7.9. SPECIFYING FUNCOIDS BY FUNCTIONS OR RELATIONS ON ATOMIC FILTERS 163

Proof.

Let

x

atoms

F

(Src

f

)

. Then

h

g

f

i

x

=

h

g

ih

f

i

x

= (theorem

848

)

F

l

G

up

g

h

G

ih

f

i

x

= (theorem

848

)

F

l

G

up

g

h

G

i

F

l

F

up

f

h

F

i

x

= (theorem

836

)

F

l

G

up

g

F

l

F

up

f

h

G

ih

F

i

x

=

F

l

h

G

ih

F

i

x

F

up

f, G

up

g

=

F

l

h

G

F

i

x

F

up

f, G

up

g

= (theorem

875

)

*

FCD

l

G

F

F

up

f, G

up

g

+

x.

Thus

g

f

=

d

FCD

n

G

F

F

up

f.G

up

g

o

.

Proposition

877

.

For

f

FCD

(

A, B

), a finite set

X

P

A

and a function

t

F

(

B

)

X

there exists (obviously unique)

g

FCD

(

A, B

) such that

h

g

i

p

=

h

f

i

p

for

p

atoms

F

(

A

)

\

atoms

X

and

h

g

i

@

{

x

}

=

t

(

x

) for

x

X

.

This funcoid

g

is determined by the formula

g

= (

f

\

(@

X

×

FCD

>

))

t

l

x

X

(@

{

x

} ×

FCD

t

(

x

))

.

Proof.

Take

g

= (

f

\

(@

X

×

FCD

>

))

t

d

q

X

(@

{

q

} ×

FCD

t

(

x

)) that is

g

=

f

u

X

× >

t

d

q

X

(@

{

q

} ×

FCD

t

(

x

)) =

f

u

X

× >

t

d

q

X

(@

{

q

} ×

FCD

t

(

x

)).

h

g

i

p

= (theorem

850

=

f

u

X

× >

p

t

d

q

X

h

@

{

q

} ×

FCD

t

(

x

)

i

p

=

(theorem

875

=

h

f

i

p

u

X

× >

p

t

d

q

X

h

@

{

q

} ×

FCD

t

(

x

)

i

p

.

So

h

g

i

@

{

x

}

= (

h

f

i

@

{

x

} u ⊥

)

t

t

(

x

) =

t

(

x

) for

x

X

.

If

p

atoms

F

(

A

)

\

atoms

X

then we have

h

g

i

p

= (

h

f

i

p

u >

)

t ⊥

=

h

f

i

p

.

Corollary

878

.

If

f

FCD

(

A, B

),

x

A

, and

Y ∈

F

(

B

), then there exists

an (obviously unique)

g

FCD

(

A, B

) such that

h

g

i

p

=

h

f

i

p

for all ultrafilters

p

except of

p

= @

{

x

}

and

h

g

i

@

{

x

}

=

Y

.

This funcoid

g

is determined by the formula

g

= (

f

\

(@

{

x

} ×

FCD

>

))

t

(

{

x

} ×

FCD

Y

)

.

Theorem

879

.

Let

A

,

B

,

C

be sets,

f

FCD

(

A, B

),

g

FCD

(

B, C

),

h

FCD

(

A, C

). Then

g

f

6

h

g

6

h

f

1

.