K
v is Relation-like NAT -defined the carrier of K -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
dom v is finite len V2 -element Element of bool NAT
v . w is set
v /. w is Element of the carrier of K
V2 /. w is Element of the carrier of V1
(AI + MAI) . (V2 /. w) is Element of the carrier of K
AI . (V2 /. w) is Element of the carrier of K
MAI . (V2 /. w) is Element of the carrier of K
(AI . (V2 /. w)) + (MAI . (V2 /. w)) is Element of the carrier of K
(MK /. w) + (MAI . (V2 /. w)) is Element of the carrier of K
(MK /. w) + (x /. w) is Element of the carrier of K
MK + x is Relation-like NAT -defined the carrier of K -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
the addF of K .: (MK,x) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(MK + x) . w is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is Element of the carrier of K
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
f is Element of the carrier of V2
V1 * f is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . (V1,f) is Element of the carrier of V2
(V1 * f) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
f |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V1 * (f |-- V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V1 multfield is Relation-like 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 Function-like total quasi_total 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:]
K378( the carrier of K, the carrier of K, the multF of K,V1,(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(V1 multfield) * (f |-- V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
rng V3 is finite set
len (f |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like Function-like total quasi_total Linear_Combination of V2
Sum A is Element of the carrier of V2
Carrier A is finite Element of bool the carrier of V2
bool the carrier of V2 is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of V2 : not A . b₁ = 0. K } is set
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((V1 * f) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
S is Relation-like Function-like total quasi_total Linear_Combination of V2
Sum S is Element of the carrier of V2
Carrier S is finite Element of bool the carrier of V2
{ b₁ where b₁ is Element of the carrier of V2 : not S . b₁ = 0. K } is set
len (V1 * (f |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
AI is Basis of V2
x is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
V1 * A is Relation-like Function-like total quasi_total Linear_Combination of V2
V3 /. x is Element of the carrier of V2
S . (V3 /. x) is Element of the carrier of K
A . (V3 /. x) is Element of the carrier of K
V1 * (A . (V3 /. x)) is Element of the carrier of K
KER is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
KER /. x is Element of the carrier of K
V1 * (KER /. x) is Element of the carrier of K
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
V3 /. x is Element of the carrier of V2
S . (V3 /. x) is Element of the carrier of K
KER is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
KER /. x is Element of the carrier of K
V1 * (KER /. x) is Element of the carrier of K
V3 /. x is Element of the carrier of V2
S . (V3 /. x) is Element of the carrier of K
KER is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
KER /. x is Element of the carrier of K
V1 * (KER /. x) is Element of the carrier of K
V3 /. x is Element of the carrier of V2
S . (V3 /. x) is Element of the carrier of K
KER is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
KER /. x is Element of the carrier of K
V1 * (KER /. x) is Element of the carrier of K
dom V3 is finite Element of bool NAT
dom (f |-- V3) is finite Element of bool NAT
(f |-- V3) . x is set
(f |-- V3) /. x is Element of the carrier of K
MAI is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
dom MAI is finite len V3 -element Element of bool NAT
MAI . x is set
MAI /. x is Element of the carrier of K
MK is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
MK . x is set
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of V1 is non empty non trivial set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
dom V3 is finite Element of bool NAT
V3 /. K is Element of the carrier of V2
(V3 /. K) |-- V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
1. (V1,(len V3)) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of V1
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
Line ((1. (V1,(len V3))),K) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width (1. (V1,(len V3))) -element FinSequence-like FinSubsequence-like Element of (width (1. (V1,(len V3)))) -tuples_on the carrier of V1
width (1. (V1,(len V3))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (1. (V1,(len V3)))) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
rng V3 is finite set
len ((V3 /. K) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like Function-like total quasi_total Linear_Combination of V2
Sum A is Element of the carrier of V2
Carrier A is finite Element of bool the carrier of V2
bool the carrier of V2 is set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{ b₁ where b₁ is Element of the carrier of V2 : not A . b₁ = 0. V1 } is set
A is Basis of V2
{(V3 /. K)} is non empty trivial finite 1 -element Element of bool the carrier of V2
Lin {(V3 /. K)} is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
AI is Relation-like Function-like total quasi_total Linear_Combination of {(V3 /. K)}
Sum AI is Element of the carrier of V2
V3 . K is set
Carrier AI is finite Element of bool the carrier of V2
{ b₁ where b₁ is Element of the carrier of V2 : not AI . b₁ = 0. V1 } is set
{(V3 . K)} is non empty trivial finite 1 -element set
Indices (1. (V1,(len V3))) is set
dom (1. (V1,(len V3))) is finite Element of bool NAT
Seg (width (1. (V1,(len V3)))) is finite width (1. (V1,(len V3))) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (V1,(len V3))) ) } is set
[:(dom (1. (V1,(len V3)))),(Seg (width (1. (V1,(len V3))))):] is Relation-like finite set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
[:(Seg (len V3)),(Seg (len V3)):] is Relation-like finite set
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[K,S] is set
{K,S} is non empty finite V32() set
{K} is non empty trivial finite V32() 1 -element set
{{K,S},{K}} is non empty finite V32() set
dom ((V3 /. K) |-- V3) is finite Element of bool NAT
AI . (V3 /. K) is Element of the carrier of V1
(AI . (V3 /. K)) * (V3 /. K) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
[:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . ((AI . (V3 /. K)),(V3 /. K)) is Element of the carrier of V2
1_ V1 is Element of the carrier of V1
1. V1 is V47(V1) Element of the carrier of V1
the OneF of V1 is Element of the carrier of V1
(1_ V1) * (V3 /. K) is Element of the carrier of V2
the lmult of V2 . ((1_ V1),(V3 /. K)) is Element of the carrier of V2
A . (V3 /. K) is Element of the carrier of V1
((V3 /. K) |-- V3) /. S is Element of the carrier of V1
(1. (V1,(len V3))) * (K,S) is Element of the carrier of V1
(Line ((1. (V1,(len V3))),K)) . S is set
((V3 /. K) |-- V3) . S is set
V3 . S is set
(1. (V1,(len V3))) * (K,S) is Element of the carrier of V1
(Line ((1. (V1,(len V3))),K)) . S is set
V3 /. S is Element of the carrier of V2
A . (V3 /. S) is Element of the carrier of V1
((V3 /. K) |-- V3) /. S is Element of the carrier of V1
((V3 /. K) |-- V3) . S is set
(Line ((1. (V1,(len V3))),K)) . S is set
((V3 /. K) |-- V3) . S is set
(Line ((1. (V1,(len V3))),K)) . S is set
((V3 /. K) |-- V3) . S is set
len (Line ((1. (V1,(len V3))),K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
(0. V1) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V2) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
(len V2) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
dom V2 is finite Element of bool NAT
len ((0. V1) |-- V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom V2 is finite Element of bool NAT
V3 is set
1. (K,(len V2)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
width (1. (K,(len V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V2 /. f is Element of the carrier of V1
(V2 /. f) - (V2 /. f) is Element of the carrier of V1
- (V2 /. f) is Element of the carrier of V1
(V2 /. f) + (- (V2 /. f)) is Element of the carrier of V1
1_ K is Element of the carrier of K
1. K is V47(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
- (1_ K) is Element of the carrier of K
(- (1_ K)) * (V2 /. f) is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . ((- (1_ K)),(V2 /. f)) is Element of the carrier of V1
(V2 /. f) + ((- (1_ K)) * (V2 /. f)) is Element of the carrier of V1
(V2 /. f) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((- (1_ K)) * (V2 /. f)) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((V2 /. f) |-- V2) + (((- (1_ K)) * (V2 /. f)) |-- V2) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: (((V2 /. f) |-- V2),(((- (1_ K)) * (V2 /. f)) |-- V2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- (1_ K)) * ((V2 /. f) |-- V2) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- (1_ K)) multfield is Relation-like 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
the multF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K378( the carrier of K, the carrier of K, the multF of K,(- (1_ K)),(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
((- (1_ K)) multfield) * ((V2 /. f) |-- V2) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((V2 /. f) |-- V2) + ((- (1_ K)) * ((V2 /. f) |-- V2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K .: (((V2 /. f) |-- V2),((- (1_ K)) * ((V2 /. f) |-- V2))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Line ((1. (K,(len V2))),f) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
(width (1. (K,(len V2)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((1. (K,(len V2))),f)) + ((- (1_ K)) * ((V2 /. f) |-- V2)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K .: ((Line ((1. (K,(len V2))),f)),((- (1_ K)) * ((V2 /. f) |-- V2))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(- (1_ K)) * (Line ((1. (K,(len V2))),f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
((- (1_ K)) multfield) * (Line ((1. (K,(len V2))),f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line ((1. (K,(len V2))),f)) + ((- (1_ K)) * (Line ((1. (K,(len V2))),f))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
the addF of K .: ((Line ((1. (K,(len V2))),f)),((- (1_ K)) * (Line ((1. (K,(len V2))),f)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
- (Line ((1. (K,(len V2))),f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
comp K is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(comp K) * (Line ((1. (K,(len V2))),f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line ((1. (K,(len V2))),f)) + (- (Line ((1. (K,(len V2))),f))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
the addF of K .: ((Line ((1. (K,(len V2))),f)),(- (Line ((1. (K,(len V2))),f)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom V2 is finite Element of bool NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
rng V2 is finite set
Seg (len V2) is finite len V2 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V2 ) } is set
card (Seg (len V2)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom V2 is finite Element of bool NAT
card (dom V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 is Basis of V1
card V3 is V21() V22() V23() cardinal set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
the carrier of V2 is non empty set
V3 is Relation-like Function-like total quasi_total Linear_Combination of V2
Carrier V3 is finite Element of bool the carrier of V2
bool the carrier of V2 is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of V2 : not V3 . b₁ = 0. K } is set
Sum V3 is Element of the carrier of V2
[: the carrier of V2, the carrier of K:] is Relation-like set
bool [: the carrier of V2, the carrier of K:] is set
g is set
V3 . g is set
A is Element of the carrier of V2
V3 . A is Element of the carrier of K
V3 . A is set
[: the carrier of V1, the carrier of K:] is Relation-like set
bool [: the carrier of V1, the carrier of K:] is set
g is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of K:]
bool the carrier of V1 is set
A is Element of the carrier of V1
A is finite Element of bool the carrier of V1
V3 . A is set
g . A is Element of the carrier of K
Funcs ( the carrier of V1, the carrier of K) is functional non empty FUNCTION_DOMAIN of the carrier of V1, the carrier of K
A is Relation-like Function-like total quasi_total Linear_Combination of V1
A | the carrier of V2 is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of K:]
Carrier A is finite Element of bool the carrier of V1
{ b₁ where b₁ is Element of the carrier of V1 : not A . b₁ = 0. K } is set
Sum A is Element of the carrier of V1
S is set
KER is Element of the carrier of V1
A . KER is Element of the carrier of K
V3 . KER is set
S is set
A . S is set
V3 . S is set
f is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of K:]
f . S is set
AI is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of K:]
AI . S is set
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of V2 is non empty set
bool the carrier of V2 is set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
V3 | K is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Seg K is finite K -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
V3 | (Seg K) is Relation-like NAT -defined Seg K -defined NAT -defined the carrier of V2 -valued Function-like finite FinSubsequence-like set
rng (V3 | K) is finite set
rng V3 is finite set
f is Basis of V2
g is Element of bool the carrier of V2
Lin g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
the carrier of (Lin g) is non empty set
A is set
bool the carrier of (Lin g) is set
A is linearly-independent Element of bool the carrier of (Lin g)
Lin A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of Lin g
the addF of (Lin g) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):]
[: the carrier of (Lin g), the carrier of (Lin g):] is Relation-like set
[:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):] is Relation-like set
bool [:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):] is set
the ZeroF of (Lin g) is Element of the carrier of (Lin g)
the lmult of (Lin g) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):]
the carrier of V1 is non empty non trivial set
[: the carrier of V1, the carrier of (Lin g):] is Relation-like set
[:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):] is Relation-like set
bool [:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):] is set
VectSpStr(# the carrier of (Lin g), the addF of (Lin g), the ZeroF of (Lin g), the lmult of (Lin g) #) is non empty strict VectSpStr over V1
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of V2 is non empty set
bool the carrier of V2 is set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
V3 /^ K is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
rng (V3 /^ K) is finite set
rng V3 is finite set
f is Basis of V2
g is Element of bool the carrier of V2
Lin g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
the carrier of (Lin g) is non empty set
A is set
bool the carrier of (Lin g) is set
V3 | K is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Seg K is finite K -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
V3 | (Seg K) is Relation-like NAT -defined Seg K -defined NAT -defined the carrier of V2 -valued Function-like finite FinSubsequence-like set
(V3 | K) ^ (V3 /^ K) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
A is linearly-independent Element of bool the carrier of (Lin g)
Lin A is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of Lin g
the addF of (Lin g) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):]
[: the carrier of (Lin g), the carrier of (Lin g):] is Relation-like set
[:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):] is Relation-like set
bool [:[: the carrier of (Lin g), the carrier of (Lin g):], the carrier of (Lin g):] is set
the ZeroF of (Lin g) is Element of the carrier of (Lin g)
the lmult of (Lin g) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):]
the carrier of V1 is non empty non trivial set
[: the carrier of V1, the carrier of (Lin g):] is Relation-like set
[:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):] is Relation-like set
bool [:[: the carrier of V1, the carrier of (Lin g):], the carrier of (Lin g):] is set
VectSpStr(# the carrier of (Lin g), the addF of (Lin g), the ZeroF of (Lin g), the lmult of (Lin g) #) is non empty strict VectSpStr over V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V2 /\ V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of V2 is non empty set
the carrier of V3 is non empty set
V2 + V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (V2 + V3) is non empty set
[#] ((0). V1) is non empty non proper Element of bool the carrier of ((0). V1)
the carrier of ((0). V1) is non empty set
bool the carrier of ((0). V1) is set
the carrier of V1 is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
card ([#] ((0). V1)) is non empty V21() V22() V23() cardinal set
dim V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(dim V2) + (dim V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (V2 + V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (V2 /\ V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(dim (V2 + V3)) + (dim (V2 /\ V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(dim (V2 + V3)) + {} is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
g is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V3
f ^ g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A is Relation-like NAT -defined the carrier of (V2 + V3) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2 + V3
rng A is finite set
AI is Element of the carrier of (V2 + V3)
S is Element of the carrier of (V2 + V3)
KER is Element of the carrier of (V2 + V3)
S + KER is Element of the carrier of (V2 + V3)
AI |-- A is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
MAI is Element of the carrier of V2
MAI |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
MK is Element of the carrier of V3
MK |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(MAI |-- f) ^ (MK |-- g) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
rng g is finite set
len (MK |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
v is Relation-like Function-like total quasi_total Linear_Combination of V3
Sum v is Element of the carrier of V3
Carrier v is finite Element of bool the carrier of V3
bool the carrier of V3 is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of V3 : not v . b₁ = 0. K } is set
w is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
Carrier w is finite Element of bool the carrier of (V2 + V3)
bool the carrier of (V2 + V3) is set
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not w . b₁ = 0. K } is set
Sum w is Element of the carrier of (V2 + V3)
w | the carrier of V3 is Relation-like Function-like Element of bool [: the carrier of (V2 + V3), the carrier of K:]
[: the carrier of (V2 + V3), the carrier of K:] is Relation-like set
bool [: the carrier of (V2 + V3), the carrier of K:] is set
A is Basis of V2 + V3
rng f is finite set
len (MAI |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
i is Relation-like Function-like total quasi_total Linear_Combination of V2
Sum i is Element of the carrier of V2
Carrier i is finite Element of bool the carrier of V2
bool the carrier of V2 is set
{ b₁ where b₁ is Element of the carrier of V2 : not i . b₁ = 0. K } is set
len (AI |-- A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A12 is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
Sum A12 is Element of the carrier of (V2 + V3)
Carrier A12 is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not A12 . b₁ = 0. K } is set
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (AI |-- A) is finite Element of bool NAT
dom A is finite Element of bool NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (MK |-- g) is finite Element of bool NAT
dom g is finite Element of bool NAT
i is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
Carrier i is finite Element of bool the carrier of (V2 + V3)
{ b₁ where b₁ is Element of the carrier of (V2 + V3) : not i . b₁ = 0. K } is set
Sum i is Element of the carrier of (V2 + V3)
i | the carrier of V2 is Relation-like Function-like Element of bool [: the carrier of (V2 + V3), the carrier of K:]
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (MAI |-- f) is finite Element of bool NAT
dom f is finite Element of bool NAT
len ((MAI |-- f) ^ (MK |-- g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len (MAI |-- f)) + (len (MK |-- g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((MAI |-- f) ^ (MK |-- g)) is finite Element of bool NAT
i + w is Relation-like Function-like total quasi_total Linear_Combination of V2 + V3
fb is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
