:: RLVECT_5 semantic presentation

REAL is non empty V12() non finite V129() V130() V131() V135() set

NAT is non empty V12() V18() V19() V20() non finite cardinal limit_cardinal V129() V130() V131() V132() V133() V134() V135() Element of K19(REAL)

K19(REAL) is non empty V12() non finite set

NAT is non empty V12() V18() V19() V20() non finite cardinal limit_cardinal V129() V130() V131() V132() V133() V134() V135() set

K19(NAT) is non empty V12() non finite set

K19(NAT) is non empty V12() non finite set

COMPLEX is non empty V12() non finite V129() V135() set

RAT is non empty V12() non finite V129() V130() V131() V132() V135() set

INT is non empty V12() non finite V129() V130() V131() V132() V133() V135() set

{} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty non negative V18() V19() V20() V22() V23() V24() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V129() V130() V131() V132() V133() V134() V135() set

2 is non empty ext-real positive non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

1 is non empty ext-real positive non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

3 is non empty ext-real positive non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

0 is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty ext-real non negative V18() V19() V20() V22() V23() V24() V25() V26() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V117() V118() V129() V130() V131() V132() V133() V134() V135() Element of NAT

card {} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty non negative V18() V19() V20() V22() V23() V24() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V129() V130() V131() V132() V133() V134() V135() set

Seg 1 is non empty V12() finite 1 -element V129() V130() V131() V132() V133() V134() Element of K19(NAT)

0 " is V25() set

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier V is finite Element of K19( the carrier of n)

Sum V is Element of the carrier of n

W is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier W is finite Element of K19( the carrier of n)

Sum W is Element of the carrier of n

x is Element of K19( the carrier of n)

(Carrier V) \/ (Carrier W) is finite Element of K19( the carrier of n)

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

- (Sum V) is Element of the carrier of n

(Sum V) + (- (Sum V)) is Element of the carrier of n

0. n is V64(n) Element of the carrier of n

V - W is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier (V - W) is finite Element of K19( the carrier of n)

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

W1 is Element of the carrier of n

(V - W) . W1 is ext-real V25() V26() Element of REAL

V . W1 is ext-real V25() V26() Element of REAL

W . W1 is ext-real V25() V26() Element of REAL

(V . W1) - (W . W1) is ext-real V25() V26() Element of REAL

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Element of K19( the carrier of n)

the ZeroF of n is Element of the carrier of n

the U5 of n is Relation-like K20( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20( the carrier of n, the carrier of n), the carrier of n))

K20( the carrier of n, the carrier of n) is Relation-like non empty set

K20(K20( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set

K19(K20(K20( the carrier of n, the carrier of n), the carrier of n)) is non empty set

the Mult of n is Relation-like K20(REAL, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20(REAL, the carrier of n), the carrier of n))

K20(REAL, the carrier of n) is Relation-like non empty V12() non finite set

K20(K20(REAL, the carrier of n), the carrier of n) is Relation-like non empty V12() non finite set

K19(K20(K20(REAL, the carrier of n), the carrier of n)) is non empty V12() non finite set

RLSStruct(# the carrier of n, the ZeroF of n, the U5 of n, the Mult of n #) is strict RLSStruct

W is Element of K19( the carrier of n)

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

x is Basis of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier V is finite Element of K19( the carrier of n)

K19( the carrier of n) is non empty set

W is Element of the carrier of n

{ b

x is Element of the carrier of n

V . x is ext-real V25() V26() Element of REAL

V is set

n is set

{V} is non empty V12() finite 1 -element set

n \ {V} is Element of K19(n)

K19(n) is non empty set

(n \ {V}) \/ {V} is non empty set

{V} \/ (n \ {V}) is non empty set

{V} \/ n is non empty set

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is Relation-like the carrier of n -defined REAL -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 V129() V130() V131() V132() V133() V134() Element of K19(NAT)

K20((dom W),(dom W)) is Relation-like finite set

K19(K20((dom W),(dom W))) is non empty finite V37() 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 total quasi_total onto bijective finite Element of K19(K20((dom W),(dom W)))

