REAL is set
NAT is V6() V7() V8() non empty V19() non finite cardinal limit_cardinal Element of K27(REAL)
K27(REAL) is non empty set
NAT is V6() V7() V8() non empty V19() non finite cardinal limit_cardinal set
K27(NAT) is non empty V19() non finite set
K27(NAT) is non empty V19() non finite set
COMPLEX is set
RAT is set
INT is set
{} is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() set
2 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
1 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
3 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
card {} is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() set
Seg 1 is non empty V19() finite 1 -element Element of K27(NAT)
0 is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() Element of NAT
GF is non empty 1-sorted
the carrier of GF is non empty set
K27( the carrier of GF) is non empty set
the non empty Element of K27( the carrier of GF) is non empty Element of K27( the carrier of GF)
n is set
GF is set
{n} is non empty V19() finite 1 -element set
GF \ {n} is Element of K27(GF)
K27(GF) is non empty set
(GF \ {n}) \/ {n} is non empty set
{n} \/ (GF \ {n}) is non empty set
{n} \/ GF is non empty set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom W is finite Element of K27(NAT)
K28((dom W),(dom W)) is Relation-like finite set
K27(K28((dom W),(dom W))) is non empty finite V28() set
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
V (#) W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W) is Element of the carrier of n
V (#) x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) x) is Element of the carrier of n
A is Relation-like dom W -defined dom W -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom W),(dom W)))
W * A is Relation-like dom W -defined the carrier of n -valued Function-like finite Element of K27(K28((dom W), the carrier of n))
K28((dom W), the carrier of n) is Relation-like set
K27(K28((dom W), the carrier of n)) is non empty set
len x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len (V (#) W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom (V (#) W) is finite Element of K27(NAT)
(V (#) W) * A is Relation-like dom W -defined the carrier of n -valued Function-like finite Element of K27(K28((dom W), the carrier of n))
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len B is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom B is finite Element of K27(NAT)
len (V (#) x) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom (V (#) x) is finite Element of K27(NAT)
dom x is finite Element of K27(NAT)
I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
A . I1 is set
dom A is finite set
rng A is finite set
x /. I1 is Element of the carrier of n
x . I1 is set
W . (A . I1) is set
I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W /. I1 is Element of the carrier of n
(V (#) x) . I1 is set
V . (W /. I1) is Element of the carrier of GF
(V . (W /. I1)) * (W /. I1) is Element of the carrier of n
(V (#) W) . (A . I1) is set
B . I1 is set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
V is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier V is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
W is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x is set
<*x*> is Relation-like NAT -defined Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like set
W ^ <*x*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
V (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W1) is Element of the carrier of n
{x} is non empty V19() finite 1 -element set
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng A is finite set
(Carrier V) /\ (rng W1) is finite Element of K27( the carrier of n)
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
B is Element of the carrier of n
<*B*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng <*B*> is non empty V19() finite 1 -element set
(rng W1) \/ (rng <*B*>) is non empty finite set
(Carrier V) /\ ((rng W1) \/ (rng <*B*>)) is finite Element of K27( the carrier of n)
{B} is non empty V19() finite 1 -element Element of K27( the carrier of n)
(rng W1) \/ {B} is non empty finite set
(Carrier V) /\ ((rng W1) \/ {B}) is finite Element of K27( the carrier of n)
(Carrier V) /\ (rng W1) is finite Element of K27( the carrier of n)
(Carrier V) /\ {B} is finite Element of K27( the carrier of n)
((Carrier V) /\ (rng W1)) \/ ((Carrier V) /\ {B}) is finite Element of K27( the carrier of n)
V (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W1) is Element of the carrier of n
V . B is Element of the carrier of GF
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
V (#) A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(V (#) W1) ^ (V (#) A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((V (#) W1) ^ (V (#) A)) is Element of the carrier of n
Sum (V (#) A) is Element of the carrier of n
(Sum (V (#) W1)) + (Sum (V (#) A)) is Element of the carrier of n
(V . B) * B is Element of the carrier of n
<*((V . B) * B)*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum <*((V . B) * B)*> is Element of the carrier of n
(0. n) + (Sum <*((V . B) * B)*>) is Element of the carrier of n
(0. GF) * B is Element of the carrier of n
W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W is finite set
V (#) W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W) is Element of the carrier of n
<*> the carrier of n is Relation-like non-empty empty-yielding NAT -defined the carrier of n -valued V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty proper finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() FinSequence of the carrier of n
K28(NAT, the carrier of n) is Relation-like non empty V19() non finite set
W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W is finite set
V (#) W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W) is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng V is finite set
K27((rng V)) is non empty finite V28() set
x is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier x is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
x (#) V is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (x (#) V) is Element of the carrier of n
Sum x is Element of the carrier of n
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
x (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (x (#) W1) is Element of the carrier of n
W1 is finite Element of K27((rng V))
W1 ` is finite Element of K27((rng V))
(rng V) \ W1 is finite set
V - (W1 `) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
W is finite Element of K27((rng V))
W \ (W1 `) is finite Element of K27((rng V))
(W1 `) ` is finite Element of K27((rng V))
(rng V) \ (W1 `) is finite set
W /\ ((W1 `) `) is finite Element of K27((rng V))
rng (V - (W1 `)) is finite set
dom W1 is finite Element of K27(NAT)
K28((dom W1),(dom W1)) is Relation-like finite set
K27(K28((dom W1),(dom W1))) is non empty finite V28() set
V - W1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng I1 is finite set
(rng V) \ (rng W1) is finite Element of K27((rng V))
dom V is finite Element of K27(NAT)
K28((dom V),(dom V)) is Relation-like finite set
K27(K28((dom V),(dom V))) is non empty finite V28() set
(V - (W1 `)) ^ (V - W1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B ^ I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
x (#) (B ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (x (#) (B ^ I1)) is Element of the carrier of n
x (#) B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
x (#) I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(x (#) B) ^ (x (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((x (#) B) ^ (x (#) I1)) is Element of the carrier of n
Sum (x (#) B) is Element of the carrier of n
Sum (x (#) I1) is Element of the carrier of n
(Sum (x (#) B)) + (Sum (x (#) I1)) is Element of the carrier of n
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(Sum (x (#) B)) + (0. n) is Element of the carrier of n
(Sum (x (#) W1)) + (0. n) is Element of the carrier of n
I2 is Relation-like dom V -defined dom V -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom V),(dom V)))
V * I2 is Relation-like dom V -defined the carrier of n -valued Function-like finite Element of K27(K28((dom V), the carrier of n))
K28((dom V), the carrier of n) is Relation-like set
K27(K28((dom V), the carrier of n)) is non empty set
I2 is Relation-like dom W1 -defined dom W1 -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom W1),(dom W1)))
W1 * I2 is Relation-like dom W1 -defined the carrier of n -valued Function-like finite Element of K27(K28((dom W1), the carrier of n))
K28((dom W1), the carrier of n) is Relation-like set
K27(K28((dom W1), the carrier of n)) is non empty set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier V is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W is finite set
(rng W) /\ (Carrier V) is finite Element of K27( the carrier of n)
V (#) W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
W1 is set
V . W1 is set
W1 is Element of the carrier of n
V . W1 is Element of the carrier of GF
K28( the carrier of n, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of n, the carrier of GF)) is non empty set
