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2. A relation

δ

P

(atoms 1

F

(

A

)

×

atoms 1

F

(

B

)

)

such that (for every

a

atoms 1

F

(

A

)

,

b

atoms 1

F

(

B

)

)

X

up

a, Y

up

b

x

atoms

A

X, y

atoms

B

Y

:

x δ y

a δ b

(8)

can be continued to the relation

[

f

]

for a unique

f

FCD

(

A

;

B

)

;

X

[

f

]

Y ⇔ ∃

x

atoms

X

, y

atoms

Y

:

x δ y

(9)

for every

X ∈

F

(

A

)

,

Y ∈

F

(

B

)

.

Proof

Existence of no more than one such funcoids and formulas (7) and (9)

follow from the previous theorem.

1. Consider the function

α

F

(

B

)

P

A

defined by the formula (for every

X

P

A

)

α

X

=

[

h

α

i

atoms

A

X.

Obviously

α

= 0

F

(

B

)

. For every

I, J

P

A

α

(

I

J

) =

[

h

α

i

atoms

A

(

I

J

)

=

[

h

α

i

(atoms

A

I

atoms

A

J

)

=

[

h

α

i

atoms

A

I

∪ h

α

i

atoms

A

J

=

[

h

α

i

atoms

A

I

[

h

α

i

atoms

A

J.

=

α

I

α

J.

Let continue

α

till a funcoid

f

(by the theorem 7):

h

f

i X

=

T

h

α

i

up

X

.

Let’s prove the reverse of (6):

\ D[

◦ h

α

i ◦

atoms

◦ ↑

A

E

up

a

=

\ D[

◦ h

α

i

E

h

atoms

i

A

up

a

\ D[

◦ h

α

i

E

{{

a

}}

=

\ n[

◦ h

α

i

{

a

}

o

=

\ n[

h

α

i {

a

}

o

=

\ n[

{

αa

}

o

=

\

{

αa

}

=

αa.

Finally,

αa

=

\ D[

◦ h

α

i ◦

atoms

◦ ↑

A

E

up

a

=

\

h

α

i

up

a

=

h

f

i

a,

so

h

f

i

is a continuation of

α

.

2. Consider the relation

δ

P

(

P

A

×

P

B

) defined by the formula (for

every

X

P

A

,

Y

P

B

)

X δ

Y

⇔ ∃

x

atoms

A

X, y

atoms

B

Y

:

x δ y.

22