W * A is Relation-like dom W -defined the carrier of n -valued Function-like finite Element of K19(K20((dom W), the carrier of n))

K20((dom W), the carrier of n) is Relation-like set

K19(K20((dom W), the carrier of n)) is non empty set

len x is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len W is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len (V (#) W) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom (V (#) W) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

(V (#) W) * A is Relation-like dom W -defined the carrier of n -valued Function-like finite Element of K19(K20((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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom B is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

len (V (#) x) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom (V (#) x) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

dom x is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

I1 is non negative V18() V19() V20() V24() finite cardinal 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

W /. I1 is Element of the carrier of n

(V (#) x) . I1 is set

V . (W /. I1) is ext-real V25() V26() Element of REAL

(V . (W /. I1)) * (W /. I1) is Element of the carrier of n

(V (#) W) . (A . I1) is set

B . I1 is set

Sum B is Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

0. n is V64(n) Element of the carrier of n

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier V is finite Element of K19( the carrier of n)

K19( 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 V12() 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 V12() 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 K19( 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 V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n

rng <*B*> is non empty V12() finite 1 -element set

(rng W1) \/ (rng <*B*>) is non empty finite set

(Carrier V) /\ ((rng W1) \/ (rng <*B*>)) is finite Element of K19( the carrier of n)

{B} is non empty V12() finite 1 -element Element of K19( the carrier of n)

(rng W1) \/ {B} is non empty finite set

(Carrier V) /\ ((rng W1) \/ {B}) is finite Element of K19( the carrier of n)

(Carrier V) /\ (rng W1) is finite Element of K19( the carrier of n)

(Carrier V) /\ {B} is finite Element of K19( the carrier of n)

((Carrier V) /\ (rng W1)) \/ ((Carrier V) /\ {B}) is finite Element of K19( 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 ext-real V25() V26() Element of REAL

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 V12() 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 * 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 Function-like one-to-one constant functional empty proper non negative V18() V19() V20() V22() V23() V24() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V129() V130() V131() V132() V133() V134() V135() FinSequence of the carrier of n

K20(NAT, the carrier of n) is Relation-like non empty V12() 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

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

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

K19((rng V)) is non empty finite V37() set

x is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier x is finite Element of K19( the carrier of n)

K19( 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 K19((rng V))

W1 ` is finite Element of K19((rng V))

V - (W1 `) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set

W is finite Element of K19((rng V))

W \ (W1 `) is finite Element of K19((rng V))

(W1 `) ` is finite Element of K19((rng V))

W /\ ((W1 `) `) is finite Element of K19((rng V))

rng (V - (W1 `)) is finite set

dom W1 is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

K20((dom W1),(dom W1)) is Relation-like finite set

K19(K20((dom W1),(dom W1))) is non empty finite V37() set

V - W1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set

(rng V) \ (rng W1) is finite Element of K19((rng 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

rng I1 is finite set

(rng I1) /\ (rng W1) is finite set

((rng V) \ (rng W1)) /\ (rng W1) is finite Element of K19((rng V))

dom V is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

K20((dom V),(dom V)) is Relation-like finite set

K19(K20((dom V),(dom V))) is non empty finite V37() 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 V64(n) 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 total quasi_total onto bijective finite Element of K19(K20((dom V),(dom V)))

V * I2 is Relation-like dom V -defined the carrier of n -valued Function-like finite Element of K19(K20((dom V), the carrier of n))

K20((dom V), the carrier of n) is Relation-like set

K19(K20((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 total quasi_total onto bijective finite Element of K19(K20((dom W1),(dom W1)))

W1 * I2 is Relation-like dom W1 -defined the carrier of n -valued Function-like finite Element of K19(K20((dom W1), the carrier of n))

K20((dom W1), the carrier of n) is Relation-like set

K19(K20((dom W1), the carrier of n)) is non empty set

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier V is finite Element of K19( the carrier of n)

K19( 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 K19( 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

W1 is set

V . W1 is set

W1 is Element of the carrier of n

V . W1 is ext-real V25() V26() Element of REAL

K20( the carrier of n,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of n,REAL)) is non empty V12() non finite set

