:: VECTSP_9 semantic presentation

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

RAT is set
INT is 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

Seg 1 is non empty V19() finite 1 -element Element of K27(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

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

Sum (V (#) W) is Element of the carrier of n

Sum (V (#) x) is Element of the carrier of n

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))

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

Carrier V is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set

x is set

rng W1 is finite set

Sum (V (#) W1) is Element of the carrier of n
{x} is non empty V19() finite 1 -element set

rng A is finite set
() /\ (rng W1) is finite Element of K27( the carrier of n)

rng W1 is finite set
B is Element of the carrier of n

rng <*B*> is non empty V19() finite 1 -element set
(rng W1) \/ () is non empty finite set
() /\ ((rng W1) \/ ()) 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
() /\ ((rng W1) \/ {B}) is finite Element of K27( the carrier of n)
() /\ (rng W1) is finite Element of K27( the carrier of n)
() /\ {B} is finite Element of K27( the carrier of n)
(() /\ (rng W1)) \/ (() /\ {B}) is finite Element of K27( 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 (#) 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

rng W is finite set

Sum (V (#) W) is Element of the carrier of n

K28(NAT, the carrier of n) is Relation-like non empty V19() non finite set

rng W is finite set

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

rng V is finite set
K27((rng V)) is non empty finite V28() set

Carrier x is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set

Sum (x (#) V) is Element of the carrier of n
Sum x is Element of the carrier of n

rng W1 is finite set

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

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

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

Sum (x (#) (B ^ I1)) is Element 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

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

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

Carrier V is finite Element of K27( the carrier of n)
K27( the carrier of n) is non empty set

rng W is finite set
(rng W) /\ () 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
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 /\ () 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 . I1 is Element of the carrier of GF
Carrier A is finite Element of K27( 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

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

rng W1 is finite set
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT

Sum (V (#) W1) is Element of the carrier of n

A is 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) \/ () is non empty finite set
{A} is non empty V19() finite 1 -element set
(rng B) \/ {A} is non empty finite set

Sum I2 is Element of the carrier of n
I1 is Element 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

Sum (V (#) B) is Element of the carrier of n

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

(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

rng x is finite set

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

rng W1 is finite set
len W1 is V6() V7() V8() V12() finite cardinal ext-real non negative V38() V58() Element of NAT

Sum (V (#) W1) is Element 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

Sum () 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

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

rng x is finite set

Sum (V (#) x) is Element of the carrier of n

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

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

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

rng W1 is finite set

Sum (x (#) W1) is Element of the carrier of V

rng W1 is finite set

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

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))

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

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

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 | 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

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

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

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

Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)

Carrier W1 is finite Element of K27( the carrier of V)
Sum W1 is Element of the carrier of V

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

Sum W1 is Element of the carrier of V
Carrier W1 is finite Element of K27( the carrier of V)

Carrier W1 is finite Element of K27( the carrier of n)
Sum W1 is Element of the carrier of n

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

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)

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) \/ () is finite Element of K27( the carrier 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

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

Sum W1 is Element of the carrier of n
Carrier W1 is finite Element of K27( the carrier of n)

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

Sum B is Element of the carrier of n
Carrier B is finite Element of K27( the carrier of n)

Carrier I1 is finite Element of K27( the carrier of V)
Sum I1 is Element of the carrier of V

Sum I2 is Element of the carrier of V
the carrier of (Lin x) is non empty set
A is set

Sum B is Element of the carrier of V
Carrier B is finite Element of K27( the carrier of V)

Carrier I1 is finite Element of K27( the carrier of n)
Sum I1 is Element of the carrier of n

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

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

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

rng I1 is finite set

Sum (W1 (#) I1) is Element of the carrier of n

rng I2 is finite set

rng I1 is finite set

rng (I2 ^ I1) is finite set
(Carrier W1) \ {A} is finite Element of K27( 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*>) ^ (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

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)

Carrier A is finite Element of K27( 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) + 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)

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) - () 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

rng x is finite set
the carrier of GF is non empty V19() set