f /. fb is Element of the carrier of V2
i . (f /. fb) is Element of the carrier of K
(MAI |-- f) /. fb is Element of the carrier of K
(MAI |-- f) . fb is set
((MAI |-- f) ^ (MK |-- g)) . fb is set
i . (f /. fb) is set
0. (V2 + V3) is V47(V2 + V3) Element of the carrier of (V2 + V3)
the ZeroF of (V2 + V3) is Element of the carrier of (V2 + V3)
fbi is Element of the carrier of (V2 + V3)
w . fbi is Element of the carrier of K
A12 . fbi is Element of the carrier of K
i . fbi is Element of the carrier of K
(i . fbi) + (0. K) is Element of the carrier of K
f . fb is set
A . fb is set
A /. fb is Element of the carrier of (V2 + V3)
(AI |-- A) /. fb is Element of the carrier of K
(AI |-- A) . fb is set
fbi is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(len (MAI |-- f)) + fbi is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
fbi is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(len (MAI |-- f)) + fbi is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g /. fbi is Element of the carrier of V3
v . (g /. fbi) is Element of the carrier of K
(MK |-- g) /. fbi is Element of the carrier of K
(MK |-- g) . fbi is set
((MAI |-- f) ^ (MK |-- g)) . fb is set
w . (g /. fbi) is set
0. (V2 + V3) is V47(V2 + V3) Element of the carrier of (V2 + V3)
the ZeroF of (V2 + V3) is Element of the carrier of (V2 + V3)
b2n is Element of the carrier of (V2 + V3)
i . b2n is Element of the carrier of K
A12 . b2n is Element of the carrier of K
w . b2n is Element of the carrier of K
(0. K) + (w . b2n) is Element of the carrier of K
g . fbi is set
A . fb is set
A /. fb is Element of the carrier of (V2 + V3)
(AI |-- A) /. fb is Element of the carrier of K
(AI |-- A) . fb is set
((MAI |-- f) ^ (MK |-- g)) . fb is set
(AI |-- A) . fb is set
((MAI |-- f) ^ (MK |-- g)) . fb is set
(AI |-- A) . fb is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
(Omega). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the ZeroF of V1 is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
the carrier of K is non empty non trivial set
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) is non empty strict VectSpStr over K
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of V3 is non empty set
f is Element of the carrier of V3
g is Element of the carrier of V1
g |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V3
f |-- A is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
rng A is finite set
len (f |-- A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like Function-like total quasi_total Linear_Combination of V3
Sum A is Element of the carrier of V3
Carrier A is finite Element of bool the carrier of V3
bool the carrier of V3 is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of V3 : not A . b₁ = 0. K } is set
AI is Relation-like Function-like total quasi_total Linear_Combination of V1
Carrier AI is finite Element of bool the carrier of V1
bool the carrier of V1 is set
{ b₁ where b₁ is Element of the carrier of V1 : not AI . b₁ = 0. K } is set
Sum AI is Element of the carrier of V1
AI | the carrier of V3 is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of K:]
[: the carrier of V1, the carrier of K:] is Relation-like set
bool [: the carrier of V1, the carrier of K:] is set
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dom A is finite Element of bool NAT
dom AI is set
(f |-- A) /. S is Element of the carrier of K
A /. S is Element of the carrier of V3
AI . (A /. S) is set
A . S is set
AI . (A . S) is set
V2 /. S is Element of the carrier of V1
AI . (V2 /. S) is Element of the carrier of K
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
V2 /\ V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of V2 is non empty set
the carrier of V3 is non empty set
V2 + V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (V2 + V3) is non empty set
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
g is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V3
f ^ g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng f is finite set
rng g is finite set
A is Basis of V2
A is Basis of V3
A \/ A is set
rng (f ^ g) is finite set
AI is set
the carrier of ((0). V1) is non empty set
the carrier of V1 is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
AI is Relation-like NAT -defined the carrier of (V2 + V3) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (V2 + V3)
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
g is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
AutMt (V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (AutMt (V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (AutMt (V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len (AutMt (V3,f,g)), width (AutMt (V3,f,g)), the carrier of K
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom f is finite Element of bool NAT
dom A is finite Element of bool NAT
Seg (len f) is finite len f -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
f /. (len f) is Element of the carrier of V1
V3 . (f /. (len f)) is Element of the carrier of V2
(V3 . (f /. (len f))) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A /. (len f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
A . (len f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (A,(len f)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len ((V3 . (f /. (len f))) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V1 is Relation-like set
K is 1-sorted
the carrier of K is set
V1 | the carrier of K is Relation-like set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
g is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
(K,V1,V2,g,V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
A is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the carrier of A is non empty set
the carrier of A is non empty set
(A,g) is set
g | the carrier of A is Relation-like Function-like set
(A,g) is set
g | the carrier of A is Relation-like Function-like set
AI is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
dim AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
S is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
dim S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the carrier of AI is non empty set
[: the carrier of A, the carrier of AI:] is Relation-like set
bool [: the carrier of A, the carrier of AI:] is set
the carrier of S is non empty set
[: the carrier of A, the carrier of S:] is Relation-like set
bool [: the carrier of A, the carrier of S:] is set
AI /\ S is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
(0). V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
KER is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of A, the carrier of AI:]
MAI is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of A, the carrier of S:]
MK is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of A
x is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of A
MK ^ x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
v is Relation-like NAT -defined the carrier of AI -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of AI
(K,A,AI,KER,MK,v) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len MK, len v, the carrier of K
len MK is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len v is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
w is Relation-like NAT -defined the carrier of S -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of S
v ^ w is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,A,S,MAI,x,w) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len x, len w, the carrier of K
len x is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len w is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
<*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> is Relation-like NAT -defined ( the carrier of K *) * -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like Matrix-yielding FinSequence of ( the carrier of K *) *
( the carrier of K *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K *
<*(K,A,AI,KER,MK,v)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(K,A,AI,KER,MK,v)] is set
{1,(K,A,AI,KER,MK,v)} is non empty finite V32() set
{{1,(K,A,AI,KER,MK,v)},{1}} is non empty finite V32() set
{[1,(K,A,AI,KER,MK,v)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(K,A,S,MAI,x,w)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(K,A,S,MAI,x,w)] is set
{1,(K,A,S,MAI,x,w)} is non empty finite V32() set
{{1,(K,A,S,MAI,x,w)},{1}} is non empty finite V32() set
{[1,(K,A,S,MAI,x,w)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(K,A,AI,KER,MK,v)*> ^ <*(K,A,S,MAI,x,w)*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
block_diagonal (<*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>,(0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of Sum (Len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>), Sum (Width <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>), the carrier of K
Len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> is Relation-like NAT -defined NAT -valued Function-like finite len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> -element FinSequence-like FinSubsequence-like Element of (len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>) -tuples_on NAT
len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> is non empty V21() V22() V23() V27() finite cardinal ext-real positive non negative V93() V94() Element of NAT
(len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
Sum (Len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Width <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> is Relation-like NAT -defined NAT -valued Function-like finite len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*> -element FinSequence-like FinSubsequence-like Element of (len <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>) -tuples_on NAT
Sum (Width <*(K,A,AI,KER,MK,v),(K,A,S,MAI,x,w)*>) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len MK) + (len x) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
AI + S is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
(Omega). V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
the addF of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:]
[: the carrier of V2, the carrier of V2:] is Relation-like set
[:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is set
the ZeroF of V2 is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
VectSpStr(# the carrier of V2, the addF of V2, the ZeroF of V2, the lmult of V2 #) is non empty strict VectSpStr over K
len (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (K,A,S,MAI,x,w) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (K,A,AI,KER,MK,v) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (K,A,S,MAI,x,w) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (K,A,AI,KER,MK,v) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
the carrier of (AI + S) is non empty set
dom (K,V1,V2,g,V3,f) is finite Element of bool NAT
Seg (len (K,V1,V2,g,V3,f)) is finite len (K,V1,V2,g,V3,f) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len (K,V1,V2,g,V3,f) ) } is set
i is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V3 /. i is Element of the carrier of V1
g . (V3 /. i) is Element of the carrier of V2
i + (len (K,A,AI,KER,MK,v)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 /. (i + (len (K,A,AI,KER,MK,v))) is Element of the carrier of V1
g . (V3 /. (i + (len (K,A,AI,KER,MK,v)))) is Element of the carrier of V2
dom V3 is finite Element of bool NAT
dom (K,A,AI,KER,MK,v) is finite Element of bool NAT
dom MK is finite Element of bool NAT
dom KER is non empty set
Line ((K,V1,V2,g,V3,f),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,V1,V2,g,V3,f) -element FinSequence-like FinSubsequence-like Element of (width (K,V1,V2,g,V3,f)) -tuples_on the carrier of K
width (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (K,V1,V2,g,V3,f)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Line ((K,A,AI,KER,MK,v),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,A,AI,KER,MK,v) -element FinSequence-like FinSubsequence-like Element of (width (K,A,AI,KER,MK,v)) -tuples_on the carrier of K
(width (K,A,AI,KER,MK,v)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(width (K,A,S,MAI,x,w)) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,A,S,MAI,x,w) -element FinSequence-like FinSubsequence-like Element of (width (K,A,S,MAI,x,w)) -tuples_on the carrier of K
(width (K,A,S,MAI,x,w)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((K,A,AI,KER,MK,v),i)) ^ ((width (K,A,S,MAI,x,w)) |-> (0. K)) is Relation-like NAT -defined the carrier of K -valued Function-like finite (width (K,A,AI,KER,MK,v)) + (width (K,A,S,MAI,x,w)) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(width (K,A,AI,KER,MK,v)) + (width (K,A,S,MAI,x,w)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Seg (len (K,A,AI,KER,MK,v)) is finite len (K,A,AI,KER,MK,v) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len (K,A,AI,KER,MK,v) ) } is set
(K,A,AI,KER,MK,v) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,A,AI,KER,MK,v) /. i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
MK /. i is Element of the carrier of A
KER . (MK /. i) is Element of the carrier of AI
(KER . (MK /. i)) |-- v is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
V3 . i is set
MK . i is set
fb is Element of the carrier of (AI + S)
(K,V1,V2,g,V3,f) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,V1,V2,g,V3,f) /. i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(g . (V3 /. i)) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A12 is Relation-like NAT -defined the carrier of (AI + S) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of AI + S
fb |-- A12 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. (AI + S) is V47(AI + S) Element of the carrier of (AI + S)
the ZeroF of (AI + S) is Element of the carrier of (AI + S)
fb + (0. (AI + S)) is Element of the carrier of (AI + S)
(fb + (0. (AI + S))) |-- A12 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. S is V47(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(0. S) |-- w is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((KER . (MK /. i)) |-- v) ^ ((0. S) |-- w) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom (K,A,S,MAI,x,w) is finite Element of bool NAT
dom x is finite Element of bool NAT
dom MAI is non empty set
Line ((K,V1,V2,g,V3,f),(i + (len (K,A,AI,KER,MK,v)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,V1,V2,g,V3,f) -element FinSequence-like FinSubsequence-like Element of (width (K,V1,V2,g,V3,f)) -tuples_on the carrier of K
(width (K,A,AI,KER,MK,v)) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,A,AI,KER,MK,v) -element FinSequence-like FinSubsequence-like Element of (width (K,A,AI,KER,MK,v)) -tuples_on the carrier of K
Line ((K,A,S,MAI,x,w),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,A,S,MAI,x,w) -element FinSequence-like FinSubsequence-like Element of (width (K,A,S,MAI,x,w)) -tuples_on the carrier of K
((width (K,A,AI,KER,MK,v)) |-> (0. K)) ^ (Line ((K,A,S,MAI,x,w),i)) is Relation-like NAT -defined the carrier of K -valued Function-like finite (width (K,A,AI,KER,MK,v)) + (width (K,A,S,MAI,x,w)) -element FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Seg (len (K,A,S,MAI,x,w)) is finite len (K,A,S,MAI,x,w) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len (K,A,S,MAI,x,w) ) } is set
V3 . (i + (len (K,A,AI,KER,MK,v))) is set
x . i is set
x /. i is Element of the carrier of A
fbi is Element of the carrier of (AI + S)
MAI . (x /. i) is Element of the carrier of S
(K,A,S,MAI,x,w) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,A,S,MAI,x,w) /. i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(MAI . (x /. i)) |-- w is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(K,V1,V2,g,V3,f) . (i + (len (K,A,AI,KER,MK,v))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,V1,V2,g,V3,f) /. (i + (len (K,A,AI,KER,MK,v))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(g . (V3 /. (i + (len (K,A,AI,KER,MK,v))))) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A12 is Relation-like NAT -defined the carrier of (AI + S) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of AI + S
fbi |-- A12 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. (AI + S) is V47(AI + S) Element of the carrier of (AI + S)
the ZeroF of (AI + S) is Element of the carrier of (AI + S)
(0. (AI + S)) + fbi is Element of the carrier of (AI + S)
((0. (AI + S)) + fbi) |-- A12 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. AI is V47(AI) Element of the carrier of AI
the ZeroF of AI is Element of the carrier of AI
(0. AI) |-- v is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((0. AI) |-- v) ^ ((MAI . (x /. i)) |-- w) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(len (K,A,AI,KER,MK,v)) + (len (K,A,S,MAI,x,w)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (K,A,AI,KER,MK,v)) + (width (K,A,S,MAI,x,w)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len v) + (len w) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
(K,V1,V2,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
id V1 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
bool [: the carrier of V1, the carrier of V1:] is set
id the carrier of V1 is Relation-like the carrier of V1 -defined the carrier of V1 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V1:]
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
(K,V1,V1,(id V1),V2,V2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
1. (K,(len V2)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
len (K,V1,V1,(id V1),V2,V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dom V2 is finite Element of bool NAT
Seg (len V2) is finite len V2 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V2 ) } is set
dom (K,V1,V1,(id V1),V2,V2) is finite Element of bool NAT
(K,V1,V1,(id V1),V2,V2) . g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,V1,V1,(id V1),V2,V2) /. g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
V2 /. g is Element of the carrier of V1
(id V1) . (V2 /. g) is Element of the carrier of V1
((id V1) . (V2 /. g)) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(V2 /. g) |-- V2 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Line ((1. (K,(len V2))),g) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
width (1. (K,(len V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (1. (K,(len V2)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(1. (K,(len V2))) . g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (1. (K,(len V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
id V1 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
bool [: the carrier of V1, the carrier of V1:] is set
id the carrier of V1 is Relation-like the carrier of V1 -defined the carrier of V1 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V1:]
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
(K,V1,V1,(id V1),V2,V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(K,V1,V1,(id V1),V3,V2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of K
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V1,(id V1),V2,V3) ~ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
1_ K is Element of the carrier of K
1. K is V47(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(K,V1,V1,(id V1),V2,V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V3, the carrier of K
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
(K,V1,V1,(id V1),V3,V2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V2, the carrier of K
Det (K,V1,V1,(id V1),V2,V3) is Element of the carrier of K
(len V2) + {} is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (id V1) is non empty set
(id V1) * (id V1) is Relation-like Function-like non empty total total quasi_total quasi_total additive Element of bool [: the carrier of V1, the carrier of V1:]
(K,V1,V1,(id V1),V2,V3) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
(K,V1,V1,((id V1) * (id V1)),V2,V2) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
1. (K,(len V2)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
Det ((K,V1,V1,(id V1),V2,V3) * g) is Element of the carrier of K
Det (K,V1,V1,(id V1),V2,V3) is Element of the carrier of K
Det g is Element of the carrier of K
(Det (K,V1,V1,(id V1),V2,V3)) * (Det g) is Element of the carrier of K
(K,V1,V1,(id V1),V2,V3) * ((K,V1,V1,(id V1),V2,V3) ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
(len V2) + {} is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of V1 is non empty non trivial set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
dom V3 is finite Element of bool NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
lmlt (g,f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
[:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: (g,f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (g,f)) is Element of the carrier of V2
(Sum (lmlt (g,f))) |-- V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
((Sum (lmlt (g,f))) |-- V3) . K is set