W1 is Relation-like the carrier of n -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of n,REAL))

W1 is Relation-like the carrier of n -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of n,REAL))

W1 is Element of the carrier of n

x is finite Element of K19( the carrier of n)

x /\ (Carrier V) is finite Element of K19( the carrier of n)

W1 . W1 is ext-real V25() V26() Element of REAL

V . W1 is ext-real V25() V26() Element of REAL

W1 . W1 is ext-real V25() V26() Element of REAL

Funcs ( the carrier of n,REAL) is functional non empty FUNCTION_DOMAIN of the carrier of n, REAL

W1 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Element of Funcs ( the carrier of n,REAL)

B is set

I1 is Element of the carrier of n

V . I1 is ext-real V25() V26() Element of REAL

A is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

A . I1 is ext-real V25() V26() Element of REAL

{ b

Carrier A is finite Element of K19( 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len W is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len (A (#) W) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom (V (#) W) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

dom (A (#) W) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

dom W is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

I2 is non negative V18() V19() V20() V24() finite cardinal set

W /. I2 is Element of the carrier of n

A . (W /. I2) is ext-real V25() V26() Element of REAL

V . (W /. I2) is set

W . I2 is set

(V (#) W) . I2 is set

V . (W /. I2) is ext-real V25() V26() Element of REAL

(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 ext-real V25() V26() Element of REAL

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

W is Element of K19( the carrier of n)

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of (Lin W) is non empty set

x is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

x + 1 is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 V12() 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 V12() finite 1 -element set

(rng B) \/ (rng <*A*>) is non empty finite set

{A} is non empty V12() finite 1 -element set

(rng B) \/ {A} is non empty finite set

I2 is Relation-like the carrier of n -defined REAL -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 V12() 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 non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len B is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len <*I1*> is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(len B) + (len <*I1*>) is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(len B) + 1 is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 REAL -valued Function-like total quasi_total Linear_Combination of W

Sum A is Element of the carrier of n

V . I1 is ext-real V25() V26() Element of REAL

(V . I1) * I2 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

A + ((V . I1) * I2) is Relation-like the carrier of n -defined REAL -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 V12() 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 Function-like one-to-one constant functional empty proper non negative V18() V19() V20() V22() V23() V24() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V129() V130() V131() V132() V133() V134() V135() FinSequence of the carrier of n

K20(NAT, the carrier of n) is Relation-like non empty V12() non finite set

0. n is V64(n) Element of the carrier of n

ZeroLC n is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Sum (ZeroLC n) is Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier V is finite Element of K19( the carrier of n)

Sum V is Element of the carrier of n

W is Element of K19( the carrier of n)

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital 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 REAL -valued Function-like total quasi_total Linear_Combination of W

Sum W1 is Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

W is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier W is finite Element of K19( the carrier of n)

K19( 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 REAL -valued Function-like Element of K19(K20( the carrier of n,REAL))

K20( the carrier of n,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of n,REAL)) is non empty V12() non finite set

Sum W is Element of the carrier of n

x is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier x is finite Element of K19( the carrier of V)

K19( the carrier of V) is non empty set

Sum x is Element of the carrier of V

dom x is set

W1 is set

W1 is Element of the carrier of V

x . W1 is ext-real V25() V26() Element of REAL

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 ext-real V25() V26() Element of REAL

x . B is set

dom W1 is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

K20((dom W1),(dom W1)) is Relation-like finite set

K19(K20((dom W1),(dom W1))) is non empty finite V37() set

A is Relation-like dom W1 -defined dom W1 -valued Function-like one-to-one total quasi_total onto bijective finite Element of K19(K20((dom W1),(dom W1)))

W1 * A is Relation-like dom W1 -defined the carrier of V -valued Function-like finite Element of K19(K20((dom W1), the carrier of V))

K20((dom W1), the carrier of V) is Relation-like set

K19(K20((dom W1), the carrier of V)) is non empty set

len (x (#) W1) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

len W1 is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom (x (#) W1) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