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of n, the carrier of GF))
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of n, the carrier of GF))
W1 is Element of the carrier of n
x is finite Element of K27( the carrier of n)
x /\ (Carrier V) is finite Element of K27( the carrier of n)
W1 . W1 is Element of the carrier of GF
W1 . W1 is Element of the carrier of GF
V . W1 is set
V . W1 is Element of the carrier of GF
Funcs ( the carrier of n, the carrier of GF) is functional non empty FUNCTION_DOMAIN of the carrier of n, the carrier of GF
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Element of Funcs ( the carrier of n, the carrier of GF)
B is set
I1 is Element of the carrier of n
V . I1 is Element of the carrier of GF
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
A . I1 is Element of the carrier of GF
Carrier A is finite Element of K27( the carrier of n)
A (#) W is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (V (#) W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len (A (#) W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom (V (#) W) is finite Element of K27(NAT)
dom (A (#) W) is finite Element of K27(NAT)
dom W is finite Element of K27(NAT)
I2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W /. I2 is Element of the carrier of n
A . (W /. I2) is Element of the carrier of GF
V . (W /. I2) is set
W . I2 is set
(V (#) W) . I2 is set
V . (W /. I2) is Element of the carrier of GF
(V . (W /. I2)) * (W /. I2) is Element of the carrier of n
(A (#) W) . I2 is set
B is set
I1 is Element of the carrier of n
A . I1 is Element of the carrier of GF
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
W is Element of K27( the carrier of n)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of (Lin W) is non empty set
x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x + 1 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
V (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W1) is Element of the carrier of n
W1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A is set
<*A*> is Relation-like NAT -defined Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like set
W1 ^ <*A*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B ^ <*A*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
rng (B ^ <*A*>) is non empty finite set
rng B is finite set
rng <*A*> is non empty V19() finite 1 -element set
(rng B) \/ (rng <*A*>) is non empty finite set
{A} is non empty V19() finite 1 -element set
(rng B) \/ {A} is non empty finite set
I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum I2 is Element of the carrier of n
I1 is Element of the carrier of n
<*I1*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B ^ <*I1*> is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (B ^ <*I1*>) is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
len B is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len <*I1*> is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
(len B) + (len <*I1*>) is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
(len B) + 1 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
V (#) B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) B) is Element of the carrier of n
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum A is Element of the carrier of n
V . I1 is Element of the carrier of GF
(V . I1) * I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
A + ((V . I1) * I2) is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
V (#) I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(V (#) B) ^ (V (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((V (#) B) ^ (V (#) I1)) is Element of the carrier of n
Sum (V (#) I1) is Element of the carrier of n
(Sum A) + (Sum (V (#) I1)) is Element of the carrier of n
(V . I1) * I1 is Element of the carrier of n
<*((V . I1) * I1)*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum <*((V . I1) * I1)*> is Element of the carrier of n
(Sum A) + (Sum <*((V . I1) * I1)*>) is Element of the carrier of n
(V . I1) * (Sum I2) is Element of the carrier of n
(Sum A) + ((V . I1) * (Sum I2)) is Element of the carrier of n
Sum ((V . I1) * I2) is Element of the carrier of n
(Sum A) + (Sum ((V . I1) * I2)) is Element of the carrier of n
Sum (A + ((V . I1) * I2)) is Element of the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng x is finite set
V (#) x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) x) is Element of the carrier of n
len x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
V (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) W1) is Element of the carrier of n
<*> the carrier of n is Relation-like non-empty empty-yielding NAT -defined the carrier of n -valued V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty proper finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() FinSequence of the carrier of n
K28(NAT, the carrier of n) is Relation-like non empty V19() non finite set
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
ZeroLC n is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Sum (ZeroLC n) is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier V is finite Element of K27( the carrier of n)
Sum V is Element of the carrier of n
W is Element of K27( the carrier of n)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of (Lin W) is non empty set
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng x is finite set
V (#) x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (V (#) x) is Element of the carrier of n
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum W1 is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
W is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier W is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
W | the carrier of V is Relation-like the carrier of n -defined the carrier of V -defined the carrier of n -defined the carrier of GF -valued Function-like Element of K27(K28( the carrier of n, the carrier of GF))
K28( the carrier of n, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of n, the carrier of GF)) is non empty set
Sum W is Element of the carrier of n
x is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier x is finite Element of K27( the carrier of V)
K27( the carrier of V) is non empty set
Sum x is Element of the carrier of V
dom x is set
W1 is set
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
W1 is Element of the carrier of V
x . W1 is Element of the carrier of GF
W . W1 is set
W1 is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
rng W1 is finite set
x (#) W1 is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
Sum (x (#) W1) is Element of the carrier of V
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng W1 is finite set
W (#) W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (W (#) W1) is Element of the carrier of n
A is set
B is Element of the carrier of n
W . B is Element of the carrier of GF
x . B is set
dom W1 is finite Element of K27(NAT)
K28((dom W1),(dom W1)) is Relation-like finite set
K27(K28((dom W1),(dom W1))) is non empty finite V28() set
A is Relation-like dom W1 -defined dom W1 -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom W1),(dom W1)))
W1 * A is Relation-like dom W1 -defined the carrier of V -valued Function-like finite Element of K27(K28((dom W1), the carrier of V))
K28((dom W1), the carrier of V) is Relation-like set
K27(K28((dom W1), the carrier of V)) is non empty set
len (x (#) W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom (x (#) W1) is finite Element of K27(NAT)
(x (#) W1) * A is Relation-like dom W1 -defined the carrier of V -valued Function-like finite Element of K27(K28((dom W1), the carrier of V))
B is Relation-like NAT -defined the carrier of V -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of V
len B is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom B is finite Element of K27(NAT)
rng B is finite set
K28(NAT, the carrier of V) is Relation-like non empty V19() non finite set
K27(K28(NAT, the carrier of V)) is non empty V19() non finite set
Sum B is Element of the carrier of V
0. V is V51(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
I1 is Relation-like NAT -defined the carrier of V -valued Function-like non empty total quasi_total Element of K27(K28(NAT, the carrier of V))
I1 . (len B) is Element of the carrier of V
I1 . 0 is Element of the carrier of V
dom I1 is non empty set
rng I1 is non empty set
K28(NAT, the carrier of n) is Relation-like non empty V19() non finite set
K27(K28(NAT, the carrier of n)) is non empty V19() non finite set
I2 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len I2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(NAT, the carrier of n))
A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
A + 1 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
I2 . (A + 1) is set
A . (A + 1) is Element of the carrier of n
A . A is Element of the carrier of n
A9 is Element of the carrier of n
(A . A) + A9 is Element of the carrier of n
I1 . (A + 1) is Element of the carrier of V
I1 . A is Element of the carrier of V
x is Element of the carrier of V
(I1 . A) + x is Element of the carrier of V