A is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom A is finite Element of bool NAT
g "*" A is Element of the carrier of V1
mlt (g,A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the multF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the multF of V1 .: (g,A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (mlt (g,A)) is Element of the carrier of V1
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len g, len V3, the carrier of V1
Indices A is set
dom A is finite Element of bool NAT
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Seg (width A) is finite width A -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom g is finite Element of bool NAT
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
Seg (len g) is finite len g -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len g ) } is set
AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[AI,K] is set
{AI,K} is non empty finite V32() set
{AI} is non empty trivial finite V32() 1 -element set
{{AI,K},{AI}} is non empty finite V32() set
f /. AI is Element of the carrier of V2
(f /. AI) |-- V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len ((f /. AI) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((f /. AI) |-- V3) is finite Element of bool NAT
Col (A,K) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of V1
(len A) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
(Col (A,K)) . AI is set
A * (AI,K) is Element of the carrier of V1
((f /. AI) |-- V3) /. K is Element of the carrier of V1
((f /. AI) |-- V3) . K is set
A . AI is set
AI is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom AI is finite Element of bool NAT
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
AI /. S is Element of the carrier of V1
Col (A,S) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of V1
mlt (g,(Col (A,S))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the multF of V1 .: (g,(Col (A,S))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (mlt (g,(Col (A,S)))) is Element of the carrier of V1
AI . S is set
dom f is finite Element of bool NAT
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Line (A,S) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of V1
(width A) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
len (Line (A,S)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f /. S is Element of the carrier of V2
(f /. S) |-- V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len ((f /. S) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (A,S)) is finite width A -element Element of bool NAT
dom ((f /. S) |-- V3) is finite Element of bool NAT
KER is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[S,KER] is set
{S,KER} is non empty finite V32() set
{S} is non empty trivial finite V32() 1 -element set
{{S,KER},{S}} is non empty finite V32() set
(Line (A,S)) . KER is set
A * (S,KER) is Element of the carrier of V1
((f /. S) |-- V3) /. KER is Element of the carrier of V1
((f /. S) |-- V3) . KER is set
lmlt (((f /. S) |-- V3),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (((f /. S) |-- V3),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (((f /. S) |-- V3),V3)) is Element of the carrier of V2
lmlt ((Line (A,S)),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (A,S)),V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line (A,S)),V3)) is Element of the carrier of V2
len (Col (A,K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((Sum (lmlt (g,f))) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((Sum (lmlt (g,f))) |-- V3) is finite Element of bool NAT
((Sum (lmlt (g,f))) |-- V3) /. K is Element of the carrier of V1
lmlt (AI,V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (AI,V3) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (AI,V3)) is Element of the carrier of V2
(Sum (lmlt (AI,V3))) |-- V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
((Sum (lmlt (AI,V3))) |-- V3) /. K is Element of the carrier of V1
AI /. K is Element of the carrier of V1
AI . K is set
mlt (g,(Col (A,K))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the multF of V1 .: (g,(Col (A,K))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (mlt (g,(Col (A,K)))) is Element of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V2, the carrier of V1:] is Relation-like set
bool [: the carrier of V2, the carrier of V1:] is set
V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of V1:]
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
(K,V2,V1,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Element of the carrier of V2
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
LineVec2Mx (A |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- f), the carrier of K
len (A |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(A |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- f), the carrier of K
[1,(A |-- f)] is set
{1,(A |-- f)} is non empty finite V32() set
{{1,(A |-- f)},{1}} is non empty finite V32() set
{[1,(A |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
V3 . A is Element of the carrier of V1
(V3 . A) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((V3 . A) |-- g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . A) |-- g), the carrier of K
len ((V3 . A) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((V3 . A) |-- g)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . A) |-- g), the carrier of K
[1,((V3 . A) |-- g)] is set
{1,((V3 . A) |-- g)} is non empty finite V32() set
{{1,((V3 . A) |-- g)},{1}} is non empty finite V32() set
{[1,((V3 . A) |-- g)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len (K,V2,V1,V3,f,g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx ((V3 . A) |-- g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (A |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (A |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (K,V2,V1,V3,f,g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom g is finite Element of bool NAT
Seg (len g) is finite len g -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len g ) } is set
dom ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) is finite Element of bool NAT
V3 * f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
len (V3 * f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
x is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
v is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[x,v] is set
{x,v} is non empty finite V32() set
{x} is non empty trivial finite V32() 1 -element set
{{x,v},{x}} is non empty finite V32() set
Indices ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) is set
Seg (width ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g))) is finite width ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) ) } is set
[:(dom ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g))),(Seg (width ((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)))):] is Relation-like finite set
Col ((K,V2,V1,V3,f,g),v) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (K,V2,V1,V3,f,g) -element FinSequence-like FinSubsequence-like Element of (len (K,V2,V1,V3,f,g)) -tuples_on the carrier of K
(len (K,V2,V1,V3,f,g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Col ((K,V2,V1,V3,f,g),v)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (V3 * f) is finite Element of bool NAT
dom f is finite Element of bool NAT
dom (K,V2,V1,V3,f,g) is finite Element of bool NAT
dom (Col ((K,V2,V1,V3,f,g),v)) is finite len (K,V2,V1,V3,f,g) -element Element of bool NAT
w is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Seg (len (K,V2,V1,V3,f,g)) is finite len (K,V2,V1,V3,f,g) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len (K,V2,V1,V3,f,g) ) } is set
(K,V2,V1,V3,f,g) . w is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(K,V2,V1,V3,f,g) /. w is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
f /. w is Element of the carrier of V2
V3 . (f /. w) is Element of the carrier of V1
f . w is set
V3 . (f . w) is set
(V3 * f) . w is set
(V3 * f) /. w is Element of the carrier of V1
(Col ((K,V2,V1,V3,f,g),v)) . w is set
(K,V2,V1,V3,f,g) * (w,v) is Element of the carrier of K
Line ((K,V2,V1,V3,f,g),w) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (K,V2,V1,V3,f,g) -element FinSequence-like FinSubsequence-like Element of (width (K,V2,V1,V3,f,g)) -tuples_on the carrier of K
(width (K,V2,V1,V3,f,g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((K,V2,V1,V3,f,g),w)) . v is set
((K,V2,V1,V3,f,g) /. w) . v is set
((V3 * f) /. w) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((V3 * f) /. w) |-- g) . v is set
(LineVec2Mx ((V3 . A) |-- g)) * (x,v) is Element of the carrier of K
Line ((LineVec2Mx ((V3 . A) |-- g)),x) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx ((V3 . A) |-- g)) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx ((V3 . A) |-- g))) -tuples_on the carrier of K
(width (LineVec2Mx ((V3 . A) |-- g))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((LineVec2Mx ((V3 . A) |-- g)),x)) . v is set
((V3 . A) |-- g) . v is set
lmlt ((A |-- f),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((A |-- f),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((A |-- f),f)) is Element of the carrier of V2
V3 . (Sum (lmlt ((A |-- f),f))) is Element of the carrier of V1
(V3 . (Sum (lmlt ((A |-- f),f)))) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((V3 . (Sum (lmlt ((A |-- f),f)))) |-- g) . v is set
lmlt ((A |-- f),(V3 * f)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: ((A |-- f),(V3 * f)) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (lmlt ((A |-- f),(V3 * f))) is Element of the carrier of V1
(Sum (lmlt ((A |-- f),(V3 * f)))) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((Sum (lmlt ((A |-- f),(V3 * f)))) |-- g) . v is set
(A |-- f) "*" (Col ((K,V2,V1,V3,f,g),v)) is Element of the carrier of K
mlt ((A |-- f),(Col ((K,V2,V1,V3,f,g),v))) 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 Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the multF of K .: ((A |-- f),(Col ((K,V2,V1,V3,f,g),v))) 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 ((A |-- f),(Col ((K,V2,V1,V3,f,g),v)))) is Element of the carrier of K
Line ((LineVec2Mx (A |-- f)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (A |-- f)) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (A |-- f))) -tuples_on the carrier of K
(width (LineVec2Mx (A |-- f))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((LineVec2Mx (A |-- f)),1)) "*" (Col ((K,V2,V1,V3,f,g),v)) is Element of the carrier of K
mlt ((Line ((LineVec2Mx (A |-- f)),1)),(Col ((K,V2,V1,V3,f,g),v))) 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 .: ((Line ((LineVec2Mx (A |-- f)),1)),(Col ((K,V2,V1,V3,f,g),v))) 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 ((LineVec2Mx (A |-- f)),1)),(Col ((K,V2,V1,V3,f,g),v)))) is Element of the carrier of K
((LineVec2Mx (A |-- f)) * (K,V2,V1,V3,f,g)) * (x,v) is Element of the carrier of K
len (LineVec2Mx ((V3 . A) |-- g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
A is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
A is Element of the carrier of V1
A . A is Element of the carrier of V2
A |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (A |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- V3), the carrier of K
len (A |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(A |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- V3), the carrier of K
[1,(A |-- V3)] is set
{1,(A |-- V3)} is non empty finite V32() set
{{1,(A |-- V3)},{1}} is non empty finite V32() set
{[1,(A |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (A |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (A |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (A |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (A |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (A |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (A |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) is Element of the carrier of V2
A is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
A is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
AI is set
A . AI is set
S is Element of the carrier of V1
S |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (S |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (S |-- V3), the carrier of K
len (S |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(S |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (S |-- V3), the carrier of K
[1,(S |-- V3)] is set
{1,(S |-- V3)} is non empty finite V32() set
{{1,(S |-- V3)},{1}} is non empty finite V32() set
{[1,(S |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (S |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (S |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (S |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (S |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (S |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (S |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (S |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx (S |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (S |-- V3)) * g),1)),f)) is Element of the carrier of V2
A . AI is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Element of the carrier of V2
g |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (g |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
len (g |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(g |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
[1,(g |-- V3)] is set
{1,(g |-- V3)} is non empty finite V32() set
{{1,(g |-- V3)},{1}} is non empty finite V32() set
{[1,(g |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
width (LineVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V2,V1,V3,f,A) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of V1:]
[: the carrier of V2, the carrier of V1:] is Relation-like set
bool [: the carrier of V2, the carrier of V1:] is set
(K,V2,V1,V3,f,A) . g is Element of the carrier of V1
((K,V2,V1,V3,f,A) . g) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,V1,V3,f,A) . g) |-- f), the carrier of K
len (((K,V2,V1,V3,f,A) . g) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(((K,V2,V1,V3,f,A) . g) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,V1,V3,f,A) . g) |-- f), the carrier of K
[1,(((K,V2,V1,V3,f,A) . g) |-- f)] is set
{1,(((K,V2,V1,V3,f,A) . g) |-- f)} is non empty finite V32() set
{{1,(((K,V2,V1,V3,f,A) . g) |-- f)},{1}} is non empty finite V32() set
{[1,(((K,V2,V1,V3,f,A) . g) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (g |-- V3)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width ((LineVec2Mx (g |-- V3)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Line (((LineVec2Mx (g |-- V3)) * A),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (g |-- V3)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (g |-- V3)) * A)) -tuples_on the carrier of K
(width ((LineVec2Mx (g |-- V3)) * A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Line (((LineVec2Mx (g |-- V3)) * A),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 .: ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f)) is Element of the carrier of V1
(Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (LineVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (g |-- V3)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f), the carrier of K
len ((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f), the carrier of K
[1,((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f)] is set
{1,((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f)} is non empty finite V32() set
{{1,((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f)},{1}} is non empty finite V32() set
{[1,((Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f))) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Element of the carrier of V1
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,A) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
(K,V1,V2,V3,f,A) . g is Element of the carrier of V2
g |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (g |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
len (g |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(g |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
[1,(g |-- V3)] is set
{1,(g |-- V3)} is non empty finite V32() set
{{1,(g |-- V3)},{1}} is non empty finite V32() set
{[1,(g |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (g |-- V3)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (g |-- V3)) * A),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (g |-- V3)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (g |-- V3)) * A)) -tuples_on the carrier of K
width ((LineVec2Mx (g |-- V3)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (g |-- V3)) * A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
width (LineVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (((LineVec2Mx (g |-- V3)) * A),1)) is finite width ((LineVec2Mx (g |-- V3)) * A) -element Element of bool NAT
lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f)) is finite Element of bool NAT
dom f is finite Element of bool NAT
(dom (Line (((LineVec2Mx (g |-- V3)) * A),1))) /\ (dom f) is finite Element of bool NAT
<*> the carrier of V2 is Relation-like non-empty empty-yielding NAT -defined the carrier of V2 -valued Function-like one-to-one constant functional empty V21() V22() V23() V25() V26() V27() finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V93() V94() FinSequence of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (g |-- V3)) * A),1)),f)) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
width V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
width V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V1 + V2 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Line ((V1 + V2),V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (V1 + V2) -element FinSequence-like FinSubsequence-like Element of (width (V1 + V2)) -tuples_on the carrier of K
width (V1 + V2) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (V1 + V2)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Line (V1,V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite width V1 -element FinSequence-like FinSubsequence-like Element of (width V1) -tuples_on the carrier of K
(width V1) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Line (V2,V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite width V2 -element FinSequence-like FinSubsequence-like Element of (width V2) -tuples_on the carrier of K
(width V2) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line (V1,V3)) + (Line (V2,V3)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: ((Line (V1,V3)),(Line (V2,V3))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
dom V1 is finite Element of bool NAT
Seg (width V1) is finite width V1 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width V1 ) } is set
[V3,(width V1)] is set
{V3,(width V1)} is non empty finite V32() set
{V3} is non empty trivial finite V32() 1 -element set
{{V3,(width V1)},{V3}} is non empty finite V32() set
Indices V1 is set
[:(dom V1),(Seg (width V1)):] is Relation-like finite set
f is Relation-like NAT -defined the carrier of K -valued Function-like finite width V1 -element FinSequence-like FinSubsequence-like Element of (width V1) -tuples_on the carrier of K
(Line (V1,V3)) + f is Relation-like NAT -defined the carrier of K -valued Function-like finite width V1 -element FinSequence-like FinSubsequence-like Element of (width V1) -tuples_on the carrier of K
the addF of K .: ((Line (V1,V3)),f) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len ((Line (V1,V3)) + f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
A is Element of the carrier of K
AI is Element of the carrier of V1
A * AI is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . (A,AI) is Element of the carrier of V1
(K,V1,V2,V3,f,g) . (A * AI) is Element of the carrier of V2
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
A * (0. V2) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . (A,(0. V2)) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . AI is Element of the carrier of V2
A * ((K,V1,V2,V3,f,g) . AI) is Element of the carrier of V2
the lmult of V2 . (A,((K,V1,V2,V3,f,g) . AI)) is Element of the carrier of V2
A is Element of the carrier of V1
AI is Element of the carrier of V1
A + AI is Element of the carrier of V1
(K,V1,V2,V3,f,g) . (A + AI) is Element of the carrier of V2
(0. V2) + (0. V2) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . A is Element of the carrier of V2
((K,V1,V2,V3,f,g) . A) + (0. V2) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . AI is Element of the carrier of V2
((K,V1,V2,V3,f,g) . A) + ((K,V1,V2,V3,f,g) . AI) is Element of the carrier of V2
A is Element of the carrier of V1
A |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
AI is Element of the carrier of V1
AI |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A + AI is Element of the carrier of V1