(x (#) W1) * A is Relation-like dom W1 -defined the carrier of V -valued Function-like finite Element of K19(K20((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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom B is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

rng B is finite set

K20(NAT, the carrier of V) is Relation-like non empty V12() non finite set

K19(K20(NAT, the carrier of V)) is non empty V12() non finite set

Sum B is Element of the carrier of V

0. V is V64(V) 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 K19(K20(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

K20(NAT, the carrier of n) is Relation-like non empty V12() non finite set

K19(K20(NAT, the carrier of n)) is non empty V12() 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

A is Relation-like NAT -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(NAT, the carrier of n))

A is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

A + 1 is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom W1 is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

len (W (#) W1) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

dom (W (#) W1) is finite V129() V130() V131() V132() V133() V134() Element of K19(NAT)

A is non negative V18() V19() V20() V24() finite cardinal 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 ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() 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 ext-real V25() V26() Element of REAL

(x . w1) * w1 is Element of the carrier of V

W . (W1 /. A) is ext-real V25() V26() Element of REAL

(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

K20((dom (x (#) W1)),(dom (x (#) W1))) is Relation-like finite set

K19(K20((dom (x (#) W1)),(dom (x (#) W1)))) is non empty finite V37() set

A is Relation-like dom (x (#) W1) -defined dom (x (#) W1) -valued Function-like one-to-one total quasi_total onto bijective finite Element of K19(K20((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 K19(K20((dom (x (#) W1)), the carrier of V))

K20((dom (x (#) W1)), the carrier of V) is Relation-like set

K19(K20((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 V64(n) Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

W is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier W is finite Element of K19( the carrier of V)

K19( the carrier of V) is non empty set

Sum W is Element of the carrier of V

K20( the carrier of V,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of V,REAL)) is non empty V12() non finite set

K19( 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 ext-real V25() V26() Element of REAL

W . B is set

A is Element of the carrier of n

A is Element of the carrier of n

K20( the carrier of n,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of n,REAL)) is non empty V12() non finite set

W1 is Relation-like the carrier of n -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of n,REAL))

W1 is Relation-like the carrier of n -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of n,REAL))

A is Element of the carrier of n

W1 is finite Element of K19( the carrier of n)

W . A is set

W1 . A is ext-real V25() V26() Element of REAL

Funcs ( the carrier of n,REAL) is functional non empty FUNCTION_DOMAIN of the carrier of n, REAL

A is Relation-like the carrier of n -defined REAL -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 REAL -valued Function-like Element of K19(K20( the carrier of n,REAL))

Carrier A is finite Element of K19( 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 ext-real V25() V26() Element of REAL

W . I2 is set

I1 is set

A . I1 is set

W . I1 is set

x is Relation-like the carrier of V -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of V,REAL))

x . I1 is set

B is Relation-like the carrier of V -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of V,REAL))

B . I1 is set

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

W is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier W is finite Element of K19( the carrier of n)

K19( the carrier of n) is non empty set

Sum W is Element of the carrier of n

K19( the carrier of V) is non empty set

K20( the carrier of V,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of V,REAL)) is non empty V12() non finite set

W | the carrier of V is Relation-like the carrier of n -defined the carrier of V -defined the carrier of n -defined REAL -valued Function-like Element of K19(K20( the carrier of n,REAL))

K20( the carrier of n,REAL) is Relation-like non empty V12() non finite set

K19(K20( the carrier of n,REAL)) is non empty V12() non finite set

W1 is Relation-like the carrier of V -defined REAL -valued Function-like non empty total quasi_total Element of K19(K20( the carrier of V,REAL))

Funcs ( the carrier of V,REAL) is functional non empty FUNCTION_DOMAIN of the carrier of V, REAL

dom W1 is non empty set

W1 is Element of the carrier of V

x is finite Element of K19( the carrier of V)

W . W1 is set

W1 . W1 is ext-real V25() V26() Element of REAL

W1 is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier W1 is finite Element of K19( the carrier of V)

Sum W1 is Element of the carrier of V

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

V is Basis of n

Lin V is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

W is Element of the carrier of n

the ZeroF of n is Element of the carrier of n

the U5 of n is Relation-like K20( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20( the carrier of n, the carrier of n), the carrier of n))

