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Theorem 36.

If

f

is a complete reloids and

S

is a set of complete reloids. Then

[TODO: The

same for funcoids?]

f

\

RLD

[

S

=

[

h

f

\

RLD

i

S:

Theorem 37.

Composition with a (co?)complete reloid is an adjoint:

Proof.

F

is a lower adjoint. Let

is its upper adjoint. Then

F

x

y

,

x

(

y

)

x

(

F

x

)

,

F

(

y

)

(

b

) =

max

f

x

2

RLD

j

F

x

b

g

(a proof circle follows this)

x

F

¡

1

b

)

F

x

F

b

F

S

RLD

R

=

T

RLD

f

F

K

j

K

2

up

S

RLD

R

g

S

RLD

h

F

 i

R

=

S

RLD

f

T

RLD

f

F

G

j

G

2

up

g

g j

g

2

R

g

Conjecture 38.

Compl

f

\

RLD

Compl

g

=

Compl

(

f

\

RLD

g

)

for every reloids

f

and

g

.

Proof.

Compl

(

f

\

RLD

g

) =

S

RLD

(

f

\

RLD

g

)

j

f

g

RLD

j

2

f

 

Compl

f

\

RLD

Compl

g

=

S

RLD

f

j

f

g

RLD

j

2

f

 

\

RLD

S

RLD

g

j

f

g

RLD

j

2

f

 

(

Compl

(

f

\

RLD

g

))

j

f

g

RLD

=(

f

\

RLD

g

)

j

f

g

RLD

(

Compl

f

\

RLD

Compl

g

)

j

f

g

RLD

=(

f

\

RLD

g

)

j

f

g

RLD

.

So enough to prove that Compl

f

\

RLD

Compl

g

is complete.

Let

A

=

atoms

Compl

RLD

f

and

B

=

atoms

Compl

RLD

g

. Then ??

Obviously Compl

f

\

RLD

Compl

g

Compl

f

\

Compl

RLD

Compl

g

Suppose it exists

a

2

atoms

RLD

(

Compl

f

\

RLD

Compl

g

)

such that

a

2

/

toms

RLD

(

Compl

f

\

Compl

RLD

Compl

g

)

. Then ??

Conjecture 39.

If

f

and

g

are reloids, then

g

f

=

[

RLD

f

G

F

j

F

2

atoms

RLD

f ; G

2

atoms

RLD

g

g

:

Proof.

g

f

= ?? =

g

S

RLD

atoms

RLD

f

= ?? =

S

RLD

h

g

 i

atoms

RLD

f

S

RLD

f

G

F

j

F

2

atoms

RLD

f ; G

2

atoms

RLD

g

g

s

S

RLD

f

g

F

j

F

2

atoms

RLD

f

g

f

??

Theorem 40.

dom

(

RLD

)

in

f

=

dom

f

and

im

(

RLD

)

in

f

=

im

f

for every funcoid

f

.

Proof.

Let an atomic f.o.

a

dom

f

. Then exists atomic f.o.

b

im

f

such that

a

FCD

b

f

.

Consequently

a

RLD

b

(

RLD

)

in

f

) 8

K

2

up

(

RLD

)

in

f

:

a

RLD

b

K

) 8

K

2

up

(

RLD

)

in

f

:

a

dom

K

,

a

T

F

h

dom

i

up

(

RLD

)

in

f

,

a

dom

(

RLD

)

in

f

.

Let now an atomic f.o.

a

dom

(

RLD

)

in

f

. Then

8

K

2

up

(

RLD

)

in

f

:

a

dom

K

What is equivalent to

8

K

2

\

f

up

(

a

RLD

b

)

j

a; b

2

atoms

F

f

; a

FCD

b

f

g

:

a

dom

K

Let

K

2

up

f

. Then

K

a

FCD

b

for every

a; b

2

atoms

F

f

where

a

FCD

b

f

that is exist ??

K

2

up

(

a

RLD

b

)

for ??

??
from what follows??

[FIXME:

b

is depended on

K

]

that exist

b

im

f

such that

8

K

2

up

(

RLD

)

in

f

:

a

RLD

b

K

that is

8

K

2

up

(

RLD

)

in

f

:

K

2

up

(

a

RLD

b

)

and thus

(

RLD

)

in

f

a

RLD

b

and consequently dom

(

RLD

)

in

f

dom

(

a

RLD

b

) =

a

.

Thus

a

dom

f

,

a

dom

(

RLD

)

in

f

for each atomic f.o.

a

from what follows dom

(

RLD

)

in

f

=

dom

f

.

12

Section 6