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom W1 is finite Element of K27(NAT)
len (W (#) W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
dom (W (#) W1) is finite Element of K27(NAT)
A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 /. A is Element of the carrier of n
A . A is set
W1 . A is set
dom A is finite set
rng A is finite set
w2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 /. w2 is Element of the carrier of V
W1 . (A . A) is set
B . A is set
(x (#) W1) . w2 is set
w1 is Element of the carrier of V
x . w1 is Element of the carrier of GF
(x . w1) * w1 is Element of the carrier of V
W . (W1 /. A) is Element of the carrier of GF
(W . (W1 /. A)) * w1 is Element of the carrier of V
(W . (W1 /. A)) * (W1 /. A) is Element of the carrier of n
(W (#) W1) . A is set
K28((dom (x (#) W1)),(dom (x (#) W1))) is Relation-like finite set
K27(K28((dom (x (#) W1)),(dom (x (#) W1)))) is non empty finite V28() set
A is Relation-like dom (x (#) W1) -defined dom (x (#) W1) -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom (x (#) W1)),(dom (x (#) W1))))
(x (#) W1) * A is Relation-like dom (x (#) W1) -defined the carrier of V -valued Function-like finite Element of K27(K28((dom (x (#) W1)), the carrier of V))
K28((dom (x (#) W1)), the carrier of V) is Relation-like set
K27(K28((dom (x (#) W1)), the carrier of V)) is non empty set
A . (len I2) is Element of the carrier of n
A . 0 is Element of the carrier of n
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
W is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier W is finite Element of K27( the carrier of V)
K27( the carrier of V) is non empty set
Sum W is Element of the carrier of V
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
K28( the carrier of V, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of V, the carrier of GF)) is non empty set
K27( the carrier of n) is non empty set
W1 is set
W . W1 is set
A is Element of the carrier of n
B is Element of the carrier of V
W . B is Element of the carrier of GF
W . B is set
A is Element of the carrier of n
A is Element of the carrier of n
K28( the carrier of n, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of n, the carrier of GF)) is non empty set
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of n, the carrier of GF))
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of n, the carrier of GF))
A is Element of the carrier of n
W1 is finite Element of K27( the carrier of n)
W . A is set
W1 . A is Element of the carrier of GF
Funcs ( the carrier of n, the carrier of GF) is functional non empty FUNCTION_DOMAIN of the carrier of n, the carrier of GF
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
A | the carrier of V is Relation-like the carrier of n -defined the carrier of V -defined the carrier of n -defined the carrier of GF -valued Function-like Element of K27(K28( the carrier of n, the carrier of GF))
Carrier A is finite Element of K27( the carrier of n)
Sum A is Element of the carrier of n
I1 is set
I2 is Element of the carrier of n
A . I2 is Element of the carrier of GF
W . I2 is set
I1 is set
A . I1 is set
W . I1 is set
x is Relation-like the carrier of V -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of V, the carrier of GF))
x . I1 is set
B is Relation-like the carrier of V -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of V, the carrier of GF))
B . I1 is set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
the carrier of GF is non empty V19() set
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
W is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier W is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
Sum W is Element of the carrier of n
K27( the carrier of V) is non empty set
K28( the carrier of V, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of V, the carrier of GF)) is non empty set
W | the carrier of V is Relation-like the carrier of n -defined the carrier of V -defined the carrier of n -defined the carrier of GF -valued Function-like Element of K27(K28( the carrier of n, the carrier of GF))
K28( the carrier of n, the carrier of GF) is Relation-like non empty set
K27(K28( the carrier of n, the carrier of GF)) is non empty set
W1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like non empty total quasi_total Element of K27(K28( the carrier of V, the carrier of GF))
Funcs ( the carrier of V, the carrier of GF) is functional non empty FUNCTION_DOMAIN of the carrier of V, the carrier of GF
dom W1 is non empty set
W1 is Element of the carrier of V
x is finite Element of K27( the carrier of V)
W . W1 is set
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
W1 . W1 is Element of the carrier of GF
W1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier W1 is finite Element of K27( the carrier of V)
Sum W1 is Element of the carrier of V
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
V is Basis of n
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W is Element of the carrier of n
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
{} n is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty proper finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() linearly-independent Element of K27( the carrier of n)
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
K27( the carrier of V) is non empty set
W is Element of K27( the carrier of V)
x is Element of K27( the carrier of n)
the carrier of GF is non empty V19() set
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)
W1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier W1 is finite Element of K27( the carrier of V)
Sum W1 is Element of the carrier of V
A is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum A is Element of the carrier of V
0. V is V51(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
K27( the carrier of V) is non empty set
W is Element of K27( the carrier of n)
x is Element of K27( the carrier of V)
the carrier of GF is non empty V19() set
0. V is V51(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
W1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Sum W1 is Element of the carrier of V
Carrier W1 is finite Element of K27( the carrier of V)
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier W1 is finite Element of K27( the carrier of n)
Sum W1 is Element of the carrier of n
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum A is Element of the carrier of n
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W is Basis of V
the carrier of n is non empty set
K27( the carrier of n) is non empty set
x is linearly-independent Element of K27( the carrier of n)
W1 is Basis of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Element of K27( the carrier of n)
W is Element of the carrier of n
{W} is non empty V19() finite 1 -element Element of K27( the carrier of n)
V \ {W} is Element of K27( the carrier of n)
K27(V) is non empty set
Lin {W} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of GF is non empty V19() set
x is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of {W}
Sum x is Element of the carrier of n
W1 is Element of K27( the carrier of n)
Lin W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W1 \/ {W} is non empty Element of K27( the carrier of n)
V \/ V is Element of K27( the carrier of n)
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1
Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)
Carrier x is finite Element of K27( the carrier of n)
(Carrier W1) \/ (Carrier x) is finite Element of K27( the carrier of n)
W1 - x is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier (W1 - x) is finite Element of K27( the carrier of n)
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
A is Element of the carrier of n
x . A is Element of the carrier of GF
A is set
B is Element of the carrier of n
W1 . B is Element of the carrier of GF
(W1 - x) . B is Element of the carrier of GF
x . B is Element of the carrier of GF
(W1 . B) - (x . B) is Element of the carrier of GF
(W1 . B) - (0. GF) is Element of the carrier of GF
- (0. GF) is Element of the carrier of GF
(W1 . B) + (- (0. GF)) is Element of the carrier of GF
(W1 . B) + (0. GF) is Element of the carrier of GF
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
A is set
- (Sum x) is Element of the carrier of n
(Sum W1) + (- (Sum x)) is Element of the carrier of n
- x is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Sum (- x) is Element of the carrier of n
(Sum W1) + (Sum (- x)) is Element of the carrier of n
W1 + (- x) is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Sum (W1 + (- x)) is Element of the carrier of n
Sum (W1 - x) is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Basis of n
W is non empty Element of K27( the carrier of n)
V \/ W is non empty Element of K27( the carrier of n)
x is set
W1 is Element of K27( the carrier of n)
W1 is Element of the carrier of n
{W1} is non empty V19() finite 1 -element Element of K27( the carrier of n)
W1 \ {W1} is Element of K27( the carrier of n)
V \ {W1} is Element of K27( the carrier of n)
A is Element of K27( the carrier of n)
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
Lin A is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