K20( the carrier of n, the carrier of n) is Relation-like non empty set

K20(K20( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set

K19(K20(K20( the carrier of n, the carrier of n), the carrier of n)) is non empty set

the Mult of n is Relation-like K20(REAL, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20(REAL, the carrier of n), the carrier of n))

K20(REAL, the carrier of n) is Relation-like non empty V12() non finite set

K20(K20(REAL, the carrier of n), the carrier of n) is Relation-like non empty V12() non finite set

K19(K20(K20(REAL, the carrier of n), the carrier of n)) is non empty V12() non finite set

RLSStruct(# the carrier of n, the ZeroF of n, the U5 of n, the Mult of n #) is strict RLSStruct

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

K19( the carrier of V) is non empty set

W is Element of K19( the carrier of V)

x is Element of K19( the carrier of n)

0. n is V64(n) Element of the carrier of n

W1 is Relation-like the carrier of n -defined REAL -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 K19( the carrier of n)

W1 is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier W1 is finite Element of K19( the carrier of V)

Sum W1 is Element of the carrier of V

A is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of W

Sum A is Element of the carrier of V

0. V is V64(V) Element of the carrier of V

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

K19( the carrier of V) is non empty set

W is Element of K19( the carrier of n)

x is Element of K19( the carrier of V)

0. V is V64(V) Element of the carrier of V

W1 is Relation-like the carrier of V -defined REAL -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 K19( the carrier of V)

W1 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier W1 is finite Element of K19( the carrier of n)

Sum W1 is Element of the carrier of n

A is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of W

Sum A is Element of the carrier of n

0. n is V64(n) Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

W is Basis of V

the carrier of n is non empty set

K19( the carrier of n) is non empty set

x is Basis of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Element of K19( the carrier of n)

W is Element of the carrier of n

{W} is non empty V12() finite 1 -element Element of K19( the carrier of n)

V \ {W} is Element of K19( the carrier of n)

K19(V) is non empty set

Lin {W} is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

x is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of {W}

Sum x is Element of the carrier of n

W1 is Element of K19( the carrier of n)

Lin W1 is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

W1 \/ {W} is non empty Element of K19( the carrier of n)

V \/ V is Element of K19( the carrier of n)

W1 is Relation-like the carrier of n -defined REAL -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 K19( the carrier of n)

Carrier x is finite Element of K19( the carrier of n)

(Carrier W1) \/ (Carrier x) is finite Element of K19( the carrier of n)

W1 - x is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier (W1 - x) is finite Element of K19( the carrier of n)

A is Element of the carrier of n

x . A is ext-real V25() V26() Element of REAL

A is set

B is Element of the carrier of n

W1 . B is ext-real V25() V26() Element of REAL

(W1 - x) . B is ext-real V25() V26() Element of REAL

x . B is ext-real V25() V26() Element of REAL

(W1 . B) - (x . B) is ext-real V25() V26() Element of REAL

(W1 . B) - 0 is ext-real V25() V26() Element of REAL

0. n is V64(n) 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 REAL -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 REAL -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

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is Basis of n

W is non empty Element of K19( the carrier of n)

V \/ W is non empty Element of K19( the carrier of n)

x is set

W1 is Element of K19( the carrier of n)

W1 is Element of the carrier of n

{W1} is non empty V12() finite 1 -element Element of K19( the carrier of n)

W1 \ {W1} is Element of K19( the carrier of n)

V \ {W1} is Element of K19( the carrier of n)

A is Element of K19( the carrier of n)

Lin V is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

Lin A is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

W is Element of K19( the carrier of n)

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

x is set

the carrier of (Lin W) is non empty set

W1 is Relation-like the carrier of n -defined REAL -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 K19( the carrier of n)

W1 is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier W1 is finite Element of K19( the carrier of V)

K19( the carrier of V) is non empty set

Sum W1 is Element of the carrier of V

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

the carrier of V is non empty set

K19( the carrier of V) is non empty set

W is Element of K19( the carrier of n)

