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Proposition 17.183.

Q

RLD

a

= 0

RLD

(

i

2

n

:

Base

(

a

i

))

if

a

i

is the non-proper lter for some

i

2

n

.

Proof.

Take

A

i

=

;

and

m

=

f

i

g

. Then

Q

i

2

n

A

i

if

i

2

m

Base

(

a

i

)

if

i

2

n

n

m

=

;

.

Example 17.184.

There exists an indexed family

a

of principal lters such that

Q

RLD

a

is non-

principal.

Proof.

Let

n

=

N

. Let Base

(

a

i

) =

R

and each

a

i

be a principal lter corresponding to a two-

element set.

Every

Q

i

2

n

A

i

if

i

2

m

Base

(

a

i

)

if

i

2

n

n

m

has at least

c

n

>

c

elements.

There are elements

Q

RLD

a

with cardinality

2

n

=

n

. They can't be elements of

Q

RLD

a

because

n

=

! <

c

.

Corollary 17.185.

There exists an indexed family

a

of principal lters such that

Q

RLD

a

=

/

Q

RLD

a

.

Proof.

Because

Q

RLD

a

is principal.

Proposition 17.186.

Pr

k

RLD

Q

RLD

x

=

x

k

for every indexed family

x

of proper lters.

Proof.

Pr

k

RLD

Q

RLD

x

=

h

Pr

k

i

GR

Q

RLD

x

=

x

k

.

17.13 Subatomic product of funcoids

Denition 17.187.

Let

f

be an indexed family of funcoids. Then

Q

(

A

)

f

(

subatomic product

)

is a funcoid

Q

i

2

dom

f

Src

f

i

!

Q

i

2

dom

f

Dst

f

i

such that for every

a

2

atoms

RLD

(

i

2

dom

f

:

Src

f

i

)

,

b

2

atoms

RLD

(

i

2

dom

f

:

Dst

f

i

)

a

"

Y

(

A

)

f

#

b

, 8

i

2

dom

f

:

Pr

i

RLD

a

[

f

i

]

Pr

i

RLD

b:

Proposition 17.188.

The funcoid

Q

(

A

)

f

exists.

Proof.

To prove that

Q

(

A

)

f

exists we need to prove (for every

a

2

atoms

RLD

(

i

2

dom

f

:

Src

f

i

)

,

b

2

atoms

RLD

(

i

2

dom

f

:

Dst

f

i

)

)

8

X

2

GR

a; Y

2

GR

b

9

x

2

atoms

"

RLD

(

i

2

dom

f

:

Src

f

i

)

X ; y

2

atoms

"

RLD

(

i

2

dom

f

:

Dst

f

i

)

Y

:

x

"

Y

(

A

)

f

#

y

)

a

"

Y

(

A

)

f

#

b:

Let

8

X

2

GR

a; Y

2

GR

b

9

x

2

atoms

"

RLD

(

i

2

dom

f

:

Src

f

i

)

X ; y

2

atoms

"

RLD

(

i

2

dom

f

:

Dst

f

i

)

Y

:

x

h Q

(

A

)

f

i

y

.

Then

8

X

2

GR

a; Y

2

GR

b

9

x

2

atoms

"

RLD

(

i

2

dom

f

:

Src

f

i

)

X ; y

2

atoms

"

RLD

(

i

2

dom

f

:

Dst

f

i

)

Y

8

i

2

dom

f

:

Pr

i

RLD

x

[

f

i

]

Pr

i

RLD

y:

Then because Pr

i

RLD

x

2

atoms

"

Src

f

i

Pr

i

X

and likewise for

y

:

8

X

2

GR

a; Y

2

GR

b

8

i

2

dom

f

9

x

2

atoms

"

Src

f

i

Pr

i

X ; y

2

atoms

"

Dst

f

i

Pr

i

Y

:

x

[

f

i

]

y

.

Thus

8

X

2

GR

a; Y

2

GR

b

8

i

2

dom

f

:

"

Src

f

i

Pr

i

X

[

f

i

]

"

Dst

f

i

Pr

i

Y

;

8

X

2

GR

a; Y

2

GR

b

8

i

2

dom

f

:

Pr

i

X

[

f

i

]

Pr

i

Y

.

230

Multifuncoids and staroids