W is Element of K27( the carrier of n)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
x is set
the carrier of (Lin W) is non empty set
the carrier of GF is non empty V19() set
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)
W1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier W1 is finite Element of K27( the carrier of V)
K27( the carrier of V) is non empty set
Sum W1 is Element of the carrier of V
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the carrier of V is non empty set
K27( the carrier of V) is non empty set
W is Element of K27( the carrier of n)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
x is Element of K27( the carrier of V)
Lin x is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of V
W1 is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
W1 is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
A is set
the carrier of (Lin W) is non empty set
the carrier of GF is non empty V19() set
B is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum B is Element of the carrier of n
Carrier B is finite Element of K27( the carrier of n)
I1 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Carrier I1 is finite Element of K27( the carrier of V)
Sum I1 is Element of the carrier of V
I2 is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Sum I2 is Element of the carrier of V
the carrier of (Lin x) is non empty set
A is set
B is Relation-like the carrier of V -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Sum B is Element of the carrier of V
Carrier B is finite Element of K27( the carrier of V)
I1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier I1 is finite Element of K27( the carrier of n)
Sum I1 is Element of the carrier of n
I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum I2 is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
W is finite Element of K27( the carrier of n)
V \/ W is finite Element of K27( the carrier of n)
Lin (V \/ W) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
x is Element of the carrier of n
{x} is non empty V19() finite 1 -element Element of K27( the carrier of n)
the carrier of GF is non empty V19() set
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V \/ W
Sum W1 is Element of the carrier of n
Lin {x} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of {x}
Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
A is set
B is Element of the carrier of n
W1 . B is Element of the carrier of GF
A is Element of the carrier of n
W1 . A is Element of the carrier of GF
A is Element of the carrier of n
W1 . A is Element of the carrier of GF
{A} is non empty V19() finite 1 -element Element of K27( the carrier of n)
(V \/ W) \ {A} is finite Element of K27( the carrier of n)
((V \/ W) \ {A}) \/ {x} is non empty finite Element of K27( the carrier of n)
Lin (((V \/ W) \ {A}) \/ {x}) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng I1 is finite set
W1 (#) I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (W1 (#) I1) is Element of the carrier of n
I1 -| A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I1 |-- A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I2 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng I2 is finite set
I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng I1 is finite set
I2 ^ I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
I1 - {A} is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (I2 ^ I1) is finite set
(Carrier W1) \ {A} is finite Element of K27( the carrier of n)
<*A*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(I1 -| A) ^ <*A*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
((I1 -| A) ^ <*A*>) ^ (I1 |-- A) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
<*A*> ^ I1 is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
I2 ^ (<*A*> ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
W1 (#) I2 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
W1 (#) (<*A*> ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) I2) ^ (W1 (#) (<*A*> ^ I1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
W1 (#) <*A*> is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
W1 (#) I1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) <*A*>) ^ (W1 (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) I2) ^ ((W1 (#) <*A*>) ^ (W1 (#) I1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) I2) ^ (W1 (#) <*A*>) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((W1 (#) I2) ^ (W1 (#) <*A*>)) ^ (W1 (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 . A) * A is Element of the carrier of n
<*((W1 . A) * A)*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V19() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) I2) ^ <*((W1 . A) * A)*> is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
((W1 (#) I2) ^ <*((W1 . A) * A)*>) ^ (W1 (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
<*((W1 . A) * A)*> ^ (W1 (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(W1 (#) I2) ^ (<*((W1 . A) * A)*> ^ (W1 (#) I1)) is Relation-like NAT -defined the carrier of n -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((W1 (#) I2) ^ (<*((W1 . A) * A)*> ^ (W1 (#) I1))) is Element of the carrier of n
Sum (W1 (#) I2) is Element of the carrier of n
Sum (<*((W1 . A) * A)*> ^ (W1 (#) I1)) is Element of the carrier of n
(Sum (W1 (#) I2)) + (Sum (<*((W1 . A) * A)*> ^ (W1 (#) I1))) is Element of the carrier of n
Sum <*((W1 . A) * A)*> is Element of the carrier of n
Sum (W1 (#) I1) is Element of the carrier of n
(Sum <*((W1 . A) * A)*>) + (Sum (W1 (#) I1)) is Element of the carrier of n
(Sum (W1 (#) I2)) + ((Sum <*((W1 . A) * A)*>) + (Sum (W1 (#) I1))) is Element of the carrier of n
(Sum (W1 (#) I1)) + ((W1 . A) * A) is Element of the carrier of n
(Sum (W1 (#) I2)) + ((Sum (W1 (#) I1)) + ((W1 . A) * A)) is Element of the carrier of n
(Sum (W1 (#) I2)) + (Sum (W1 (#) I1)) is Element of the carrier of n
((Sum (W1 (#) I2)) + (Sum (W1 (#) I1))) + ((W1 . A) * A) is Element of the carrier of n
(W1 (#) I2) ^ (W1 (#) I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((W1 (#) I2) ^ (W1 (#) I1)) is Element of the carrier of n
(Sum ((W1 (#) I2) ^ (W1 (#) I1))) + ((W1 . A) * A) is Element of the carrier of n
W1 (#) (I2 ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (W1 (#) (I2 ^ I1)) is Element of the carrier of n
(Sum (W1 (#) (I2 ^ I1))) + ((W1 . A) * A) is Element of the carrier of n
(rng (I2 ^ I1)) /\ (Carrier W1) is finite Element of K27( the carrier of n)
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier A is finite Element of K27( the carrier of n)
A (#) (I2 ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(rng I1) \ {A} is finite Element of K27((rng I1))
K27((rng I1)) is non empty finite V28() set
(W1 . A) " is Element of the carrier of GF
A9 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of (V \/ W) \ {A}
- A9 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
(- A9) + W1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
((W1 . A) ") * ((- A9) + W1) is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of n
Carrier ((- A9) + W1) is finite Element of K27( the carrier of n)
Carrier (- A9) is finite Element of K27( the carrier of n)
Carrier W1 is finite Element of K27( the carrier of n)
(Carrier (- A9)) \/ (Carrier W1) is finite Element of K27( the carrier of n)
Carrier A9 is finite Element of K27( the carrier of n)
(Carrier A9) \/ (Carrier W1) is finite Element of K27( the carrier of n)
Carrier (((W1 . A) ") * ((- A9) + W1)) is finite Element of K27( the carrier of n)
A9 (#) (I2 ^ I1) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (A9 (#) (I2 ^ I1)) is Element of the carrier of n
Sum A9 is Element of the carrier of n
((W1 . A) ") * x is Element of the carrier of n
((W1 . A) ") * (Sum A9) is Element of the carrier of n
((W1 . A) ") * ((W1 . A) * A) is Element of the carrier of n
(((W1 . A) ") * (Sum A9)) + (((W1 . A) ") * ((W1 . A) * A)) is Element of the carrier of n
(((W1 . A) ") * (Sum A9)) + A is Element of the carrier of n
(((W1 . A) ") * x) - (((W1 . A) ") * (Sum A9)) is Element of the carrier of n
x - (Sum A9) is Element of the carrier of n
((W1 . A) ") * (x - (Sum A9)) is Element of the carrier of n
- (Sum A9) is Element of the carrier of n
(- (Sum A9)) + x is Element of the carrier of n
((W1 . A) ") * ((- (Sum A9)) + x) is Element of the carrier of n
Sum (- A9) is Element of the carrier of n
(Sum (- A9)) + (Sum W1) is Element of the carrier of n
((W1 . A) ") * ((Sum (- A9)) + (Sum W1)) is Element of the carrier of n
Sum ((- A9) + W1) is Element of the carrier of n
((W1 . A) ") * (Sum ((- A9) + W1)) is Element of the carrier of n
Sum (((W1 . A) ") * ((- A9) + W1)) is Element of the carrier of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V + 1 is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W - (V + 1) is ext-real V38() V58() set
x is finite Element of K27( the carrier of n)
card x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 is finite Element of K27( the carrier of n)
card W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
Lin x is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W1 is set
A is Element of the carrier of n