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

x is Element of K19( the carrier of V)

Lin x is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of V

W1 is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

W1 is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

A is set

the carrier of (Lin W) is non empty set

B is Relation-like the carrier of n -defined REAL -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 K19( the carrier of n)

I1 is Relation-like the carrier of V -defined REAL -valued Function-like total quasi_total Linear_Combination of V

Carrier I1 is finite Element of K19( the carrier of V)

Sum I1 is Element of the carrier of V

I2 is Relation-like the carrier of V -defined REAL -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 REAL -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 K19( the carrier of V)

I1 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier I1 is finite Element of K19( the carrier of n)

Sum I1 is Element of the carrier of n

I2 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of W

Sum I2 is Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

V is finite Element of K19( the carrier of n)

W is finite Element of K19( the carrier of n)

V \/ W is finite Element of K19( the carrier of n)

Lin (V \/ W) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

Lin W is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

x is Element of the carrier of n

{x} is non empty V12() finite 1 -element Element of K19( the carrier of n)

W1 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of V \/ W

Sum W1 is Element of the carrier of n

Carrier W1 is finite Element of K19( the carrier of n)

W1 is set

A is Element of the carrier of n

W1 . A is ext-real V25() V26() Element of REAL

W1 is Element of the carrier of n

W1 . W1 is ext-real V25() V26() Element of REAL

W1 is Element of the carrier of n

W1 . W1 is ext-real V25() V26() Element of REAL

{W1} is non empty V12() finite 1 -element Element of K19( the carrier of n)

(V \/ W) \ {W1} is finite Element of K19( the carrier of n)

((V \/ W) \ {W1}) \/ {x} is non empty finite Element of K19( the carrier of n)

Lin (((V \/ W) \ {W1}) \/ {x}) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace 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

rng B is finite set

W1 (#) 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 (W1 (#) B) is Element of the carrier of n

B -| W1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set

B |-- 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

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 ^ I2 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n

B - {W1} is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set

rng (I1 ^ I2) is finite set

(Carrier W1) \ {W1} is finite Element of K19( the carrier of n)

<*W1*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n

(B -| W1) ^ <*W1*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set

((B -| W1) ^ <*W1*>) ^ (B |-- W1) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set

<*W1*> ^ I2 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

I1 ^ (<*W1*> ^ I2) 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 (#) 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 (#) (<*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 (#) I1) ^ (W1 (#) (<*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 (#) <*W1*> 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 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n

(W1 (#) <*W1*>) ^ (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 (#) I1) ^ ((W1 (#) <*W1*>) ^ (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 (#) I1) ^ (W1 (#) <*W1*>) 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) ^ (W1 (#) <*W1*>)) ^ (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 . W1) * W1 is Element of the carrier of n

<*((W1 . W1) * W1)*> is Relation-like NAT -defined the carrier of n -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of n

(W1 (#) I1) ^ <*((W1 . W1) * W1)*> 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 (#) I1) ^ <*((W1 . W1) * W1)*>) ^ (W1 (#) I2) 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 . W1) * W1)*> ^ (W1 (#) I2) 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 (#) I1) ^ (<*((W1 . W1) * W1)*> ^ (W1 (#) I2)) 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 (#) I1) ^ (<*((W1 . W1) * W1)*> ^ (W1 (#) I2))) is Element of the carrier of n

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

Sum (<*((W1 . W1) * W1)*> ^ (W1 (#) I2)) is Element of the carrier of n

(Sum (W1 (#) I1)) + (Sum (<*((W1 . W1) * W1)*> ^ (W1 (#) I2))) is Element of the carrier of n

Sum <*((W1 . W1) * W1)*> is Element of the carrier of n

Sum (W1 (#) I2) is Element of the carrier of n

(Sum <*((W1 . W1) * W1)*>) + (Sum (W1 (#) I2)) is Element of the carrier of n

(Sum (W1 (#) I1)) + ((Sum <*((W1 . W1) * W1)*>) + (Sum (W1 (#) I2))) is Element of the carrier of n