(A + AI) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((A + AI) |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A + AI) |-- V3), the carrier of K
len ((A + AI) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((A + AI) |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A + AI) |-- V3), the carrier of K
[1,((A + AI) |-- V3)] is set
{1,((A + AI) |-- V3)} is non empty finite V32() set
{{1,((A + AI) |-- V3)},{1}} is non empty finite V32() set
{[1,((A + AI) |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
LineVec2Mx (A |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- V3), the carrier of K
len (A |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(A |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- V3), the carrier of K
[1,(A |-- V3)] is set
{1,(A |-- V3)} is non empty finite V32() set
{{1,(A |-- V3)},{1}} is non empty finite V32() set
{[1,(A |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
LineVec2Mx (AI |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (AI |-- V3), the carrier of K
len (AI |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (AI |-- V3), the carrier of K
[1,(AI |-- V3)] is set
{1,(AI |-- V3)} is non empty finite V32() set
{{1,(AI |-- V3)},{1}} is non empty finite V32() set
{[1,(AI |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx ((A + AI) |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((A + AI) |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((A + AI) |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx ((A + AI) |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx ((A + AI) |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(LineVec2Mx (A |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (A |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (A |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (A |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (A |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (A |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(LineVec2Mx (AI |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (AI |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (AI |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (AI |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (AI |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (AI |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (LineVec2Mx (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (A |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (A |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (A |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (A |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx ((A + AI) |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
lmlt ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f)) is finite Element of bool NAT
dom (Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)) is finite width ((LineVec2Mx ((A + AI) |-- V3)) * g) -element Element of bool NAT
dom f is finite Element of bool NAT
(dom (Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1))) /\ (dom f) is finite Element of bool NAT
dom (Line (((LineVec2Mx (A |-- V3)) * g),1)) is finite width ((LineVec2Mx (A |-- V3)) * g) -element Element of bool NAT
(dom (Line (((LineVec2Mx (A |-- V3)) * g),1))) /\ (dom f) is finite Element of bool NAT
lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) is finite Element of bool NAT
width (LineVec2Mx (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is finite width ((LineVec2Mx (AI |-- V3)) * g) -element Element of bool NAT
(dom (Line (((LineVec2Mx (AI |-- V3)) * g),1))) /\ (dom f) is finite Element of bool NAT
lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is finite Element of bool NAT
len (lmlt ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(A |-- V3) + (AI |-- V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: ((A |-- V3),(AI |-- V3)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((A |-- V3) + (AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A |-- V3) + (AI |-- V3)), the carrier of K
len ((A |-- V3) + (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((A |-- V3) + (AI |-- V3))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A |-- V3) + (AI |-- V3)), the carrier of K
[1,((A |-- V3) + (AI |-- V3))] is set
{1,((A |-- V3) + (AI |-- V3))} is non empty finite V32() set
{{1,((A |-- V3) + (AI |-- V3))},{1}} is non empty finite V32() set
{[1,((A |-- V3) + (AI |-- V3))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (A |-- V3)) + (LineVec2Mx (AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
((LineVec2Mx (A |-- V3)) * g) + ((LineVec2Mx (AI |-- V3)) * g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
(Line (((LineVec2Mx (A |-- V3)) * g),1)) + (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K .: ((Line (((LineVec2Mx (A |-- V3)) * g),1)),(Line (((LineVec2Mx (AI |-- V3)) * g),1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the addF of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:]
[: the carrier of V2, the carrier of V2:] is Relation-like set
[:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is set
the addF of V2 .: ((lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)),(lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f))) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
i is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) /. i is Element of the carrier of V2
(lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) . i is set
(lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) /. i is Element of the carrier of V2
(lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) . i is set
(lmlt ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f)) . i is set
((lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) /. i) + ((lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) /. i) is Element of the carrier of V2
len (lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum (lmlt ((Line (((LineVec2Mx ((A + AI) |-- V3)) * g),1)),f)) is Element of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f)) is Element of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is Element of the carrier of V2
(Sum (lmlt ((Line (((LineVec2Mx (A |-- V3)) * g),1)),f))) + (Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f))) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . (A + AI) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . A is Element of the carrier of V2
((K,V1,V2,V3,f,g) . A) + (Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f))) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . AI is Element of the carrier of V2
((K,V1,V2,V3,f,g) . A) + ((K,V1,V2,V3,f,g) . AI) is Element of the carrier of V2
AI is Element of the carrier of V1
AI |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A is Element of the carrier of K
A * AI is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . (A,AI) is Element of the carrier of V1
(A * AI) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((A * AI) |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A * AI) |-- V3), the carrier of K
len ((A * AI) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((A * AI) |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((A * AI) |-- V3), the carrier of K
[1,((A * AI) |-- V3)] is set
{1,((A * AI) |-- V3)} is non empty finite V32() set
{{1,((A * AI) |-- V3)},{1}} is non empty finite V32() set
{[1,((A * AI) |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
LineVec2Mx (AI |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (AI |-- V3), the carrier of K
len (AI |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (AI |-- V3), the carrier of K
[1,(AI |-- V3)] is set
{1,(AI |-- V3)} is non empty finite V32() set
{{1,(AI |-- V3)},{1}} is non empty finite V32() set
{[1,(AI |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx ((A * AI) |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((A * AI) |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((A * AI) |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx ((A * AI) |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx ((A * AI) |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(LineVec2Mx (AI |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (AI |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (AI |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (AI |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (AI |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (AI |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
width (LineVec2Mx ((A * AI) |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)) is finite width ((LineVec2Mx ((A * AI) |-- V3)) * g) -element Element of bool NAT
dom (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is finite width ((LineVec2Mx (AI |-- V3)) * g) -element Element of bool NAT
A * (AI |-- V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A multfield is Relation-like 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
the multF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K378( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * (AI |-- V3) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (A * (AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A * (AI |-- V3)), the carrier of K
len (A * (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(A * (AI |-- V3))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A * (AI |-- V3)), the carrier of K
[1,(A * (AI |-- V3))] is set
{1,(A * (AI |-- V3))} is non empty finite V32() set
{{1,(A * (AI |-- V3))},{1}} is non empty finite V32() set
{[1,(A * (AI |-- V3))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
A * (LineVec2Mx (AI |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (AI |-- V3), the carrier of K
A * ((LineVec2Mx (AI |-- V3)) * g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
lmlt ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f)) is finite Element of bool NAT
(dom (Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1))) /\ (dom f) is finite Element of bool NAT
(dom (Line (((LineVec2Mx (AI |-- V3)) * g),1))) /\ (dom f) is finite Element of bool NAT
lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is finite Element of bool NAT
len (LineVec2Mx (AI |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (AI |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A * (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (AI |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (AI |-- V3)) * g)) -tuples_on the carrier of K
(A multfield) * (Line (((LineVec2Mx (AI |-- V3)) * g),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
w is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) . w is set
(lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) /. w is Element of the carrier of V2
(Line (((LineVec2Mx (AI |-- V3)) * g),1)) . w is set
(Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w is Element of the carrier of K
f . w is set
f /. w is Element of the carrier of V2
(Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)) . w is set
(Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)) /. w is Element of the carrier of K
(lmlt ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f)) . w is set
the lmult of V2 . (((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)) /. w),(f /. w)) is Element of the carrier of V2
A * ((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w) is Element of the carrier of K
(A * ((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w)) * (f /. w) is Element of the carrier of V2
the lmult of V2 . ((A * ((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w)),(f /. w)) is Element of the carrier of V2
((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w) * (f /. w) is Element of the carrier of V2
the lmult of V2 . (((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w),(f /. w)) is Element of the carrier of V2
A * (((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w) * (f /. w)) is Element of the carrier of V2
the lmult of V2 . (A,(((Line (((LineVec2Mx (AI |-- V3)) * g),1)) /. w) * (f /. w))) is Element of the carrier of V2
A * ((lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) /. w) is Element of the carrier of V2
the lmult of V2 . (A,((lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) /. w)) is Element of the carrier of V2
len (lmlt ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Sum (lmlt ((Line (((LineVec2Mx ((A * AI) |-- V3)) * g),1)),f)) is Element of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)) is Element of the carrier of V2
A * (Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f))) is Element of the carrier of V2
the lmult of V2 . (A,(Sum (lmlt ((Line (((LineVec2Mx (AI |-- V3)) * g),1)),f)))) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . (A * AI) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . AI is Element of the carrier of V2
A * ((K,V1,V2,V3,f,g) . AI) is Element of the carrier of V2
the lmult of V2 . (A,((K,V1,V2,V3,f,g) . AI)) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
g is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
(K,V1,V2,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V2,f,g,(K,V1,V2,V3,f,g)) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
rng f is finite set
S is set
Seg (len f) is finite len f -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
AI is Basis of V1
card AI is V21() V22() V23() cardinal set
dom f is finite Element of bool NAT
card (dom f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(Omega). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the ZeroF of V1 is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) is non empty strict VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
0. V1 is V47(V1) Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
V3 . S is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(0. K) * (0. V1) is Element of the carrier of V1
the lmult of V1 . ((0. K),(0. V1)) is Element of the carrier of V1
V3 . ((0. K) * (0. V1)) is Element of the carrier of V2
V3 . (0. V1) is Element of the carrier of V2
(0. K) * (V3 . (0. V1)) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . ((0. K),(V3 . (0. V1))) is Element of the carrier of V2
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
(K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . S is set
V3 * f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
(K,V1,V2,f,g,(K,V1,V2,V3,f,g)) * f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
rng f is finite set
bool the carrier of V1 is set
dom V3 is non empty set
AI is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (K,V1,V2,f,g,(K,V1,V2,V3,f,g)) is non empty set
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom AI is finite Element of bool NAT
dom S is finite Element of bool NAT
KER is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dom f is finite Element of bool NAT
f . KER is set
f /. KER is Element of the carrier of V1
(K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER) is Element of the carrier of V2
((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g), the carrier of K
len (((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g), the carrier of K
[1,(((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g)] is set
{1,(((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g)} is non empty finite V32() set
{{1,(((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g)},{1}} is non empty finite V32() set
{[1,(((K,V1,V2,f,g,(K,V1,V2,V3,f,g)) . (f /. KER)) |-- g)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(f /. KER) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((f /. KER) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((f /. KER) |-- f), the carrier of K
len ((f /. KER) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((f /. KER) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((f /. KER) |-- f), the carrier of K
[1,((f /. KER) |-- f)] is set
{1,((f /. KER) |-- f)} is non empty finite V32() set
{{1,((f /. KER) |-- f)},{1}} is non empty finite V32() set
{[1,((f /. KER) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx ((f /. KER) |-- f)) * (K,V1,V2,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
V3 . (f /. KER) is Element of the carrier of V2
(V3 . (f /. KER)) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((V3 . (f /. KER)) |-- g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . (f /. KER)) |-- g), the carrier of K
len ((V3 . (f /. KER)) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((V3 . (f /. KER)) |-- g)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . (f /. KER)) |-- g), the carrier of K
[1,((V3 . (f /. KER)) |-- g)] is set
{1,((V3 . (f /. KER)) |-- g)} is non empty finite V32() set
{{1,((V3 . (f /. KER)) |-- g)},{1}} is non empty finite V32() set
{[1,((V3 . (f /. KER)) |-- g)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx ((V3 . (f /. KER)) |-- g)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx ((V3 . (f /. KER)) |-- g)) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx ((V3 . (f /. KER)) |-- g))) -tuples_on the carrier of K
width (LineVec2Mx ((V3 . (f /. KER)) |-- g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx ((V3 . (f /. KER)) |-- g))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
AI . KER is set
S . KER is set
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of V1 is non empty non trivial set
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
V2 is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V1 *
dom V2 is finite Element of bool NAT
width V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
V3 is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V1 *
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Line (V2,K) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width V2 -element FinSequence-like FinSubsequence-like Element of (width V2) -tuples_on the carrier of V1
(width V2) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
LineVec2Mx (Line (V2,K)) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (V2,K)), the carrier of V1
len (Line (V2,K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of V1,(Line (V2,K))) is Relation-like NAT -defined the carrier of V1 * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (V2,K)), the carrier of V1
[1,(Line (V2,K))] is set
{1,(Line (V2,K))} is non empty finite V32() set
{{1,(Line (V2,K))},{1}} is non empty finite V32() set
{[1,(Line (V2,K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (Line (V2,K))) * V3 is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V1 *
V2 * V3 is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V1 *
Line ((V2 * V3),K) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width (V2 * V3) -element FinSequence-like FinSubsequence-like Element of (width (V2 * V3)) -tuples_on the carrier of V1
width (V2 * V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (V2 * V3)) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
LineVec2Mx (Line ((V2 * V3),K)) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((V2 * V3),K)), the carrier of V1
len (Line ((V2 * V3),K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of V1,(Line ((V2 * V3),K))) is Relation-like NAT -defined the carrier of V1 * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((V2 * V3),K)), the carrier of V1
[1,(Line ((V2 * V3),K))] is set
{1,(Line ((V2 * V3),K))} is non empty finite V32() set
{{1,(Line ((V2 * V3),K))},{1}} is non empty finite V32() set
{[1,(Line ((V2 * V3),K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
width V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (Line ((V2 * V3),K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (Line (V2,K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width ((LineVec2Mx (Line (V2,K))) * V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (Line (V2,K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (Line (V2,K))) * V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (V2 * V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (V2 * V3) is finite Element of bool NAT
A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[A,A] is set
{A,A} is non empty finite V32() set
{A} is non empty trivial finite V32() 1 -element set
{{A,A},{A}} is non empty finite V32() set
Indices ((LineVec2Mx (Line (V2,K))) * V3) is set
dom ((LineVec2Mx (Line (V2,K))) * V3) is finite Element of bool NAT
Seg (width ((LineVec2Mx (Line (V2,K))) * V3)) is finite width ((LineVec2Mx (Line (V2,K))) * V3) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width ((LineVec2Mx (Line (V2,K))) * V3) ) } is set
[:(dom ((LineVec2Mx (Line (V2,K))) * V3)),(Seg (width ((LineVec2Mx (Line (V2,K))) * V3))):] is Relation-like finite set
Seg (width (V2 * V3)) is finite width (V2 * V3) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (V2 * V3) ) } is set
[K,A] is set
{K,A} is non empty finite V32() set
{K} is non empty trivial finite V32() 1 -element set
{{K,A},{K}} is non empty finite V32() set
Indices (V2 * V3) is set
[:(dom (V2 * V3)),(Seg (width (V2 * V3))):] is Relation-like finite set