{A} is non empty V19() finite 1 -element Element of K27( the carrier of n)
W1 \ {A} is finite Element of K27( the carrier of n)
K27(W1) is non empty finite V28() set
card (W1 \ {A}) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card {A} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
(card W1) - (card {A}) is ext-real V38() V58() set
(V + 1) - 1 is ext-real V38() V58() set
W - V is ext-real V38() V58() set
I1 is finite Element of K27( the carrier of n)
card I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
(W1 \ {A}) \/ I1 is finite Element of K27( the carrier of n)
Lin ((W1 \ {A}) \/ I1) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
Lin (W1 \ {A}) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
V - V is ext-real V38() V58() set
I2 is finite Element of K27( the carrier of n)
card I2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
(W1 \ {A}) \/ I2 is finite Element of K27( the carrier of n)
Lin ((W1 \ {A}) \/ I2) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I1 \/ (W1 \ {A}) is finite Element of K27( the carrier of n)
I2 is Element of the carrier of n
{I2} is non empty V19() finite 1 -element Element of K27( the carrier of n)
(I1 \/ (W1 \ {A})) \ {I2} is finite Element of K27( the carrier of n)
((I1 \/ (W1 \ {A})) \ {I2}) \/ {A} is non empty finite Element of K27( the carrier of n)
Lin (((I1 \/ (W1 \ {A})) \ {I2}) \/ {A}) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I1 \ {I2} is finite Element of K27( the carrier of n)
(W1 \ {A}) \ {I2} is finite Element of K27( the carrier of n)
((W1 \ {A}) \ {I2}) \/ {A} is non empty finite Element of K27( the carrier of n)
(W1 \ {A}) \/ {A} is non empty finite Element of K27( the carrier of n)
(I1 \ {I2}) \/ (((W1 \ {A}) \ {I2}) \/ {A}) is non empty finite Element of K27( the carrier of n)
(I1 \ {I2}) \/ ((W1 \ {A}) \/ {A}) is non empty finite Element of K27( the carrier of n)
W1 \/ (I1 \ {I2}) is finite Element of K27( the carrier of n)
K27(I1) is non empty finite V28() set
card (I1 \ {I2}) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card {I2} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
(card I1) - (card {I2}) is ext-real V38() V58() set
(W - V) - 1 is ext-real V38() V58() set
(I1 \ {I2}) \/ ((W1 \ {A}) \ {I2}) is finite Element of K27( the carrier of n)
((I1 \ {I2}) \/ ((W1 \ {A}) \ {I2})) \/ {A} is non empty finite Element of K27( the carrier of n)
Lin (W1 \/ (I1 \ {I2})) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
(I1 \ {I2}) \/ {I2} is non empty finite Element of K27( the carrier of n)
A is set
the carrier of (Lin (W1 \/ (I1 \ {I2}))) is non empty set
the carrier of (Lin ((W1 \ {A}) \/ I1)) is non empty set
V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V - 0 is ext-real non negative V38() V58() set
W is finite Element of K27( the carrier of n)
card W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
x is finite Element of K27( the carrier of n)
card x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
x \/ W is finite Element of K27( the carrier of n)
V is finite Element of K27( the carrier of n)
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
W is finite Element of K27( the carrier of n)
card W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
(card V) - (card W) is ext-real V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
W is Basis of n
x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng x is finite set
the carrier of GF is non empty V19() set
W1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom W1 is finite Element of K27(NAT)
{ (Carrier b1) where b1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W : ex b2 being V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set st
( b2 in dom W1 & Sum b1 = W1 . b2 ) } is set
union { (Carrier b1) where b1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W : ex b2 being V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set st
( b2 in dom W1 & Sum b1 = W1 . b2 ) } is set
B is set
I1 is set
I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier I2 is finite Element of K27( the carrier of n)
I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
Sum I2 is Element of the carrier of n
W1 . I1 is set
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
B is Element of K27( the carrier of n)
Lin B is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I1 is Element of the carrier of n
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of V
Sum I2 is Element of the carrier of n
Carrier I2 is finite Element of K27( the carrier of n)
the carrier of (Lin B) is non empty set
I1 is set
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
A is Element of the carrier of n
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum A is Element of the carrier of n
A9 is set
W1 . A9 is set
x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 . x is set
Carrier A is finite Element of K27( the carrier of n)
A9 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 . A9 is set
A9 is set
I1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of B
Sum I1 is Element of the carrier of n
the carrier of VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty set
the carrier of ((Omega). n) is non empty set
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
W \ B is Element of K27( the carrier of n)
I1 is set
I1 is Element of K27( the carrier of n)
I1 \ B is Element of K27( the carrier of n)
B \/ (I1 \ B) is Element of K27( the carrier of n)
B \/ I1 is Element of K27( the carrier of n)
I2 is non empty Element of K27( the carrier of n)
B \/ I2 is non empty Element of K27( the carrier of n)
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
Seg (len W1) is finite len W1 -element Element of K27(NAT)
I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 . I1 is set
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
I2 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Sum I2 is Element of the carrier of n
Carrier I2 is finite Element of K27( the carrier of n)
I1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier I1 is finite Element of K27( the carrier of n)
Sum I1 is Element of the carrier of n
I1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom I1 is finite Element of K27(NAT)
I1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom I1 is finite Element of K27(NAT)
I2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 . I2 is set
I1 is set
A is set
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier A is finite Element of K27( the carrier of n)
Sum A is Element of the carrier of n
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier A is finite Element of K27( the carrier of n)
Sum A is Element of the carrier of n
A9 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier A9 is finite Element of K27( the carrier of n)
Sum A9 is Element of the carrier of n
A9 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier A9 is finite Element of K27( the carrier of n)
Sum A9 is Element of the carrier of n
I2 is set
I1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier I1 is finite Element of K27( the carrier of n)
Sum I1 is Element of the carrier of n
A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 . A is set
A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 . A is set
I1 . A is set
A is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier A is finite Element of K27( the carrier of n)
Sum A is Element of the carrier of n
rng I1 is finite set
I2 is set
I1 is Relation-like the carrier of n -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W
Carrier I1 is finite Element of K27( the carrier of n)
A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
Sum I1 is Element of the carrier of n
W1 . A is set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Element of K27( the carrier of n)
W is Basis of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
V is Basis of n
card V is V6() V7() V8() cardinal set
W is Basis of n
card W is V6() V7() V8() cardinal set
the carrier of n is non empty set
K27( the carrier of n) is non empty set
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
x is finite Element of K27( the carrier of n)
card x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 is finite Element of K27( the carrier of n)
card W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
(0). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of V is non empty set
K27( the carrier of V) is non empty set
{} the carrier of V is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty proper finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() linearly-independent Element of K27( the carrier of V)
the carrier of n is non empty set
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty V19() finite 1 -element Element of K27( the carrier of n)
K27( the carrier of n) is non empty set
0. V is V51(V) Element of the carrier of V
the ZeroF of V is Element of the carrier of V
{(0. V)} is non empty V19() finite 1 -element Element of K27( the carrier of V)
(0). V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of V
the carrier of ((0). V) is non empty set
W is finite Element of K27( the carrier of V)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of V