(Sum (W1 (#) I2)) + ((W1 . W1) * W1) is Element of the carrier of n

(Sum (W1 (#) I1)) + ((Sum (W1 (#) I2)) + ((W1 . W1) * W1)) is Element of the carrier of n

(Sum (W1 (#) I1)) + (Sum (W1 (#) I2)) is Element of the carrier of n

((Sum (W1 (#) I1)) + (Sum (W1 (#) I2))) + ((W1 . W1) * W1) is Element of the carrier of n

(W1 (#) I1) ^ (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

Sum ((W1 (#) I1) ^ (W1 (#) I2)) is Element of the carrier of n

(Sum ((W1 (#) I1) ^ (W1 (#) I2))) + ((W1 . W1) * W1) is Element of the carrier of n

W1 (#) (I1 ^ I2) 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 ^ I2)) is Element of the carrier of n

(Sum (W1 (#) (I1 ^ I2))) + ((W1 . W1) * W1) is Element of the carrier of n

Lin {x} is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

A is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of {x}

Sum A is Element of the carrier of n

(rng (I1 ^ I2)) /\ (Carrier W1) is finite Element of K19( the carrier of n)

A is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier A is finite Element of K19( the carrier of n)

A (#) (I1 ^ 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 B) \ {W1} is finite Element of K19((rng B))

K19((rng B)) is non empty finite V37() set

(W1 . W1) " is ext-real V25() V26() Element of REAL

A9 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of (V \/ W) \ {W1}

- A9 is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

(- A9) + A is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

((W1 . W1) ") * ((- A9) + A) is Relation-like the carrier of n -defined REAL -valued Function-like total quasi_total Linear_Combination of n

Carrier (((W1 . W1) ") * ((- A9) + A)) is finite Element of K19( the carrier of n)

Carrier ((- A9) + A) is finite Element of K19( the carrier of n)

Carrier (- A9) is finite Element of K19( the carrier of n)

Carrier A is finite Element of K19( the carrier of n)

(Carrier (- A9)) \/ (Carrier A) is finite Element of K19( the carrier of n)

Carrier A9 is finite Element of K19( the carrier of n)

(Carrier A9) \/ (Carrier A) is finite Element of K19( the carrier of n)

A9 (#) (I1 ^ I2) 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 (#) (I1 ^ I2)) is Element of the carrier of n

Sum A9 is Element of the carrier of n

((W1 . W1) ") * x is Element of the carrier of n

((W1 . W1) ") * (Sum A9) is Element of the carrier of n

((W1 . W1) ") * ((W1 . W1) * W1) is Element of the carrier of n

(((W1 . W1) ") * (Sum A9)) + (((W1 . W1) ") * ((W1 . W1) * W1)) is Element of the carrier of n

((W1 . W1) ") * (W1 . W1) is ext-real V25() V26() Element of REAL

(((W1 . W1) ") * (W1 . W1)) * W1 is Element of the carrier of n

(((W1 . W1) ") * (Sum A9)) + ((((W1 . W1) ") * (W1 . W1)) * W1) is Element of the carrier of n

1 * W1 is Element of the carrier of n

(((W1 . W1) ") * (Sum A9)) + (1 * W1) is Element of the carrier of n

(((W1 . W1) ") * (Sum A9)) + W1 is Element of the carrier of n

(((W1 . W1) ") * x) - (((W1 . W1) ") * (Sum A9)) is Element of the carrier of n

x - (Sum A9) is Element of the carrier of n

((W1 . W1) ") * (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 . W1) ") * ((- (Sum A9)) + x) is Element of the carrier of n

Sum (- A9) is Element of the carrier of n

(Sum (- A9)) + (Sum A) is Element of the carrier of n

((W1 . W1) ") * ((Sum (- A9)) + (Sum A)) is Element of the carrier of n

Sum ((- A9) + A) is Element of the carrier of n

((W1 . W1) ") * (Sum ((- A9) + A)) is Element of the carrier of n

Sum (((W1 . W1) ") * ((- A9) + A)) is Element of the carrier of n

n is non empty V83() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct

the carrier of n is non empty set

K19( the carrier of n) is non empty set

the ZeroF of n is Element of the carrier of n

the U5 of n is Relation-like K20( the carrier of n, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20( the carrier of n, the carrier of n), the carrier of n))