Seg (width V3) is finite width V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width V3 ) } is set
[:(Seg 1),(Seg (width V3)):] is Relation-like finite set
((LineVec2Mx (Line (V2,K))) * V3) * (A,A) is Element of the carrier of V1
Line ((LineVec2Mx (Line (V2,K))),1) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width (LineVec2Mx (Line (V2,K))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (V2,K)))) -tuples_on the carrier of V1
(width (LineVec2Mx (Line (V2,K)))) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
Col (V3,A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of V1
(len V3) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
(Line ((LineVec2Mx (Line (V2,K))),1)) "*" (Col (V3,A)) is Element of the carrier of V1
mlt ((Line ((LineVec2Mx (Line (V2,K))),1)),(Col (V3,A))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the multF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the multF of V1 .: ((Line ((LineVec2Mx (Line (V2,K))),1)),(Col (V3,A))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (mlt ((Line ((LineVec2Mx (Line (V2,K))),1)),(Col (V3,A)))) is Element of the carrier of V1
(Line (V2,K)) "*" (Col (V3,A)) is Element of the carrier of V1
mlt ((Line (V2,K)),(Col (V3,A))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
the multF of V1 .: ((Line (V2,K)),(Col (V3,A))) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V1
Sum (mlt ((Line (V2,K)),(Col (V3,A)))) is Element of the carrier of V1
(V2 * V3) * (K,A) is Element of the carrier of V1
(Line ((V2 * V3),K)) . A is set
Line ((LineVec2Mx (Line ((V2 * V3),K))),A) is Relation-like NAT -defined the carrier of V1 -valued Function-like finite width (LineVec2Mx (Line ((V2 * V3),K))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line ((V2 * V3),K)))) -tuples_on the carrier of V1
(width (LineVec2Mx (Line ((V2 * V3),K)))) -tuples_on the carrier of V1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
(Line ((LineVec2Mx (Line ((V2 * V3),K))),A)) . A is set
(LineVec2Mx (Line ((V2 * V3),K))) * (A,A) is Element of the carrier of V1
len (LineVec2Mx (Line ((V2 * V3),K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
(K,V1,V2,(K,V1,V2,V3,f,g),V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
1. (K,(len V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of K
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (K,V1,V2,(K,V1,V2,V3,f,g),V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (1. (K,(len V3))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Seg (len V3) is finite len V3 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V3 ) } is set
dom (1. (K,(len V3))) is finite Element of bool NAT
(K,V1,V2,(K,V1,V2,V3,f,g),V3,f) /. S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
dom V3 is finite Element of bool NAT
V3 /. S is Element of the carrier of V1
(K,V1,V2,V3,f,g) . (V3 /. S) is Element of the carrier of V2
((K,V1,V2,V3,f,g) . (V3 /. S)) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
KER is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx KER is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len KER, the carrier of K
len KER is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,KER) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len KER, the carrier of K
[1,KER] is set
{1,KER} is non empty finite V32() set
{{1,KER},{1}} is non empty finite V32() set
{[1,KER]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(V3 /. S) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((V3 /. S) |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 /. S) |-- V3), the carrier of K
len ((V3 /. S) |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((V3 /. S) |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 /. S) |-- V3), the carrier of K
[1,((V3 /. S) |-- V3)] is set
{1,((V3 /. S) |-- V3)} is non empty finite V32() set
{{1,((V3 /. S) |-- V3)},{1}} is non empty finite V32() set
{[1,((V3 /. S) |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx ((V3 /. S) |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line ((1. (K,(len V3))),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V3))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V3)))) -tuples_on the carrier of K
width (1. (K,(len V3))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (1. (K,(len V3)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((1. (K,(len V3))),S)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((1. (K,(len V3))),S)), the carrier of K
len (Line ((1. (K,(len V3))),S)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(Line ((1. (K,(len V3))),S))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((1. (K,(len V3))),S)), the carrier of K
[1,(Line ((1. (K,(len V3))),S))] is set
{1,(Line ((1. (K,(len V3))),S))} is non empty finite V32() set
{{1,(Line ((1. (K,(len V3))),S))},{1}} is non empty finite V32() set
{[1,(Line ((1. (K,(len V3))),S))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (Line ((1. (K,(len V3))),S))) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
(1. (K,(len V3))) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((1. (K,(len V3))) * g),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((1. (K,(len V3))) * g) -element FinSequence-like FinSubsequence-like Element of (width ((1. (K,(len V3))) * g)) -tuples_on the carrier of K
width ((1. (K,(len V3))) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((1. (K,(len V3))) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((1. (K,(len V3))) * g),S)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((1. (K,(len V3))) * g),S)), the carrier of K
len (Line (((1. (K,(len V3))) * g),S)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(Line (((1. (K,(len V3))) * g),S))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((1. (K,(len V3))) * g),S)), the carrier of K
[1,(Line (((1. (K,(len V3))) * g),S))] is set
{1,(Line (((1. (K,(len V3))) * g),S))} is non empty finite V32() set
{{1,(Line (((1. (K,(len V3))) * g),S))},{1}} is non empty finite V32() set
{[1,(Line (((1. (K,(len V3))) * g),S))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line (g,S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width g -element FinSequence-like FinSubsequence-like Element of (width g) -tuples_on the carrier of K
width g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width g) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (g,S)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (g,S)), the carrier of K
len (Line (g,S)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(Line (g,S))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (g,S)), the carrier of K
[1,(Line (g,S))] is set
{1,(Line (g,S))} is non empty finite V32() set
{{1,(Line (g,S))},{1}} is non empty finite V32() set
{[1,(Line (g,S))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx (Line (g,S))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (g,S))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (g,S)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (g,S))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx (Line (g,S)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
g . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (K,V1,V2,(K,V1,V2,V3,f,g),V3,f) is finite Element of bool NAT
(K,V1,V2,(K,V1,V2,V3,f,g),V3,f) . S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
V2 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of V2 is non empty non trivial set
the carrier of V2 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V2
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V3 is Relation-like NAT -defined the carrier of V2 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of K,V1, the carrier of V2
f is Relation-like NAT -defined the carrier of V2 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V2 *
V3 + f is Relation-like NAT -defined the carrier of V2 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of V2 *
width (V3 + f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (V3 + f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
((len V3),(len f),K,g,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,((len V3),(len f),K,g,A)) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
(K,V1,V2,V3,f,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
(K,V1,V2,V3,f,g) + (K,V1,V2,V3,f,A) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
MAI is set
(K,V1,V2,V3,f,((len V3),(len f),K,g,A)) . MAI is set
0. V2 is V47(V2) Element of the carrier of V2
the ZeroF of V2 is Element of the carrier of V2
(0. V2) + (0. V2) is Element of the carrier of V2
MK is Element of the carrier of V1
(K,V1,V2,V3,f,g) . MK is Element of the carrier of V2
((K,V1,V2,V3,f,g) . MK) + (0. V2) is Element of the carrier of V2
(K,V1,V2,V3,f,A) . MK is Element of the carrier of V2
((K,V1,V2,V3,f,g) . MK) + ((K,V1,V2,V3,f,A) . MK) is Element of the carrier of V2
((K,V1,V2,V3,f,g) + (K,V1,V2,V3,f,A)) . MAI is set
MK is Element of the carrier of V1
MK |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (MK |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (MK |-- V3), the carrier of K
len (MK |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(MK |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (MK |-- V3), the carrier of K
[1,(MK |-- V3)] is set
{1,(MK |-- V3)} is non empty finite V32() set
{{1,(MK |-- V3)},{1}} is non empty finite V32() set
{[1,(MK |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
width (LineVec2Mx (MK |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(LineVec2Mx (MK |-- V3)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (MK |-- V3)) * A),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MK |-- V3)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MK |-- V3)) * A)) -tuples_on the carrier of K
width ((LineVec2Mx (MK |-- V3)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (MK |-- V3)) * A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (MK |-- V3)) * A),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (((LineVec2Mx (MK |-- V3)) * A),1)) is finite width ((LineVec2Mx (MK |-- V3)) * A) -element Element of bool NAT
dom f is finite Element of bool NAT
dom (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) is finite Element of bool NAT
len (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (MK |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(LineVec2Mx (MK |-- V3)) * g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
len ((LineVec2Mx (MK |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Line (((LineVec2Mx (MK |-- V3)) * g),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MK |-- V3)) * g) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MK |-- V3)) * g)) -tuples_on the carrier of K
width ((LineVec2Mx (MK |-- V3)) * g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (MK |-- V3)) * g)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
width g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (MK |-- V3)) * g),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Line (((LineVec2Mx (MK |-- V3)) * g),1)) is finite width ((LineVec2Mx (MK |-- V3)) * g) -element Element of bool NAT
dom (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) is finite Element of bool NAT
len (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the addF of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:]
[: the carrier of V2, the carrier of V2:] is Relation-like set
[:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is set
the addF of V2 .: ((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)),(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
dom ((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is finite Element of bool NAT
(dom f) /\ (dom f) is finite Element of bool NAT
len ((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
W is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) /. W is Element of the carrier of V2
(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) . W is set
(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) /. W is Element of the carrier of V2
(lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) . W is set
((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) . W is set
((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) /. W) + ((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) /. W) is Element of the carrier of V2
(K,V1,V2,V3,f,((len V3),(len f),K,g,A)) . MAI is set
(LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A))) -tuples_on the carrier of K
width ((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line (((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * ((len V3),(len f),K,g,A)),1)),f)) is Element of the carrier of V2
((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line ((((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)) -element FinSequence-like FinSubsequence-like Element of (width (((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A))) -tuples_on the carrier of K
width (((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: ((Line ((((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)),1)),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line ((((LineVec2Mx (MK |-- V3)) * g) + ((LineVec2Mx (MK |-- V3)) * A)),1)),f)) is Element of the carrier of V2
(Line (((LineVec2Mx (MK |-- V3)) * g),1)) + (Line (((LineVec2Mx (MK |-- V3)) * A),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K .: ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),(Line (((LineVec2Mx (MK |-- V3)) * A),1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt (((Line (((LineVec2Mx (MK |-- V3)) * g),1)) + (Line (((LineVec2Mx (MK |-- V3)) * A),1))),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 .: (((Line (((LineVec2Mx (MK |-- V3)) * g),1)) + (Line (((LineVec2Mx (MK |-- V3)) * A),1))),f) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt (((Line (((LineVec2Mx (MK |-- V3)) * g),1)) + (Line (((LineVec2Mx (MK |-- V3)) * A),1))),f)) is Element of the carrier of V2
Sum ((lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) + (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is Element of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f)) is Element of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f)) is Element of the carrier of V2
(Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * g),1)),f))) + (Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is Element of the carrier of V2
(K,V1,V2,V3,f,g) . MK is Element of the carrier of V2
((K,V1,V2,V3,f,g) . MK) + (Sum (lmlt ((Line (((LineVec2Mx (MK |-- V3)) * A),1)),f))) is Element of the carrier of V2
(K,V1,V2,V3,f,A) . MK is Element of the carrier of V2
((K,V1,V2,V3,f,g) . MK) + ((K,V1,V2,V3,f,A) . MK) is Element of the carrier of V2
((K,V1,V2,V3,f,g) + (K,V1,V2,V3,f,A)) . MAI is set
(K,V1,V2,V3,f,((len V3),(len f),K,g,A)) . MAI is set
((K,V1,V2,V3,f,g) + (K,V1,V2,V3,f,A)) . MAI is set
(K,V1,V2,V3,f,((len V3),(len f),K,g,A)) . MAI is set
((K,V1,V2,V3,f,g) + (K,V1,V2,V3,f,A)) . MAI is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is Element of the carrier of K
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V3 is non empty set
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
(K,V2,V3,f,g,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V3:]
[: the carrier of V2, the carrier of V3:] is Relation-like set
bool [: the carrier of V2, the carrier of V3:] is set
V1 * (K,V2,V3,f,g,A) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of V3:]
V1 * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
(K,V2,V3,f,g,(V1 * A)) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V3:]
KER is set
MAI is Element of the carrier of V2
MAI |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (MAI |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (MAI |-- f), the carrier of K
len (MAI |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(MAI |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (MAI |-- f), the carrier of K
[1,(MAI |-- f)] is set
{1,(MAI |-- f)} is non empty finite V32() set
{{1,(MAI |-- f)},{1}} is non empty finite V32() set
{[1,(MAI |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (MAI |-- f)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
width ((LineVec2Mx (MAI |-- f)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Line (((LineVec2Mx (MAI |-- f)) * A),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MAI |-- f)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MAI |-- f)) * A)) -tuples_on the carrier of K
(width ((LineVec2Mx (MAI |-- f)) * A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MAI |-- f)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MAI |-- f)) * A)) -tuples_on the carrier of K
V1 multfield is Relation-like 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 Function-like total quasi_total 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:]
K378( the carrier of K, the carrier of K, the multF of K,V1,(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(V1 multfield) * (Line (((LineVec2Mx (MAI |-- f)) * A),1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
lmlt ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:]
[: the carrier of K, the carrier of V3:] is Relation-like set
[:[: the carrier of K, the carrier of V3:], the carrier of V3:] is Relation-like set
bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:] is set
the lmult of V3 .: ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
width (LineVec2Mx (MAI |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (MAI |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len ((LineVec2Mx (MAI |-- f)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) is finite Element of bool NAT
dom (Line (((LineVec2Mx (MAI |-- f)) * A),1)) is finite width ((LineVec2Mx (MAI |-- f)) * A) -element Element of bool NAT
dom g is finite Element of bool NAT
(dom (Line (((LineVec2Mx (MAI |-- f)) * A),1))) /\ (dom g) is finite Element of bool NAT
len (V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (Line (((LineVec2Mx (MAI |-- f)) * A),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))) is finite width ((LineVec2Mx (MAI |-- f)) * A) -element Element of bool NAT
dom (lmlt ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g)) is finite Element of bool NAT
(dom (V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1)))) /\ (dom g) is finite Element of bool NAT
len (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (lmlt ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
w is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) . w is set
(lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) /. w is Element of the carrier of V3
(Line (((LineVec2Mx (MAI |-- f)) * A),1)) . w is set
(Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w is Element of the carrier of K
g . w is set
g /. w is Element of the carrier of V3
(V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))) . w is set
V1 * ((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w) is Element of the carrier of K
(lmlt ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g)) . w is set
(V1 * ((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w)) * (g /. w) is Element of the carrier of V3
the lmult of V3 . ((V1 * ((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w)),(g /. w)) is Element of the carrier of V3
((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w) * (g /. w) is Element of the carrier of V3
the lmult of V3 . (((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w),(g /. w)) is Element of the carrier of V3
V1 * (((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w) * (g /. w)) is Element of the carrier of V3
the lmult of V3 . (V1,(((Line (((LineVec2Mx (MAI |-- f)) * A),1)) /. w) * (g /. w))) is Element of the carrier of V3
V1 * ((lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) /. w) is Element of the carrier of V3
the lmult of V3 . (V1,((lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) /. w)) is Element of the carrier of V3
(K,V2,V3,f,g,(V1 * A)) . KER is set
(LineVec2Mx (MAI |-- f)) * (V1 * A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (MAI |-- f)) * (V1 * A)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (MAI |-- f)) * (V1 * A)) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (MAI |-- f)) * (V1 * A))) -tuples_on the carrier of K
width ((LineVec2Mx (MAI |-- f)) * (V1 * A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (MAI |-- f)) * (V1 * A))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (MAI |-- f)) * (V1 * A)),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line (((LineVec2Mx (MAI |-- f)) * (V1 * A)),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * (V1 * A)),1)),g)) is Element of the carrier of V3
V1 * ((LineVec2Mx (MAI |-- f)) * A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line ((V1 * ((LineVec2Mx (MAI |-- f)) * A)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (V1 * ((LineVec2Mx (MAI |-- f)) * A)) -element FinSequence-like FinSubsequence-like Element of (width (V1 * ((LineVec2Mx (MAI |-- f)) * A))) -tuples_on the carrier of K
