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21.12. SUBATOMIC PRODUCT OF FUNCOIDS

353

Proof.

Let

a

,

b

be ultrafilters on

Q

i

n

Src

f

i

and

Q

i

n

Dst

g

i

correspondingly,

a

(

A

)

Y

i

n

(

g

i

f

i

)

b

i

dom

f

:

D

Pr

i

E

a

[

g

i

f

i

]

D

Pr

i

E

b

i

dom

f

C

atoms

F

(Dst

f

i

)

:

D

Pr

i

E

a

[

f

i

]

C

C

[

g

i

]

D

Pr

i

E

b

i

dom

f

c

atoms

RLD

(

λi

n

:Dst

f

)

:

D

Pr

i

E

a

[

f

i

]

D

Pr

i

E

c

D

Pr

i

E

c

[

g

i

]

D

Pr

i

E

b

c

atoms

RLD

(

λi

n

:Dst

f

)

i

dom

f

:

D

Pr

i

E

a

[

f

i

]

D

Pr

i

E

c

D

Pr

i

E

c

[

g

i

]

D

Pr

i

E

b

c

atoms

RLD

(

λi

n

:Dst

f

)

:

a

(

A

)

Y

f

c

c

(

A

)

Y

g

b

a

(

A

)

Y

g

(

A

)

Y

f

b.

But

i

dom

f

C

atoms

F

(Dst

f

i

)

:

D

Pr

i

E

a

[

f

i

]

C

C

[

g

i

]

D

Pr

i

E

b

implies

C

Y

i

n

atoms

F

(Dst

f

i

)

i

dom

f

:

D

Pr

i

E

a

[

f

i

]

C

i

C

i

[

g

i

]

D

Pr

i

E

b

.

Take

c

atoms

Q

RLD

C

. Then

i

dom

f

:

D

Pr

i

E

a

[

f

i

] Pr

i

c

Pr

i

c

[

g

i

]

D

Pr

i

E

b

that is

i

dom

f

:

D

Pr

i

E

a

[

f

i

]

D

Pr

i

E

c

D

Pr

i

E

[

g

i

]

D

Pr

i

E

b

We have

a

h

Q

(

A

)

i

n

(

g

i

f

i

)

i

b

a

h

Q

(

A

)

g

Q

(

A

)

f

i

b

.

Corollary

1786

.

Q

(

A

)

f

k

1

. . .

Q

(

A

)

f

0

=

Q

(

A

)

i

n

(

f

k

1

. . .

f

0

) for

every

n

-indexed families

f

0

, . . . , f

n

1

of composable funcoids.

Proposition

1787

.

Q

RLD

a

h

Q

(

A

)

f

i

Q

RLD

b

⇔ ∀

i

dom

f

:

a

i

[

f

i

]

b

i

for

an indexed family

f

of funcoids and indexed families

a

and

b

of filters where

a

i

F

(Src

f

i

),

b

i

F

(Dst

f

i

) for every

i

dom

f

.

Proof.

If

a

i

=

or

b

i

=

for some

i

our theorem is obvious. We will take

a

i

6

=

and

b

i

6

=

, thus there exist

x

atoms

RLD

Y

a,

y

atoms

RLD

Y

b.