K20( the carrier of n, the carrier of n) is Relation-like non empty set

K20(K20( the carrier of n, the carrier of n), the carrier of n) is Relation-like non empty set

K19(K20(K20( the carrier of n, the carrier of n), the carrier of n)) is non empty set

the Mult of n is Relation-like K20(REAL, the carrier of n) -defined the carrier of n -valued Function-like non empty total quasi_total Element of K19(K20(K20(REAL, the carrier of n), the carrier of n))

K20(REAL, the carrier of n) is Relation-like non empty V12() non finite set

K20(K20(REAL, the carrier of n), the carrier of n) is Relation-like non empty V12() non finite set

K19(K20(K20(REAL, the carrier of n), the carrier of n)) is non empty V12() non finite set

RLSStruct(# the carrier of n, the ZeroF of n, the U5 of n, the Mult of n #) is strict RLSStruct

V is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

V + 1 is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

W is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

W - (V + 1) is ext-real V25() V26() Element of REAL

x is finite Element of K19( the carrier of n)

card x is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

W1 is finite Element of K19( the carrier of n)

card W1 is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

Lin x is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

W1 is set

A is Element of the carrier of n

{A} is non empty V12() finite 1 -element Element of K19( the carrier of n)

W1 \ {A} is finite Element of K19( the carrier of n)

K19(W1) is non empty finite V37() set

card (W1 \ {A}) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

card {A} is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(card W1) - (card {A}) is set

(V + 1) - 1 is ext-real V25() V26() Element of REAL

W - V is ext-real V25() V26() Element of REAL

I1 is finite Element of K19( the carrier of n)

card I1 is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(W1 \ {A}) \/ I1 is finite Element of K19( the carrier of n)

Lin ((W1 \ {A}) \/ I1) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

Lin (W1 \ {A}) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

V - V is ext-real V25() V26() Element of REAL

I2 is finite Element of K19( the carrier of n)

card I2 is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(W1 \ {A}) \/ I2 is finite Element of K19( the carrier of n)

Lin ((W1 \ {A}) \/ I2) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

I1 \/ (W1 \ {A}) is finite Element of K19( the carrier of n)

I2 is Element of the carrier of n

{I2} is non empty V12() finite 1 -element Element of K19( the carrier of n)

(I1 \/ (W1 \ {A})) \ {I2} is finite Element of K19( the carrier of n)

((I1 \/ (W1 \ {A})) \ {I2}) \/ {A} is non empty finite Element of K19( the carrier of n)

Lin (((I1 \/ (W1 \ {A})) \ {I2}) \/ {A}) is non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n

I1 \ {I2} is finite Element of K19( the carrier of n)

(W1 \ {A}) \ {I2} is finite Element of K19( the carrier of n)

((W1 \ {A}) \ {I2}) \/ {A} is non empty finite Element of K19( the carrier of n)

(W1 \ {A}) \/ {A} is non empty finite Element of K19( the carrier of n)

(I1 \ {I2}) \/ (((W1 \ {A}) \ {I2}) \/ {A}) is non empty finite Element of K19( the carrier of n)

(I1 \ {I2}) \/ ((W1 \ {A}) \/ {A}) is non empty finite Element of K19( the carrier of n)

W1 \/ (I1 \ {I2}) is finite Element of K19( the carrier of n)

K19(I1) is non empty finite V37() set

card (I1 \ {I2}) is ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

card {I2} is non empty ext-real non negative V18() V19() V20() V24() V25() V26() finite cardinal V117() V118() V129() V130() V131() V132() V133() V134() Element of NAT

(card I1) - (card {I2}) is set

(W - V) - 1 is ext-real V25() V26() Element of REAL

(I1 \ {I2}) \/ ((W1 \ {A}) \ {I2}) is finite Element of K19( the carrier of n)

((I1 \ {I2}) \/ ((W1 \ {A}) \ {I2})) \/ {A} is non empty finite Element of K19( the carrier of n)

Lin (