width (V1 * ((LineVec2Mx (MAI |-- f)) * A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (V1 * ((LineVec2Mx (MAI |-- f)) * A))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((V1 * ((LineVec2Mx (MAI |-- f)) * A)),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line ((V1 * ((LineVec2Mx (MAI |-- f)) * A)),1)),g) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line ((V1 * ((LineVec2Mx (MAI |-- f)) * A)),1)),g)) is Element of the carrier of V3
Sum (lmlt ((V1 * (Line (((LineVec2Mx (MAI |-- f)) * A),1))),g)) is Element of the carrier of V3
Sum (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)) is Element of the carrier of V3
V1 * (Sum (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g))) is Element of the carrier of V3
the lmult of V3 . (V1,(Sum (lmlt ((Line (((LineVec2Mx (MAI |-- f)) * A),1)),g)))) is Element of the carrier of V3
(K,V2,V3,f,g,A) . MAI is Element of the carrier of V3
V1 * ((K,V2,V3,f,g,A) . MAI) is Element of the carrier of V3
the lmult of V3 . (V1,((K,V2,V3,f,g,A) . MAI)) is Element of the carrier of V3
(V1 * (K,V2,V3,f,g,A)) . KER is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
(K,V1,V2,(K,V1,V2,V3,f,A),V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V3 is non empty set
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V2,f,g,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len g, len A, the carrier of K
len AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A * AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
(K,V2,V3,g,A,AI) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V3:]
[: the carrier of V2, the carrier of V3:] is Relation-like set
bool [: the carrier of V2, the carrier of V3:] is set
(K,V2,V3,g,A,AI) * (K,V1,V2,f,g,A) is Relation-like Function-like non empty total total quasi_total quasi_total additive Element of bool [: the carrier of V1, the carrier of V3:]
[: the carrier of V1, the carrier of V3:] is Relation-like set
bool [: the carrier of V1, the carrier of V3:] is set
MAI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len A, the carrier of K
(K,V1,V3,f,A,MAI) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V3:]
x is set
v is Element of the carrier of V1
v |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (v |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (v |-- f), the carrier of K
len (v |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(v |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (v |-- f), the carrier of K
[1,(v |-- f)] is set
{1,(v |-- f)} is non empty finite V32() set
{{1,(v |-- f)},{1}} is non empty finite V32() set
{[1,(v |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (LineVec2Mx (v |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (LineVec2Mx (v |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(LineVec2Mx (v |-- f)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
len ((LineVec2Mx (v |-- f)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((LineVec2Mx (v |-- f)) * A) is finite Element of bool NAT
width ((LineVec2Mx (v |-- f)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Line (((LineVec2Mx (v |-- f)) * A),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (v |-- f)) * A) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (v |-- f)) * A)) -tuples_on the carrier of K
(width ((LineVec2Mx (v |-- f)) * A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Line (((LineVec2Mx (v |-- f)) * A),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom ((K,V2,V3,g,A,AI) * (K,V1,V2,f,g,A)) is non empty set
((K,V2,V3,g,A,AI) * (K,V1,V2,f,g,A)) . x is set
(K,V1,V2,f,g,A) . v is Element of the carrier of V2
(K,V2,V3,g,A,AI) . ((K,V1,V2,f,g,A) . v) is Element of the carrier of V3
lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 .: ((Line (((LineVec2Mx (v |-- f)) * A),1)),g) is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V2
Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g)) is Element of the carrier of V2
(K,V2,V3,g,A,AI) . (Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) is Element of the carrier of V3
(Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g), the carrier of K
len ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g), the carrier of K
[1,((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)] is set
{1,((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)} is non empty finite V32() set
{{1,((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)},{1}} is non empty finite V32() set
{[1,((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI)) -tuples_on the carrier of K
width ((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:]
[: the carrier of K, the carrier of V3:] is Relation-like set
[:[: the carrier of K, the carrier of V3:], the carrier of V3:] is Relation-like set
bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:] is set
the lmult of V3 .: ((Line (((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line (((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * A),1)),g))) |-- g)) * AI),1)),A)) is Element of the carrier of V3
LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((LineVec2Mx (v |-- f)) * A),1)), the carrier of K
K470( the carrier of K,(Line (((LineVec2Mx (v |-- f)) * A),1))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((LineVec2Mx (v |-- f)) * A),1)), the carrier of K
[1,(Line (((LineVec2Mx (v |-- f)) * A),1))] is set
{1,(Line (((LineVec2Mx (v |-- f)) * A),1))} is non empty finite V32() set
{{1,(Line (((LineVec2Mx (v |-- f)) * A),1))},{1}} is non empty finite V32() set
{[1,(Line (((LineVec2Mx (v |-- f)) * A),1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI)) -tuples_on the carrier of K
width ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line (((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line (((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line (((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * A),1))) * AI),1)),A)) is Element of the carrier of V3
((LineVec2Mx (v |-- f)) * A) * AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line ((((LineVec2Mx (v |-- f)) * A) * AI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (((LineVec2Mx (v |-- f)) * A) * AI) -element FinSequence-like FinSubsequence-like Element of (width (((LineVec2Mx (v |-- f)) * A) * AI)) -tuples_on the carrier of K
width (((LineVec2Mx (v |-- f)) * A) * AI) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (((LineVec2Mx (v |-- f)) * A) * AI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)), the carrier of K
len (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)), the carrier of K
[1,(Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))] is set
{1,(Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))} is non empty finite V32() set
{{1,(Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))},{1}} is non empty finite V32() set
{[1,(Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)))) -tuples_on the carrier of K
width (LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line ((LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line ((LineVec2Mx (Line ((((LineVec2Mx (v |-- f)) * A) * AI),1))),1)),A)) is Element of the carrier of V3
(LineVec2Mx (v |-- f)) * MAI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Line (((LineVec2Mx (v |-- f)) * MAI),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((LineVec2Mx (v |-- f)) * MAI) -element FinSequence-like FinSubsequence-like Element of (width ((LineVec2Mx (v |-- f)) * MAI)) -tuples_on the carrier of K
width ((LineVec2Mx (v |-- f)) * MAI) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width ((LineVec2Mx (v |-- f)) * MAI)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((LineVec2Mx (v |-- f)) * MAI),1)), the carrier of K
len (Line (((LineVec2Mx (v |-- f)) * MAI),1)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(Line (((LineVec2Mx (v |-- f)) * MAI),1))) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (Line (((LineVec2Mx (v |-- f)) * MAI),1)), the carrier of K
[1,(Line (((LineVec2Mx (v |-- f)) * MAI),1))] is set
{1,(Line (((LineVec2Mx (v |-- f)) * MAI),1))} is non empty finite V32() set
{{1,(Line (((LineVec2Mx (v |-- f)) * MAI),1))},{1}} is non empty finite V32() set
{[1,(Line (((LineVec2Mx (v |-- f)) * MAI),1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1)))) -tuples_on the carrier of K
width (LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
lmlt ((Line ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line ((LineVec2Mx (Line (((LineVec2Mx (v |-- f)) * MAI),1))),1)),A)) is Element of the carrier of V3
lmlt ((Line (((LineVec2Mx (v |-- f)) * MAI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
the lmult of V3 .: ((Line (((LineVec2Mx (v |-- f)) * MAI),1)),A) is Relation-like NAT -defined the carrier of V3 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V3
Sum (lmlt ((Line (((LineVec2Mx (v |-- f)) * MAI),1)),A)) is Element of the carrier of V3
(K,V1,V3,f,A,MAI) . x is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Element of the carrier of V2
g |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V2,V1,V3,f,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V1:]
[: the carrier of V2, the carrier of V1:] is Relation-like set
bool [: the carrier of V2, the carrier of V1:] is set
ker (K,V2,V1,V3,f,A) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
A @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Space_of_Solutions_of (A @) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (A @)) -VectSp_over K
width (A @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (A @)) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (A @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
LineVec2Mx (g |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
len (g |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(g |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (g |-- V3), the carrier of K
[1,(g |-- V3)] is set
{1,(g |-- V3)} is non empty finite V32() set
{{1,(g |-- V3)},{1}} is non empty finite V32() set
{[1,(g |-- V3)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
width (LineVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(len f) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len f -element FinSequence-like FinSubsequence-like Element of (len f) -tuples_on the carrier of K
(len f) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len ((len f) |-> (0. K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
LineVec2Mx ((len f) |-> (0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((len f) |-> (0. K)), the carrier of K
K470( the carrier of K,((len f) |-> (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 tabular V171() Matrix of 1, len ((len f) |-> (0. K)), the carrier of K
[1,((len f) |-> (0. K))] is set
{1,((len f) |-> (0. K))} is non empty finite V32() set
{{1,((len f) |-> (0. K))},{1}} is non empty finite V32() set
{[1,((len f) |-> (0. K))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
width (LineVec2Mx ((len f) |-> (0. K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
ColVec2Mx ((len f) |-> (0. K)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len ((len f) |-> (0. K)),1, the carrier of K
K470( the carrier of K,((len f) |-> (0. K))) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
width (ColVec2Mx ((len f) |-> (0. K))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
ColVec2Mx (g |-- V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len (g |-- V3),1, the carrier of K
K470( the carrier of K,(g |-- V3)) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
len (ColVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width (ColVec2Mx (g |-- V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V2,V1,V3,f,A) . g is Element of the carrier of V1
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
(LineVec2Mx (g |-- V3)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
(0. V1) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((0. V1) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((0. V1) |-- f), the carrier of K
len ((0. V1) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((0. V1) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((0. V1) |-- f), the carrier of K
[1,((0. V1) |-- f)] is set
{1,((0. V1) |-- f)} is non empty finite V32() set
{{1,((0. V1) |-- f)},{1}} is non empty finite V32() set
{[1,((0. V1) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(A @) * (ColVec2Mx (g |-- V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Solutions_of ((A @),(ColVec2Mx ((len f) |-> (0. K)))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K * : ( len b₁ = width (A @) & width b₁ = width (ColVec2Mx ((len f) |-> (0. K))) & (A @) * b₁ = ColVec2Mx ((len f) |-> (0. K)) ) } is set
Solutions_of ((A @),((len f) |-> (0. K))) is Element of bool the carrier of ((width (A @)) -VectSp_over K)
the carrier of ((width (A @)) -VectSp_over K) is non empty set
bool the carrier of ((width (A @)) -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 : ColVec2Mx b₁ in Solutions_of ((A @),(ColVec2Mx ((len f) |-> (0. K)))) } is set
the carrier of (Space_of_Solutions_of (A @)) is non empty set
the carrier of (Space_of_Solutions_of (A @)) is non empty set
Solutions_of ((A @),((len f) |-> (0. K))) is Element of bool the carrier of ((width (A @)) -VectSp_over K)
the carrier of ((width (A @)) -VectSp_over K) is non empty set
bool the carrier of ((width (A @)) -VectSp_over K) is set
Solutions_of ((A @),(ColVec2Mx ((len f) |-> (0. K)))) is set
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K * : ( len b₁ = width (A @) & width b₁ = width (ColVec2Mx ((len f) |-> (0. K))) & (A @) * b₁ = ColVec2Mx ((len f) |-> (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 : ColVec2Mx b₁ in Solutions_of ((A @),(ColVec2Mx ((len f) |-> (0. K)))) } is set
MAI is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ColVec2Mx MAI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len MAI,1, the carrier of K
len MAI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,MAI) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len MAI, the carrier of K
[1,MAI] is set
{1,MAI} is non empty finite V32() set
{{1,MAI},{1}} is non empty finite V32() set
{[1,MAI]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K470( the carrier of K,MAI) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
(LineVec2Mx (g |-- V3)) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
((LineVec2Mx (g |-- V3)) * A) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
MAI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
len MAI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width MAI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(A @) * MAI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
width ((LineVec2Mx (g |-- V3)) * A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
(0. V1) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx ((0. V1) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((0. V1) |-- f), the carrier of K
len ((0. V1) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((0. V1) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((0. V1) |-- f), the carrier of K
[1,((0. V1) |-- f)] is set
{1,((0. V1) |-- f)} is non empty finite V32() set
{{1,((0. V1) |-- f)},{1}} is non empty finite V32() set
{[1,((0. V1) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(K,V2,V1,V3,f,A) . g is Element of the carrier of V1
((K,V2,V1,V3,f,A) . g) |-- f is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,V1,V3,f,A) . g) |-- f), the carrier of K
len (((K,V2,V1,V3,f,A) . g) |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(((K,V2,V1,V3,f,A) . g) |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,V1,V3,f,A) . g) |-- f), the carrier of K
[1,(((K,V2,V1,V3,f,A) . g) |-- f)] is set
{1,(((K,V2,V1,V3,f,A) . g) |-- f)} is non empty finite V32() set
{{1,(((K,V2,V1,V3,f,A) . g) |-- f)},{1}} is non empty finite V32() set
{[1,(((K,V2,V1,V3,f,A) . g) |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f)) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f))) -tuples_on the carrier of K
width (LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx (((K,V2,V1,V3,f,A) . g) |-- f))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the carrier of V1 is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
V2 is set
(Omega). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
the carrier of K is non empty non trivial set
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) is non empty strict VectSpStr over K
the carrier of ((Omega). V1) is non empty set
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of ((0). V1) is non empty set
V2 is Element of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
V3 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
ker V3 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
f is set
dom V3 is non empty set
V3 . f is set
g is set
V3 . g is set
A is Element of the carrier of V1
A is Element of the carrier of V1
A - A is Element of the carrier of V1
- A is Element of the carrier of V1
A + (- A) is Element of the carrier of V1
the carrier of ((0). V1) is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
A + (- A) is Element of the carrier of V1
- (- A) is Element of the carrier of V1
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
f is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
V3 + f is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
the carrier of K is non empty non trivial set
g is Element of the carrier of K
A is Element of the carrier of V1
g * A is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . (g,A) is Element of the carrier of V1
(V3 + f) . (g * A) is Element of the carrier of V2
V3 . (g * A) is Element of the carrier of V2
f . (g * A) is Element of the carrier of V2
(V3 . (g * A)) + (f . (g * A)) is Element of the carrier of V2
V3 . A is Element of the carrier of V2
g * (V3 . A) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . (g,(V3 . A)) is Element of the carrier of V2
(g * (V3 . A)) + (f . (g * A)) is Element of the carrier of V2
f . A is Element of the carrier of V2
g * (f . A) is Element of the carrier of V2
the lmult of V2 . (g,(f . A)) is Element of the carrier of V2
(g * (V3 . A)) + (g * (f . A)) is Element of the carrier of V2
(V3 . A) + (f . A) is Element of the carrier of V2
g * ((V3 . A) + (f . A)) is Element of the carrier of V2
the lmult of V2 . (g,((V3 . A) + (f . A))) is Element of the carrier of V2
(V3 + f) . A is Element of the carrier of V2
g * ((V3 + f) . A) is Element of the carrier of V2
the lmult of V2 . (g,((V3 + f) . A)) is Element of the carrier of V2
g is Element of the carrier of V1
A is Element of the carrier of V1
g + A is Element of the carrier of V1
(V3 + f) . (g + A) is Element of the carrier of V2
V3 . (g + A) is Element of the carrier of V2
f . (g + A) is Element of the carrier of V2
(V3 . (g + A)) + (f . (g + A)) is Element of the carrier of V2
V3 . g is Element of the carrier of V2
V3 . A is Element of the carrier of V2
(V3 . g) + (V3 . A) is Element of the carrier of V2
((V3 . g) + (V3 . A)) + (f . (g + A)) is Element of the carrier of V2
f . g is Element of the carrier of V2
f . A is Element of the carrier of V2
(f . g) + (f . A) is Element of the carrier of V2
((V3 . g) + (V3 . A)) + ((f . g) + (f . A)) is Element of the carrier of V2
(V3 . A) + ((f . g) + (f . A)) is Element of the carrier of V2
(V3 . g) + ((V3 . A) + ((f . g) + (f . A))) is Element of the carrier of V2
(f . A) + (V3 . A) is Element of the carrier of V2
(f . g) + ((f . A) + (V3 . A)) is Element of the carrier of V2
(V3 . g) + ((f . g) + ((f . A) + (V3 . A))) is Element of the carrier of V2
(V3 . g) + (f . g) is Element of the carrier of V2
(V3 . A) + (f . A) is Element of the carrier of V2
((V3 . g) + (f . g)) + ((V3 . A) + (f . A)) is Element of the carrier of V2
(V3 + f) . g is Element of the carrier of V2
((V3 + f) . g) + ((V3 . A) + (f . A)) is Element of the carrier of V2
(V3 + f) . A is Element of the carrier of V2
((V3 + f) . g) + ((V3 + f) . A) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
the carrier of K is non empty non trivial set
V3 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
f is Element of the carrier of K
f * V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
g is Element of the carrier of K
A is Element of the carrier of V1
g * A is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . (g,A) is Element of the carrier of V1
(f * V3) . (g * A) is Element of the carrier of V2
V3 . (g * A) is Element of the carrier of V2
f * (V3 . (g * A)) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . (f,(V3 . (g * A))) is Element of the carrier of V2
V3 . A is Element of the carrier of V2
g * (V3 . A) is Element of the carrier of V2
the lmult of V2 . (g,(V3 . A)) is Element of the carrier of V2
f * (g * (V3 . A)) is Element of the carrier of V2
the lmult of V2 . (f,(g * (V3 . A))) is Element of the carrier of V2
f * g is Element of the carrier of K
(f * g) * (V3 . A) is Element of the carrier of V2
the lmult of V2 . ((f * g),(V3 . A)) is Element of the carrier of V2
f * (V3 . A) is Element of the carrier of V2
