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8.1. FUNCOID INDUCED BY A RELOID

148

Proof.

h

(

FCD

)(

g

f

)

i

X

=

l

Dst

g

h

H

i

X

H

GR(

g

f

)

=

l

Dst

g

h

H

i

X

H

GR

d

n

RLD

(

G

F

)

F

xyGR

f,G

xyGR

g

o

.

Obviously

l

RLD

(

G

F

)

F

xyGR

f, G

xyGR

g

=

l

RLD

xyGR

l

RLD

(

G

F

)

F

xyGR

f, G

xyGR

g

;

from this by lemma

802

(taking into account that

(

G

F

)

F

xyGR

f, G

xyGR

g

and

xyGR

l

RLD

(

G

F

)

F

xyGR

f, G

xyGR

g

are filter bases)

l

Dst

g

h

H

i

X

H

GR

d

n

RLD

(

G

F

)

F

xyGR

f,G

xyGR

g

o

=

Dst

g

h

G

F

i

X

F

GR

f, G

GR

g

.

On the other side

h

((

FCD

)

g

)

((

FCD

)

f

)

i

X

=

h

(

FCD

)

g

ih

(

FCD

)

f

i

X

=

h

(

FCD

)

g

i

l

Dst

g

h

F

i

X

F

xyGR

f

=

l

FCD

G

d

n

Dst

g

h

F

i

X

F

xyGR

f

o

G

xyGR

g

.

Let’s prove that

n

h

F

i

X

F

xyGR

f

o

is a filter base. If

A, B

n

h

F

i

X

F

xyGR

f

o

then

A

=

h

F

1

i

X

,

B

=

h

F

2

i

X

where

F

1

, F

2

xyGR

f

.

A

B

⊇ h

F

1

u

F

2

i

X

n

h

F

i

X

F

xyGR

f

o

.

So

n

h

F

i

X

F

xyGR

f

o

is really a filter base.

By theorem

599

we have

FCD

G

l

Dst

g

h

F

i

X

F

xyGR

f

=

l

Dst

g

h

G

i

h

F

i

X

F

xyGR

f

.