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
the Basis of V is Basis of V
x is Basis of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
(0). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(0). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
card V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x is Basis of n
card x is V6() V7() V8() cardinal set
V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
K27( the carrier of n) is non empty set
x is finite Element of K27( the carrier of n)
card x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the Basis of V is Basis of V
the carrier of V is non empty set
K27( the carrier of V) is non empty set
x is Element of K27( the carrier of V)
card x is V6() V7() V8() cardinal set
the carrier of n is non empty set
K27( the carrier of n) is non empty set
W1 is linearly-independent Element of K27( the carrier of n)
A is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
the Basis of A is Basis of A
Lin the Basis of A is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of A
the carrier of A is non empty set
the U5 of A is Relation-like K28( the carrier of A, the carrier of A) -defined the carrier of A -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of A, the carrier of A), the carrier of A))
K28( the carrier of A, the carrier of A) is Relation-like non empty set
K28(K28( the carrier of A, the carrier of A), the carrier of A) is Relation-like non empty set
K27(K28(K28( the carrier of A, the carrier of A), the carrier of A)) is non empty set
the ZeroF of A is Element of the carrier of A
the lmult of A is Relation-like K28( the carrier of GF, the carrier of A) -defined the carrier of A -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of A), the carrier of A))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of A) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of A), the carrier of A) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of A), the carrier of A)) is non empty set
VectSpStr(# the carrier of A, the U5 of A, the ZeroF of A, the lmult of A #) is non empty strict VectSpStr over GF
W1 is finite Element of K27( the carrier of n)
card W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
I1 is finite Element of K27( the carrier of n)
card I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is Element of K27( the carrier of n)
card V is V6() V7() V8() cardinal set
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,(Lin V)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x is set
the carrier of (Lin V) is non empty set
K27( the carrier of (Lin V)) is non empty set
x is linearly-independent Element of K27( the carrier of (Lin V))
Lin x is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of Lin V
W1 is Basis of Lin V
card W1 is V6() V7() V8() cardinal set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,((Omega). n)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(Omega). V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
the carrier of n is non empty set
K27( the carrier of n) is non empty set
W is finite Element of K27( the carrier of n)
the Basis of V is Basis of V
W1 is Basis of n
the carrier of V is non empty set
card the Basis of V is V6() V7() V8() cardinal set
card W1 is V6() V7() V8() cardinal set
A is finite Element of K27( the carrier of n)
card A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 is finite Element of K27( the carrier of n)
card W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
K27( the carrier of V) is non empty set
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
B is Element of K27( the carrier of n)
Lin B is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
I1 is Element of K27( the carrier of V)
Lin I1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
the U5 of V is Relation-like K28( the carrier of V, the carrier of V) -defined the carrier of V -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of V, the carrier of V), the carrier of V))
K28( the carrier of V, the carrier of V) is Relation-like non empty set
K28(K28( the carrier of V, the carrier of V), the carrier of V) is Relation-like non empty set
K27(K28(K28( the carrier of V, the carrier of V), the carrier of V)) is non empty set
the ZeroF of V is Element of the carrier of V
the lmult of V is Relation-like K28( the carrier of GF, the carrier of V) -defined the carrier of V -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of V), the carrier of V))
K28( the carrier of GF, the carrier of V) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of V), the carrier of V) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of V), the carrier of V)) is non empty set
VectSpStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) is non empty strict VectSpStr over GF
x is finite Element of K27( the carrier of V)
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
Lin x is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
W1 is Element of K27( the carrier of n)
card W1 is V6() V7() V8() cardinal set
W1 is Element of K27( the carrier of V)
Lin W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
(GF,(Lin W1)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
card W1 is V6() V7() V8() cardinal set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(0). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
the carrier of n is non empty set
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
W is finite Element of K27( the carrier of n)
card W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
{} the carrier of n is Relation-like non-empty empty-yielding NAT -defined V6() V7() V8() V10() V11() V12() Function-like one-to-one constant functional empty proper finite finite-yielding V28() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V38() V58() linearly-independent Element of K27( the carrier of n)
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
Lin W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
0. n is V51(n) Element of the carrier of n
{(0. n)} is non empty V19() finite 1 -element Element of K27( the carrier of n)
Lin V is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
0. n is V51(n) Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
card V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W is set
{W} is non empty V19() finite 1 -element set
x is Element of the carrier of n
{x} is non empty V19() finite 1 -element Element of K27( the carrier of n)
Lin {x} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
V is Element of the carrier of n
{V} is non empty V19() finite 1 -element Element of K27( the carrier of n)
Lin {V} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card {V} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
K27( the carrier of n) is non empty set
V is finite Element of K27( the carrier of n)
card V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W is set
x is Element of the carrier of n
{x} is non empty V19() finite 1 -element Element of K27( the carrier of n)
card {x} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
W1 is set
W1 is Element of the carrier of n
{x,W1} is non empty finite Element of K27( the carrier of n)
A is set
{x,W1,A} is non empty finite set
B is set
card {x,W1,A} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
A is set
Lin {x,W1} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
V is Element of the carrier of n
W is Element of the carrier of n
{V,W} is non empty finite Element of K27( the carrier of n)
Lin {V,W} is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card {V,W} is V6() V7() V8() V12() non empty finite cardinal ext-real positive non negative V38() V58() Element of NAT
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
V + W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,(V + W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V /\ W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,(V /\ W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(V + W)) + (GF,(V /\ W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,V) + (GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() VectSpStr over GF
W1 is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
W1 is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
W1 /\ W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
the carrier of (W1 /\ W1) is non empty set
K27( the carrier of (W1 /\ W1)) is non empty set
A is finite Element of K27( the carrier of (W1 /\ W1))
B is Basis of W1
I1 is Basis of W1
the carrier of W1 is non empty set
K27( the carrier of W1) is non empty set
the carrier of W1 is non empty set
K27( the carrier of W1) is non empty set
I1 is finite Element of K27( the carrier of W1)
the carrier of x is non empty set
K27( the carrier of x) is non empty set
I2 is finite Element of K27( the carrier of W1)
I1 /\ I2 is finite Element of K27( the carrier of W1)
A is set
Lin I1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1
the U5 of W1 is Relation-like K28( the carrier of W1, the carrier of W1) -defined the carrier of W1 -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of W1, the carrier of W1), the carrier of W1))
K28( the carrier of W1, the carrier of W1) is Relation-like non empty set