the lmult of V2 . (f,(V3 . A)) is Element of the carrier of V2
g * (f * (V3 . A)) is Element of the carrier of V2
the lmult of V2 . (g,(f * (V3 . A))) is Element of the carrier of V2
(f * V3) . A is Element of the carrier of V2
g * ((f * V3) . A) is Element of the carrier of V2
the lmult of V2 . (g,((f * V3) . A)) is Element of the carrier of V2
g is Element of the carrier of V1
A is Element of the carrier of V1
g + A is Element of the carrier of V1
(f * V3) . (g + A) is Element of the carrier of V2
V3 . (g + A) is Element of the carrier of V2
f * (V3 . (g + A)) is Element of the carrier of V2
the lmult of V2 . (f,(V3 . (g + A))) is Element of the carrier of V2
V3 . g is Element of the carrier of V2
V3 . A is Element of the carrier of V2
(V3 . g) + (V3 . A) is Element of the carrier of V2
f * ((V3 . g) + (V3 . A)) is Element of the carrier of V2
the lmult of V2 . (f,((V3 . g) + (V3 . A))) is Element of the carrier of V2
f * (V3 . g) is Element of the carrier of V2
the lmult of V2 . (f,(V3 . g)) is Element of the carrier of V2
f * (V3 . A) is Element of the carrier of V2
the lmult of V2 . (f,(V3 . A)) is Element of the carrier of V2
(f * (V3 . g)) + (f * (V3 . A)) is Element of the carrier of V2
(f * V3) . g is Element of the carrier of V2
((f * V3) . g) + (f * (V3 . A)) is Element of the carrier of V2
(f * V3) . A is Element of the carrier of V2
((f * V3) . g) + ((f * V3) . A) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V3 is non empty set
[: the carrier of V2, the carrier of V3:] is Relation-like set
bool [: the carrier of V2, the carrier of V3:] is set
f is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
g is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V3:]
f (#) g is Relation-like Function-like set
[: the carrier of V1, the carrier of V3:] is Relation-like set
bool [: the carrier of V1, the carrier of V3:] is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
the_rank_of g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
g @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Space_of_Solutions_of (g @) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (g @)) -VectSp_over K
width (g @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (g @)) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(Omega). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the addF of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
[:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of V1, the carrier of V1:], the carrier of V1:] is set
the ZeroF of V1 is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) is non empty strict VectSpStr over K
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
ker (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (ker (K,V1,V2,V3,f,g)) is non empty set
0. V1 is V47(V1) Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
width g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
ker (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (ker (K,V1,V2,V3,f,g)) is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
AI is set
dim (Space_of_Solutions_of (g @)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the_rank_of (g @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) - (the_rank_of (g @)) is ext-real V93() V94() set
(len V3) - (len V3) is ext-real V93() V94() set
(Omega). (Space_of_Solutions_of (g @)) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of Space_of_Solutions_of (g @)
the carrier of (Space_of_Solutions_of (g @)) is non empty set
the addF of (Space_of_Solutions_of (g @)) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of (Space_of_Solutions_of (g @)), the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):]
[: the carrier of (Space_of_Solutions_of (g @)), the carrier of (Space_of_Solutions_of (g @)):] is Relation-like set
[:[: the carrier of (Space_of_Solutions_of (g @)), the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):] is Relation-like set
bool [:[: the carrier of (Space_of_Solutions_of (g @)), the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):] is set
the ZeroF of (Space_of_Solutions_of (g @)) is Element of the carrier of (Space_of_Solutions_of (g @))
the lmult of (Space_of_Solutions_of (g @)) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):]
[: the carrier of K, the carrier of (Space_of_Solutions_of (g @)):] is Relation-like set
[:[: the carrier of K, the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):] is Relation-like set
bool [:[: the carrier of K, the carrier of (Space_of_Solutions_of (g @)):], the carrier of (Space_of_Solutions_of (g @)):] is set
VectSpStr(# the carrier of (Space_of_Solutions_of (g @)), the addF of (Space_of_Solutions_of (g @)), the ZeroF of (Space_of_Solutions_of (g @)), the lmult of (Space_of_Solutions_of (g @)) #) is non empty strict VectSpStr over K
(0). (Space_of_Solutions_of (g @)) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of Space_of_Solutions_of (g @)
S is Element of the carrier of V1
S |-- V3 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 ((0). (Space_of_Solutions_of (g @))) is non empty set
(len V3) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(0). ((len V3) -VectSp_over K) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (len V3) -VectSp_over K
the carrier of ((0). ((len V3) -VectSp_over K)) is non empty set
the carrier of ((len V3) -VectSp_over K) is non empty set
0. ((len V3) -VectSp_over K) is Relation-like Function-like V47((len V3) -VectSp_over K) Element of the carrier of ((len V3) -VectSp_over K)
the ZeroF of ((len V3) -VectSp_over K) is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
{(0. ((len V3) -VectSp_over K))} is functional non empty trivial finite 1 -element Element of bool the carrier of ((len V3) -VectSp_over K)
bool the carrier of ((len V3) -VectSp_over K) is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(len V3) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len V3 -element FinSequence-like FinSubsequence-like Element of (len V3) -tuples_on the carrier of K
(len V3) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(0. V1) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ker (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (ker (K,V1,V2,V3,f,g)) is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
ker (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of (ker (K,V1,V2,V3,f,g)) is non empty set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
{(0. V1)} is non empty trivial finite 1 -element Element of bool the carrier of V1
bool the carrier of V1 is set
(0). V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
the carrier of ((0). V1) is non empty set
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
1. (V1,K) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of K,K, the carrier of V1
the carrier of V1 is non empty non trivial set
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
the_rank_of (1. (V1,K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
1_ V1 is Element of the carrier of V1
1. V1 is V47(V1) Element of the carrier of V1
the OneF of V1 is Element of the carrier of V1
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
K + {} is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
Det (1. (V1,K)) is Element of the carrier of V1
K is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V1 is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
1. (V1,K) is Relation-like NAT -defined the carrier of V1 * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of K,K, the carrier of V1
the carrier of V1 is non empty non trivial set
the carrier of V1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of V1
MX2FinS (1. (V1,K)) is Relation-like NAT -defined the carrier of (K -VectSp_over V1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (K -VectSp_over V1)
K -VectSp_over V1 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over V1
the carrier of (K -VectSp_over V1) is non empty set
the_rank_of (1. (V1,K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Indices (1. (V1,K)) is set
dom (1. (V1,K)) is finite Element of bool NAT
width (1. (V1,K)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Seg (width (1. (V1,K))) is finite width (1. (V1,K)) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (V1,K)) ) } is set
[:(dom (1. (V1,K))),(Seg (width (1. (V1,K)))):] is Relation-like finite set
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[V3,f] is set
{V3,f} is non empty finite V32() set
{V3} is non empty trivial finite V32() 1 -element set
{{V3,f},{V3}} is non empty finite V32() set
(1. (V1,K)) * (V3,f) is Element of the carrier of V1
lines (1. (V1,K)) is finite Element of bool the carrier of (K -VectSp_over V1)
bool the carrier of (K -VectSp_over V1) is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V2) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of ((len V2) -VectSp_over K) is non empty set
1. (K,(len V2)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V2, len V2, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
MX2FinS (1. (K,(len V2))) is Relation-like NAT -defined the carrier of ((len V2) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of ((len V2) -VectSp_over K)
V3 is Relation-like NAT -defined the carrier of ((len V2) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of (len V2) -VectSp_over K
f is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
f |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
rng V3 is finite set
len (f |-- V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A is Relation-like Function-like total quasi_total Linear_Combination of (len V2) -VectSp_over K
Sum A is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
Carrier A is finite Element of bool the carrier of ((len V2) -VectSp_over K)
bool the carrier of ((len V2) -VectSp_over K) is set
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
{ b₁ where b₁ is Element of the carrier of ((len V2) -VectSp_over K) : not A . b₁ = 0. K } is set
(len V2) -tuples_on 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 len V2 -element FinSequence-like FinSubsequence-like Element of (len V2) -tuples_on the carrier of K
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dim ((len V2) -VectSp_over K) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dom V3 is finite Element of bool NAT
dom (f |-- V3) is finite Element of bool NAT
the_rank_of (1. (K,(len V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A (#) V3 is Relation-like NAT -defined the carrier of ((len V2) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of ((len V2) -VectSp_over K)
FinS2MX (A (#) V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len (A (#) V3), len V2, the carrier of K
len (A (#) V3) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Indices (1. (K,(len V2))) is set
dom (1. (K,(len V2))) is finite Element of bool NAT
width (1. (K,(len V2))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Seg (width (1. (K,(len V2)))) is finite width (1. (K,(len V2))) -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,(len V2))) ) } is set
[:(dom (1. (K,(len V2)))),(Seg (width (1. (K,(len V2))))):] is Relation-like finite set
Seg (len V2) is finite len V2 -element V136() Element of bool NAT
{ b₁ where b₁ is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT : ( 1 <= b₁ & b₁ <= len V2 ) } is set
[:(Seg (len V2)),(Seg (len V2)):] is Relation-like finite set
S is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[S,S] is set
{S,S} is non empty finite V32() set
{S} is non empty trivial finite V32() 1 -element set
{{S,S},{S}} is non empty finite V32() set
Line ((1. (K,(len V2))),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
(width (1. (K,(len V2)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(Line ((1. (K,(len V2))),S)) . S is set
(1. (K,(len V2))) * (S,S) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V47(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
Col ((FinS2MX (A (#) V3)),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (FinS2MX (A (#) V3)) -element FinSequence-like FinSubsequence-like Element of (len (FinS2MX (A (#) V3))) -tuples_on the carrier of K
len (FinS2MX (A (#) V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len (FinS2MX (A (#) V3))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len (Col ((FinS2MX (A (#) V3)),S)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dom (Col ((FinS2MX (A (#) V3)),S)) is finite len (FinS2MX (A (#) V3)) -element Element of bool NAT
dom (FinS2MX (A (#) V3)) is finite Element of bool NAT
width (FinS2MX (A (#) V3)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
KER is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
[KER,S] is set
{KER,S} is non empty finite V32() set
{KER} is non empty trivial finite V32() 1 -element set
{{KER,S},{KER}} is non empty finite V32() set
Line ((FinS2MX (A (#) V3)),KER) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (FinS2MX (A (#) V3)) -element FinSequence-like FinSubsequence-like Element of (width (FinS2MX (A (#) V3))) -tuples_on the carrier of K
(width (FinS2MX (A (#) V3))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
(A (#) V3) . KER is set
V3 /. KER is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
A . (V3 /. KER) is Element of the carrier of K
(A . (V3 /. KER)) * (V3 /. KER) is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
the lmult of ((len V2) -VectSp_over K) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of ((len V2) -VectSp_over K):], the carrier of ((len V2) -VectSp_over K):]
[: the carrier of K, the carrier of ((len V2) -VectSp_over K):] is Relation-like set
[:[: the carrier of K, the carrier of ((len V2) -VectSp_over K):], the carrier of ((len V2) -VectSp_over K):] is Relation-like set
bool [:[: the carrier of K, the carrier of ((len V2) -VectSp_over K):], the carrier of ((len V2) -VectSp_over K):] is set
the lmult of ((len V2) -VectSp_over K) . ((A . (V3 /. KER)),(V3 /. KER)) is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
(Col ((FinS2MX (A (#) V3)),S)) . KER is set
(FinS2MX (A (#) V3)) * (KER,S) is Element of the carrier of K
(Line ((FinS2MX (A (#) V3)),KER)) . S is set
Line ((1. (K,(len V2))),KER) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
(Line ((1. (K,(len V2))),KER)) . S is set
(1. (K,(len V2))) * (KER,S) is Element of the carrier of K
V3 . KER is set
(A . (V3 /. KER)) * (Line ((1. (K,(len V2))),KER)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
(A . (V3 /. KER)) multfield is Relation-like 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 Function-like total quasi_total 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:]
K378( the carrier of K, the carrier of K, the multF of K,(A . (V3 /. KER)),(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
((A . (V3 /. KER)) multfield) * (Line ((1. (K,(len V2))),KER)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((A . (V3 /. KER)) * (Line ((1. (K,(len V2))),KER))) . S is set
(A . (V3 /. KER)) * (0. K) is Element of the carrier of K
(Col ((FinS2MX (A (#) V3)),S)) . S is set
Sum (Col ((FinS2MX (A (#) V3)),S)) is Element of the carrier of K
f . S is set
Line ((FinS2MX (A (#) V3)),S) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (FinS2MX (A (#) V3)) -element FinSequence-like FinSubsequence-like Element of (width (FinS2MX (A (#) V3))) -tuples_on the carrier of K
(A (#) V3) . S is set
V3 /. S is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
A . (V3 /. S) is Element of the carrier of K
(A . (V3 /. S)) * (V3 /. S) is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
the lmult of ((len V2) -VectSp_over K) . ((A . (V3 /. S)),(V3 /. S)) is Relation-like Function-like Element of the carrier of ((len V2) -VectSp_over K)
(FinS2MX (A (#) V3)) * (S,S) is Element of the carrier of K
(Line ((FinS2MX (A (#) V3)),S)) . S is set
V3 . S is set
(A . (V3 /. S)) * (Line ((1. (K,(len V2))),S)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (1. (K,(len V2))) -element FinSequence-like FinSubsequence-like Element of (width (1. (K,(len V2)))) -tuples_on the carrier of K
(A . (V3 /. S)) multfield is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K378( the carrier of K, the carrier of K, the multF of K,(A . (V3 /. S)),(id the carrier of K)) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
((A . (V3 /. S)) multfield) * (Line ((1. (K,(len V2))),S)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((A . (V3 /. S)) * (Line ((1. (K,(len V2))),S))) . S is set
(A . (V3 /. S)) * (1_ K) is Element of the carrier of K
A . S is set
(f |-- V3) /. S is Element of the carrier of K
(f |-- V3) . S is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V2, the carrier of V1:] is Relation-like set
bool [: the carrier of V2, the carrier of V1:] is set
V3 is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of V1:]
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len g) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of ((len g) -VectSp_over K) is non empty set
1. (K,(len g)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len g, len g, the carrier of K
MX2FinS (1. (K,(len g))) is Relation-like NAT -defined the carrier of ((len g) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of ((len g) -VectSp_over K)
(K,V2,V1,V3,f,g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len g, the carrier of K
A is Element of the carrier of V2
V3 . A is Element of the carrier of V1
(V3 . A) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A is Relation-like NAT -defined the carrier of ((len g) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of (len g) -VectSp_over K
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len f, len A, the carrier of K
(K,V2,((len g) -VectSp_over K),f,A,AI) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of ((len g) -VectSp_over K):]
[: the carrier of V2, the carrier of ((len g) -VectSp_over K):] is Relation-like set
bool [: the carrier of V2, the carrier of ((len g) -VectSp_over K):] is set
(K,V2,((len g) -VectSp_over K),f,A,AI) . A is Relation-like Function-like Element of the carrier of ((len g) -VectSp_over K)
dim V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(Omega). V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
the addF of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:]
[: the carrier of V2, the carrier of V2:] is Relation-like set
[:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of V2, the carrier of V2:], the carrier of V2:] is set
the ZeroF of V2 is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
VectSpStr(# the carrier of V2, the addF of V2, the ZeroF of V2, the lmult of V2 #) is non empty strict VectSpStr over K
(0). V2 is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
0. V2 is V47(V2) Element of the carrier of V2
{(0. V2)} is non empty trivial finite 1 -element Element of bool the carrier of V2
bool the carrier of V2 is set
S is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V1:]
ker S is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
0. V1 is V47(V1) Element of the carrier of V1
the ZeroF of V1 is Element of the carrier of V1
(0. V1) |-- g is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
(len g) |-> (0. K) is Relation-like NAT -defined the carrier of K -valued Function-like finite len g -element FinSequence-like FinSubsequence-like Element of (len g) -tuples_on the carrier of K
(len g) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. ((len g) -VectSp_over K) is Relation-like Function-like V47((len g) -VectSp_over K) Element of the carrier of ((len g) -VectSp_over K)
the ZeroF of ((len g) -VectSp_over K) is Relation-like Function-like Element of the carrier of ((len g) -VectSp_over K)
((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A), the carrier of K
len (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A), the carrier of K
[1,(((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)] is set
{1,(((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)} is non empty finite V32() set
{{1,(((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)},{1}} is non empty finite V32() set
{[1,(((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
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
LineVec2Mx (A |-- f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- f), the carrier of K
len (A |-- f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,(A |-- f)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len (A |-- f), the carrier of K
[1,(A |-- f)] is set
{1,(A |-- f)} is non empty finite V32() set
{{1,(A |-- f)},{1}} is non empty finite V32() set
{[1,(A |-- f)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(LineVec2Mx (A |-- f)) * AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
LineVec2Mx ((V3 . A) |-- g) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . A) |-- g), the carrier of K
len ((V3 . A) |-- g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K470( the carrier of K,((V3 . A) |-- g)) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular V171() Matrix of 1, len ((V3 . A) |-- g), the carrier of K
