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
{ b1 where b1 is Element of the carrier of n : not V . b1 = 0 } is set
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
{ b1 where b1 is Element of the carrier of n : not A . b1 = 0 } is set
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 (