K28(K28( the carrier of W1, the carrier of W1), the carrier of W1) is Relation-like non empty set
K27(K28(K28( the carrier of W1, the carrier of W1), the carrier of W1)) is non empty set
the ZeroF of W1 is Element of the carrier of W1
the lmult of W1 is Relation-like K28( the carrier of GF, the carrier of W1) -defined the carrier of W1 -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of W1), the carrier of W1))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of W1) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of W1), the carrier of W1) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of W1), the carrier of W1)) is non empty set
VectSpStr(# the carrier of W1, the U5 of W1, the ZeroF of W1, the lmult of W1 #) is non empty strict VectSpStr over GF
x is set
{A} is non empty V19() finite 1 -element set
A \/ {A} is non empty finite set
w1 is set
A is linearly-independent Element of K27( the carrier of x)
x is Element of K27( the carrier of x)
A9 is Element of the carrier of x
Lin I2 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1
the U5 of W1 is Relation-like K28( the carrier of W1, the carrier of W1) -defined the carrier of W1 -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of W1, the carrier of W1), the carrier of W1))
K28( the carrier of W1, the carrier of W1) is Relation-like non empty set
K28(K28( the carrier of W1, the carrier of W1), the carrier of W1) is Relation-like non empty set
K27(K28(K28( the carrier of W1, the carrier of W1), the carrier of W1)) is non empty set
the ZeroF of W1 is Element of the carrier of W1
the lmult of W1 is Relation-like K28( the carrier of GF, the carrier of W1) -defined the carrier of W1 -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of W1), the carrier of W1))
K28( the carrier of GF, the carrier of W1) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of W1), the carrier of W1) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of W1), the carrier of W1)) is non empty set
VectSpStr(# the carrier of W1, the U5 of W1, the ZeroF of W1, the lmult of W1 #) is non empty strict VectSpStr over GF
the carrier of W1 /\ the carrier of W1 is set
the U5 of (W1 /\ W1) is Relation-like K28( the carrier of (W1 /\ W1), the carrier of (W1 /\ W1)) -defined the carrier of (W1 /\ W1) -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of (W1 /\ W1), the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1)))
K28( the carrier of (W1 /\ W1), the carrier of (W1 /\ W1)) is Relation-like non empty set
K28(K28( the carrier of (W1 /\ W1), the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1)) is Relation-like non empty set
K27(K28(K28( the carrier of (W1 /\ W1), the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1))) is non empty set
the ZeroF of (W1 /\ W1) is Element of the carrier of (W1 /\ W1)
the lmult of (W1 /\ W1) is Relation-like K28( the carrier of GF, the carrier of (W1 /\ W1)) -defined the carrier of (W1 /\ W1) -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1)))
K28( the carrier of GF, the carrier of (W1 /\ W1)) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1)) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of (W1 /\ W1)), the carrier of (W1 /\ W1))) is non empty set
VectSpStr(# the carrier of (W1 /\ W1), the U5 of (W1 /\ W1), the ZeroF of (W1 /\ W1), the lmult of (W1 /\ W1) #) is non empty strict VectSpStr over GF
Lin A is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1 /\ W1
w1 is Element of K27( the carrier of x)
Lin w1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
x \ {A} is Element of K27( the carrier of x)
A \ {A} is finite Element of K27( the carrier of (W1 /\ W1))
I1 \/ I2 is finite set
A is set
A9 is Element of the carrier of x
W1 + W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
the carrier of (W1 + W1) is non empty set
K27( the carrier of (W1 + W1)) is non empty set
A is finite Element of K27( the carrier of (W1 + W1))
card A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card I1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card I2 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
(card I1) + (card I2) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
card A is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
((card I1) + (card I2)) - (card A) is ext-real V38() V58() set
0. (W1 + W1) is V51(W1 + W1) Element of the carrier of (W1 + W1)
the ZeroF of (W1 + W1) is Element of the carrier of (W1 + W1)
w2 is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of A
Sum w2 is Element of the carrier of (W1 + W1)
Carrier w2 is finite Element of K27( the carrier of (W1 + W1))
(Carrier w2) \ I1 is finite Element of K27( the carrier of (W1 + W1))
(Carrier w2) /\ I1 is finite Element of K27( the carrier of W1)
K1 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
rng K1 is finite set
w2 (#) K1 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
Sum (w2 (#) K1) is Element of the carrier of (W1 + W1)
K27((rng K1)) is non empty finite V28() set
w2 is finite Element of K27((rng K1))
w2 ` is finite Element of K27((rng K1))
(rng K1) \ w2 is finite set
K1 - (w2 `) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K1 - w2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
rng K2 is finite set
(rng K2) /\ (Carrier w2) is finite Element of K27( the carrier of (W1 + W1))
w2 (#) K2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
L1 is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1 + W1
Carrier L1 is finite Element of K27( the carrier of (W1 + W1))
L1 (#) K2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
Sum (w2 (#) K2) is Element of the carrier of (W1 + W1)
Sum L1 is Element of the carrier of (W1 + W1)
L2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
rng L2 is finite set
(rng L2) /\ (Carrier w2) is finite Element of K27( the carrier of (W1 + W1))
w2 (#) L2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
L is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1 + W1
Carrier L is finite Element of K27( the carrier of (W1 + W1))
L (#) L2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
Sum (w2 (#) L2) is Element of the carrier of (W1 + W1)
Sum L is Element of the carrier of (W1 + W1)
L is finite Element of K27((rng K1))
L \ (w2 `) is finite Element of K27((rng K1))
(w2 `) ` is finite Element of K27((rng K1))
(rng K1) \ (w2 `) is finite set
L /\ ((w2 `) `) is finite Element of K27((rng K1))
(Carrier w2) /\ (Carrier w2) is finite Element of K27( the carrier of (W1 + W1))
I1 /\ ((Carrier w2) /\ (Carrier w2)) is finite Element of K27( the carrier of (W1 + W1))
w1 is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1 + W1
the carrier of w1 is non empty set
K1 is Relation-like the carrier of w1 -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of w1
Carrier K1 is finite Element of K27( the carrier of w1)
K27( the carrier of w1) is non empty set
Sum K1 is Element of the carrier of w1
(Carrier w2) \ ((Carrier w2) /\ I1) is finite Element of K27( the carrier of (W1 + W1))
(Carrier L1) /\ (Carrier L) is finite Element of K27( the carrier of (W1 + W1))
I1 /\ ((Carrier w2) \ I1) is finite Element of K27( the carrier of (W1 + W1))
(Carrier w2) /\ (I1 /\ ((Carrier w2) \ I1)) is finite Element of K27( the carrier of (W1 + W1))
(Carrier w2) /\ {} is Relation-like finite Element of K27( the carrier of (W1 + W1))
x is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1 + W1
the carrier of x is non empty set
K2 is Relation-like the carrier of x -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Carrier K2 is finite Element of K27( the carrier of x)
K27( the carrier of x) is non empty set
Sum K2 is Element of the carrier of x
dom K1 is finite Element of K27(NAT)
K28((dom K1),(dom K1)) is Relation-like finite set
K27(K28((dom K1),(dom K1))) is non empty finite V28() set
(K1 - (w2 `)) ^ (K1 - w2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
K2 ^ L2 is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
w2 (#) (K2 ^ L2) is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
Sum (w2 (#) (K2 ^ L2)) is Element of the carrier of (W1 + W1)
(w2 (#) K2) ^ (w2 (#) L2) is Relation-like NAT -defined the carrier of (W1 + W1) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (W1 + W1)
Sum ((w2 (#) K2) ^ (w2 (#) L2)) is Element of the carrier of (W1 + W1)
(Sum L1) + (Sum L) is Element of the carrier of (W1 + W1)
KI is Relation-like dom K1 -defined dom K1 -valued Function-like one-to-one finite total quasi_total onto bijective Element of K27(K28((dom K1),(dom K1)))
K1 * KI is Relation-like dom K1 -defined the carrier of (W1 + W1) -valued Function-like finite Element of K27(K28((dom K1), the carrier of (W1 + W1)))
K28((dom K1), the carrier of (W1 + W1)) is Relation-like set
K27(K28((dom K1), the carrier of (W1 + W1))) is non empty set
- (Sum L) is Element of the carrier of (W1 + W1)
- (Sum K2) is Element of the carrier of x
KI is Relation-like the carrier of (W1 /\ W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of A
Sum KI is Element of the carrier of (W1 /\ W1)
(Carrier L1) \/ (Carrier L) is finite Element of K27( the carrier of (W1 + W1))
L1 + L is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1 + W1