[1,((V3 . A) |-- g)] is set
{1,((V3 . A) |-- g)} is non empty finite V32() set
{{1,((V3 . A) |-- g)},{1}} is non empty finite V32() set
{[1,((V3 . A) |-- g)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Line ((LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)) -element FinSequence-like FinSubsequence-like Element of (width (LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A))) -tuples_on the carrier of K
width (LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (LineVec2Mx (((K,V2,((len g) -VectSp_over K),f,A,AI) . A) |-- A))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
K is non empty right_complementable unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
f is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V2:]
(V3,f) is set
the carrier of V3 is non empty set
f | the carrier of V3 is Relation-like Function-like set
[: the carrier of V3, the carrier of V2:] is Relation-like set
bool [: the carrier of V3, the carrier of V2:] is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed Subspace of V1
f is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
(V3,f) is set
the carrier of V3 is non empty set
f | the carrier of V3 is Relation-like Function-like set
[: the carrier of V3, the carrier of V2:] is Relation-like set
bool [: the carrier of V3, the carrier of V2:] is set
(K,V1,V2,V3,f) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V3, the carrier of V2:]
dom (K,V1,V2,V3,f) is non empty set
the carrier of K is non empty non trivial set
A is Element of the carrier of V3
A is Element of the carrier of K
A * A is Element of the carrier of V3
the lmult of V3 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:]
[: the carrier of K, the carrier of V3:] is Relation-like set
[:[: the carrier of K, the carrier of V3:], the carrier of V3:] is Relation-like set
bool [:[: the carrier of K, the carrier of V3:], the carrier of V3:] is set
the lmult of V3 . (A,A) is Element of the carrier of V3
AI is Element of the carrier of V1
A * AI is Element of the carrier of V1
the lmult of V1 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:]
[: the carrier of K, the carrier of V1:] is Relation-like set
[:[: the carrier of K, the carrier of V1:], the carrier of V1:] is Relation-like set
bool [:[: the carrier of K, the carrier of V1:], the carrier of V1:] is set
the lmult of V1 . (A,AI) is Element of the carrier of V1
(K,V1,V2,V3,f) . (A * A) is Element of the carrier of V2
f . (A * AI) is Element of the carrier of V2
f . AI is Element of the carrier of V2
A * (f . AI) is Element of the carrier of V2
the lmult of V2 is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:]
[: the carrier of K, the carrier of V2:] is Relation-like set
[:[: the carrier of K, the carrier of V2:], the carrier of V2:] is Relation-like set
bool [:[: the carrier of K, the carrier of V2:], the carrier of V2:] is set
the lmult of V2 . (A,(f . AI)) is Element of the carrier of V2
(K,V1,V2,V3,f) . A is Element of the carrier of V2
A * ((K,V1,V2,V3,f) . A) is Element of the carrier of V2
the lmult of V2 . (A,((K,V1,V2,V3,f) . A)) is Element of the carrier of V2
A is Element of the carrier of V3
A is Element of the carrier of V3
A + A is Element of the carrier of V3
AI is Element of the carrier of V1
S is Element of the carrier of V1
AI + S is Element of the carrier of V1
(K,V1,V2,V3,f) . (A + A) is Element of the carrier of V2
f . (AI + S) is Element of the carrier of V2
f . AI is Element of the carrier of V2
f . S is Element of the carrier of V2
(f . AI) + (f . S) is Element of the carrier of V2
(K,V1,V2,V3,f) . A is Element of the carrier of V2
((K,V1,V2,V3,f) . A) + (f . S) is Element of the carrier of V2
(K,V1,V2,V3,f) . A is Element of the carrier of V2
((K,V1,V2,V3,f) . A) + ((K,V1,V2,V3,f) . A) is Element of the carrier of V2
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
id V1 is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V1:]
[: the carrier of V1, the carrier of V1:] is Relation-like set
bool [: the carrier of V1, the carrier of V1:] is set
id the carrier of V1 is Relation-like the carrier of V1 -defined the carrier of V1 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V1:]
(len V3) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of ((len V3) -VectSp_over K) is non empty set
1. (K,(len V3)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of K
MX2FinS (1. (K,(len V3))) is Relation-like NAT -defined the carrier of ((len V3) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of ((len V3) -VectSp_over K)
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,A) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
nullity (K,V1,V2,V3,f,A) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker (K,V1,V2,V3,f,A) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker (K,V1,V2,V3,f,A)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the_rank_of A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) - (the_rank_of A) is ext-real V93() V94() set
A is Relation-like NAT -defined the carrier of ((len V3) -VectSp_over K) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of (len V3) -VectSp_over K
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dim ((len V3) -VectSp_over K) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(K,V1,V1,(id V1),V3,V3) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of K
0. K is V47(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V47(K) Element of the carrier of K
the OneF of K is Element of the carrier of K
(len V3) + {} is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
Det (1. (K,(len V3))) is Element of the carrier of K
AI is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len A, the carrier of K
the_rank_of AI is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
A @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() FinSequence of the carrier of K *
Space_of_Solutions_of (A @) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (width (A @)) -VectSp_over K
width (A @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(width (A @)) -VectSp_over K is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
(K,V1,((len V3) -VectSp_over K),V3,A,AI) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of ((len V3) -VectSp_over K):]
[: the carrier of V1, the carrier of ((len V3) -VectSp_over K):] is Relation-like set
bool [: the carrier of V1, the carrier of ((len V3) -VectSp_over K):] is set
(K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of (ker (K,V1,V2,V3,f,A)), the carrier of ((len V3) -VectSp_over K):]
the carrier of (ker (K,V1,V2,V3,f,A)) is non empty set
[: the carrier of (ker (K,V1,V2,V3,f,A)), the carrier of ((len V3) -VectSp_over K):] is Relation-like set
bool [: the carrier of (ker (K,V1,V2,V3,f,A)), the carrier of ((len V3) -VectSp_over K):] is set
(K,V1,((len V3) -VectSp_over K),V3,A,AI) | the carrier of (ker (K,V1,V2,V3,f,A)) is Relation-like Function-like set
dom (K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) is non empty set
im (K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (len V3) -VectSp_over K
the carrier of (im (K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI))) is non empty set
the carrier of (Space_of_Solutions_of (A @)) is non empty set
x is set
v is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
w is Element of the carrier of (ker (K,V1,V2,V3,f,A))
(K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) . w is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
(K,V1,((len V3) -VectSp_over K),V3,A,AI) . w is set
W is Element of the carrier of V1
(id V1) . W is Element of the carrier of V1
((id V1) . W) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
W |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
width A is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
rank (K,V1,((len V3) -VectSp_over K),V3,A,AI) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
im (K,V1,((len V3) -VectSp_over K),V3,A,AI) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (len V3) -VectSp_over K
dim (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(Omega). ((len V3) -VectSp_over K) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of (len V3) -VectSp_over K
the addF of ((len V3) -VectSp_over K) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of ((len V3) -VectSp_over K), the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):]
[: the carrier of ((len V3) -VectSp_over K), the carrier of ((len V3) -VectSp_over K):] is Relation-like set
[:[: the carrier of ((len V3) -VectSp_over K), the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):] is Relation-like set
bool [:[: the carrier of ((len V3) -VectSp_over K), the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):] is set
the ZeroF of ((len V3) -VectSp_over K) is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
the lmult of ((len V3) -VectSp_over K) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):]
[: the carrier of K, the carrier of ((len V3) -VectSp_over K):] is Relation-like set
[:[: the carrier of K, the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):] is Relation-like set
bool [:[: the carrier of K, the carrier of ((len V3) -VectSp_over K):], the carrier of ((len V3) -VectSp_over K):] is set
VectSpStr(# the carrier of ((len V3) -VectSp_over K), the addF of ((len V3) -VectSp_over K), the ZeroF of ((len V3) -VectSp_over K), the lmult of ((len V3) -VectSp_over K) #) is non empty strict VectSpStr over K
(Omega). (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of im (K,V1,((len V3) -VectSp_over K),V3,A,AI)
the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is non empty set
the addF of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):]
[: the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is Relation-like set
[:[: the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is Relation-like set
bool [:[: the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is set
the ZeroF of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is Element of the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI))
the lmult of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) is Relation-like Function-like total quasi_total Element of bool [:[: the carrier of K, the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):]
[: the carrier of K, the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is Relation-like set
[:[: the carrier of K, the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is Relation-like set
bool [:[: the carrier of K, the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):], the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)):] is set
VectSpStr(# the carrier of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the addF of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the ZeroF of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)), the lmult of (im (K,V1,((len V3) -VectSp_over K),V3,A,AI)) #) is non empty strict VectSpStr over K
x is set
v is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
w is Element of the carrier of V1
(K,V1,((len V3) -VectSp_over K),V3,A,AI) . w is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
(id V1) . w is Element of the carrier of V1
((id V1) . w) |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
w |-- V3 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
W is Element of the carrier of (ker (K,V1,V2,V3,f,A))
(K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) . W is Relation-like Function-like Element of the carrier of ((len V3) -VectSp_over K)
rank (K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (im (K,V1,((len V3) -VectSp_over K),(ker (K,V1,V2,V3,f,A)),(K,V1,((len V3) -VectSp_over K),V3,A,AI))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the_rank_of (A @) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(len V3) - (the_rank_of (A @)) is ext-real V93() V94() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
g is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
rank g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
im g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
dim (im g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(K,V1,V2,g,V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
the_rank_of (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
len (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
nullity g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(rank g) + (nullity g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
width (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
dim V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
nullity g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(rank g) + (nullity g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(K,V1,V2,V3,f,(K,V1,V2,g,V3,f)) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
nullity (K,V1,V2,V3,f,(K,V1,V2,g,V3,f)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker (K,V1,V2,V3,f,(K,V1,V2,g,V3,f)) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker (K,V1,V2,V3,f,(K,V1,V2,g,V3,f))) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(len V3) - (the_rank_of (K,V1,V2,g,V3,f)) is ext-real V93() V94() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
g is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
rank (K,V1,V2,V3,f,g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
im (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
dim (im (K,V1,V2,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the_rank_of g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V2,(K,V1,V2,V3,f,g),V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
the_rank_of (K,V1,V2,(K,V1,V2,V3,f,g),V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
0. K is V47(K) Element of the carrier of K
the carrier of K is non empty non trivial set
the ZeroF of K is Element of the carrier of K
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
dim V2 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
V3 is Relation-like NAT -defined the carrier of V1 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V1
len V3 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
f is Relation-like NAT -defined the carrier of V2 -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of V2
g is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
ker g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
(K,V1,V2,g,V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len V3, the carrier of K
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
Det (K,V1,V2,g,V3,f) is Element of the carrier of K
len f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
(K,V1,V2,g,V3,f) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular V171() Matrix of len V3, len f, the carrier of K
rank g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
im g is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
dim (im g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
nullity g is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (ker g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(rank g) + (nullity g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the_rank_of (K,V1,V2,g,V3,f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V3 is non empty set
[: the carrier of V2, the carrier of V3:] is Relation-like set
bool [: the carrier of V2, the carrier of V3:] is set
f is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
im f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
the carrier of (im f) is non empty set
g is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V2, the carrier of V3:]
g * f is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of V1, the carrier of V3:]
[: the carrier of V1, the carrier of V3:] is Relation-like set
bool [: the carrier of V1, the carrier of V3:] is set
(K,V2,V3,(im f),g) is Relation-like Function-like non empty total quasi_total Element of bool [: the carrier of (im f), the carrier of V3:]
[: the carrier of (im f), the carrier of V3:] is Relation-like set
bool [: the carrier of (im f), the carrier of V3:] is set
g | the carrier of (im f) is Relation-like Function-like set
(K,V2,V3,(im f),g) * f is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of V3:]
dom f is non empty set
[#] V1 is non empty non proper Element of bool the carrier of V1
bool the carrier of V1 is set
[#] (im f) is non empty non proper Element of bool the carrier of (im f)
bool the carrier of (im f) is set
f .: (dom f) is set
rng f is non empty set
id (rng f) is Relation-like rng f -defined rng f -valued Function-like one-to-one non empty total quasi_total Element of bool [:(rng f),(rng f):]
[:(rng f),(rng f):] is Relation-like set
bool [:(rng f),(rng f):] is set
g * (id (rng f)) is Relation-like rng f -defined Function-like Element of bool [:(rng f), the carrier of V3:]
[:(rng f), the carrier of V3:] is Relation-like set
bool [:(rng f), the carrier of V3:] is set
(g * (id (rng f))) * f is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of V3:]
(id (rng f)) * f is Relation-like rng f -valued Function-like Element of bool [: the carrier of V1,(rng f):]
[: the carrier of V1,(rng f):] is Relation-like set
bool [: the carrier of V1,(rng f):] is set
g * ((id (rng f)) * f) is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of V3:]
K is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed doubleLoopStr
V1 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V1 is non empty set
V2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V2 is non empty set
[: the carrier of V1, the carrier of V2:] is Relation-like set
bool [: the carrier of V1, the carrier of V2:] is set
V3 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional VectSpStr over K
the carrier of V3 is non empty set
[: the carrier of V2, the carrier of V3:] is Relation-like set
bool [: the carrier of V2, the carrier of V3:] is set
f is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V2:]
im f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V2
rank f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (im f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
nullity f is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker f is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
g is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of V2, the carrier of V3:]
(K,V2,V3,(im f),g) is Relation-like Function-like non empty total quasi_total additive homogeneous Element of bool [: the carrier of (im f), the carrier of V3:]
the carrier of (im f) is non empty set
[: the carrier of (im f), the carrier of V3:] is Relation-like set
bool [: the carrier of (im f), the carrier of V3:] is set
g | the carrier of (im f) is Relation-like Function-like set
(K,V1,V2,V3,f,g) is Relation-like Function-like non empty total quasi_total quasi_total additive homogeneous Element of bool [: the carrier of V1, the carrier of V3:]
[: the carrier of V1, the carrier of V3:] is Relation-like set
bool [: the carrier of V1, the carrier of V3:] is set
rank (K,V1,V2,V3,f,g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
im (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V3
dim (im (K,V1,V2,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
nullity (K,V1,V2,V3,f,g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
ker (K,V1,V2,V3,f,g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V1
dim (ker (K,V1,V2,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
the carrier of (im (K,V1,V2,V3,f,g)) is non empty set
[#] (im (K,V1,V2,V3,f,g)) is non empty non proper Element of bool the carrier of (im (K,V1,V2,V3,f,g))
bool the carrier of (im (K,V1,V2,V3,f,g)) is set
[#] V1 is non empty non proper Element of bool the carrier of V1
bool the carrier of V1 is set
(K,V1,V2,V3,f,g) .: ([#] V1) is Element of bool the carrier of V3
bool the carrier of V3 is set
(K,V2,V3,(im f),g) * f is Relation-like Function-like Element of bool [: the carrier of V1, the carrier of V3:]
((K,V2,V3,(im f),g) * f) .: ([#] V1) is set
f .: ([#] V1) is Element of bool the carrier of V2
bool the carrier of V2 is set
(K,V2,V3,(im f),g) .: (f .: ([#] V1)) is set
[#] (im f) is non empty non proper Element of bool the carrier of (im f)
bool the carrier of (im f) is set
(K,V2,V3,(im f),g) .: ([#] (im f)) is Element of bool the carrier of V3
im (K,V2,V3,(im f),g) is non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional Subspace of V3
[#] (im (K,V2,V3,(im f),g)) is non empty non proper Element of bool the carrier of (im (K,V2,V3,(im f),g))
the carrier of (im (K,V2,V3,(im f),g)) is non empty set
bool the carrier of (im (K,V2,V3,(im f),g)) is set
rank (K,V2,V3,(im f),g) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim (im (K,V2,V3,(im f),g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(nullity f) + (rank f) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
dim V1 is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set
(nullity (K,V1,V2,V3,f,g)) + (rank (K,V1,V2,V3,f,g)) is V21() V22() V23() V27() finite cardinal ext-real non negative V93() V94() set