Carrier (L1 + L) is finite Element of K27( the carrier of (W1 + W1))
LI is set
0. GF is V51(GF) Element of the carrier of GF
the ZeroF of GF is Element of the carrier of GF
K is Element of the carrier of (W1 + W1)
(L1 + L) . K is Element of the carrier of GF
L1 . K is Element of the carrier of GF
L . K is Element of the carrier of GF
(L1 . K) + (L . K) is Element of the carrier of GF
K is Element of the carrier of (W1 + W1)
L1 . K is Element of the carrier of GF
K is Element of the carrier of (W1 + W1)
L . K is Element of the carrier of GF
A \/ I2 is finite set
Carrier KI is finite Element of K27( the carrier of (W1 /\ W1))
LI is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1 + W1
Carrier LI is finite Element of K27( the carrier of (W1 + W1))
Sum LI is Element of the carrier of (W1 + W1)
A9 is linearly-independent Element of K27( the carrier of (W1 + W1))
LI + L is Relation-like the carrier of (W1 + W1) -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1 + W1
Carrier (LI + L) is finite Element of K27( the carrier of (W1 + W1))
(Carrier LI) \/ (Carrier L) is finite Element of K27( the carrier of (W1 + W1))
Sum (LI + L) is Element of the carrier of (W1 + W1)
K is Relation-like the carrier of W1 -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of W1
Carrier K is finite Element of K27( the carrier of W1)
Sum K is Element of the carrier of W1
0. W1 is V51(W1) Element of the carrier of W1
(Sum LI) + (Sum L) is Element of the carrier of (W1 + W1)
K is Relation-like the carrier of W1 -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of I2
Sum K is Element of the carrier of W1
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x is set
w1 is Element of the carrier of x
w2 is Element of the carrier of x
w1 + w2 is Element of the carrier of x
w1 is Element of the carrier of W1
K1 is Relation-like the carrier of W1 -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of I1
Sum K1 is Element of the carrier of W1
w2 is Element of the carrier of W1
K2 is Relation-like the carrier of W1 -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of I2
Sum K2 is Element of the carrier of W1
Carrier K2 is finite Element of K27( the carrier of W1)
L2 is Relation-like the carrier of x -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Carrier L2 is finite Element of K27( the carrier of x)
Sum L2 is Element of the carrier of x
Carrier K1 is finite Element of K27( the carrier of W1)
L1 is Relation-like the carrier of x -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Carrier L1 is finite Element of K27( the carrier of x)
Sum L1 is Element of the carrier of x
L1 + L2 is Relation-like the carrier of x -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of x
Carrier (L1 + L2) is finite Element of K27( the carrier of x)
(Carrier L1) \/ (Carrier L2) is finite Element of K27( the carrier of x)
A9 is Element of K27( the carrier of x)
L is Relation-like the carrier of x -defined the carrier of GF -valued Function-like total quasi_total Linear_Combination of A9
Sum L is Element of the carrier of x
Lin A9 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of x
the carrier of (Lin A9) is non empty set
Lin A is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() Subspace of W1 + W1
(GF,(W1 /\ W1)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(W1 + W1)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(W1 + W1)) + (GF,(W1 /\ W1)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
- (card A) is ext-real non positive V38() V58() set
((card I1) + (card I2)) + (- (card A)) is ext-real V38() V58() set
(((card I1) + (card I2)) + (- (card A))) + (card A) is ext-real V38() V58() set
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,W1) + (GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,V) + (GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
((GF,V) + (GF,W)) - (GF,n) is ext-real V38() V58() set
V /\ W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,(V /\ W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V + W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,(V + W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(V /\ W)) - (GF,n) is ext-real V38() V58() set
(GF,n) + ((GF,(V /\ W)) - (GF,n)) is ext-real V38() V58() set
(GF,(V + W)) + (GF,(V /\ W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
((GF,(V + W)) + (GF,(V /\ W))) - (GF,n) is ext-real V38() V58() set
(GF,(V + W)) + ((GF,(V /\ W)) - (GF,n)) is ext-real V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,n) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,V) + (GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
the U5 of n is Relation-like K28( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of n, the carrier of n), the carrier of n))
K28( the carrier of n, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of n, the carrier of n), the carrier of n)) is non empty set
the ZeroF of n is Element of the carrier of n
the lmult of n is Relation-like K28( the carrier of GF, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n))
the carrier of GF is non empty V19() set
K28( the carrier of GF, the carrier of n) is Relation-like non empty set
K28(K28( the carrier of GF, the carrier of n), the carrier of n) is Relation-like non empty set
K27(K28(K28( the carrier of GF, the carrier of n), the carrier of n)) is non empty set
VectSpStr(# the carrier of n, the U5 of n, the ZeroF of n, the lmult of n #) is non empty strict VectSpStr over GF
V + W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
V /\ W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(0). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(Omega). (V /\ W) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V /\ W
(0). (V /\ W) is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V /\ W
(GF,(V /\ W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(V + W)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,(V + W)) + 0 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
(Omega). n is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,((Omega). n)) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of V is non empty set
K27( the carrier of V) is non empty set
W is finite Element of K27( the carrier of V)
card W is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
K27(W) is non empty finite V28() set
x is finite Element of K27(W)
card x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT
W1 is Element of K27( the carrier of V)
Lin W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
W is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
W1 is non empty V72() V109(W) V110(W) V111(W) V112(W) V116() V117() V118() finite-dimensional VectSpStr over W
n is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 is non empty V72() strict V109(W) V110(W) V111(W) V112(W) V116() V117() V118() finite-dimensional Subspace of W1
(W,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
x is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(W,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
V is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of n is non empty set
K27( the carrier of n) is non empty set
{ (Lin b1) where b1 is Element of K27( the carrier of n) : ( b1 is linearly-independent & card b1 = V ) } is set
x is set
W1 is Element of K27( the carrier of n)
Lin W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card W1 is V6() V7() V8() cardinal set
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 is Element of K27( the carrier of n)
Lin W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card W1 is V6() V7() V8() cardinal set
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
the carrier of W1 is non empty set
K27( the carrier of W1) is non empty set
W1 is finite Element of K27( the carrier of W1)
A is Element of K27( the carrier of W1)
Lin A is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of W1
B is linearly-independent Element of K27( the carrier of n)
Lin B is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
card B is V6() V7() V8() cardinal set
x is set
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of n
W1 is set
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,V,n) is set
W is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
(GF,W) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,V) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
(GF,V,n) is set
W is set
x is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
(GF,x) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
GF is non empty non degenerated V49() V72() V92() V96() V98() V100() right-distributive left-distributive right_unital well-unital V106() left_unital V116() V117() V118() L11()
n is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
V is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional VectSpStr over GF
(GF,V,n) is set
W is non empty V72() V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V
(GF,W,n) is set
x is set
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of W
(GF,W1) is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() set
W1 is non empty V72() strict V109(GF) V110(GF) V111(GF) V112(GF) V116() V117() V118() finite-dimensional Subspace of V