background image

2.

g

f

1

|

(

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

)

\{

Z

B

f

A

)

}

is injective.

Really,

1. Let

y

Z

B

f

A

)

for some

y

dom

ϕ

A

. Then

f

1

y

ZC

A

because otherwise

fx

Z

B

f

A

)

for some

x

ZC

A

. Consequently

g f

1

y

Z

C

g

A

)

.

2.

f

1

|

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}\{

Z

B

f

A

)

}

is obviously injective.

g

|

h

f

1

i

((

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

)

\{

Z

B

f

A

)

}

)

is injective because

f

1

y

ZC

A

for

y

Z

B

f

A

)

.

Thus

g

f

1

|

(

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

)

\{

Z

B

f

A

)

}

is injective.

As shown by the below theorems, every two quasi-cartesian functions are equivalent up to a
bijection:

Theorem 19.

ϕ

A

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

=

g

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

.

Proof.

If

x

∈ {

x

X

A

dom

A

|

ρ

A

x

=

A

} \

ZC

A

then

[TODO: Make clear that multivalued functions

are not applied below. Rewrite the proof for clarity.]

ϕ

A

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x

=

g

id

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

f

1

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x

=

g

id

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x

=

gx

=

g

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x.

If

x

∈ {

x

X

A

dom

A

|

ρ

A

x

=

A

} ∩

ZC

A

then

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x

=

Z

B

Υ

f

A

and

g

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

x

=

Z

C

Υ

g

A

;

f

1

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

{

x

}

=

{

x

X

A

dom

A

|

ρ

A

x

=

A

} ∩

ZC

A

. Thus it is easy to show that

g

id

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

f

1

f

|

{

x

X

A

dom

A

|

ρ

A

x

=

A

}

{

x

}

=

{

Z

C

Υ

g

A

}

.

Now let also

f

and

g

be with injective aggregation.

Let

Φ =

g

f

1

.

Lemma 20.

The set of all

h

f

i{

x

X

0

dom

A

|

ρ

0

x

=

A

}

, for

A

being small indexed families of

forms, is a partition of the set

im

f

where

f

is a quasi-cartesian function with injective aggregation

S

1

S

2

.

Proof.

Let denote this set

S

. That

S

S

=

im

f

is obvious.

Suppose

A

=

h

f

i{

x

X

0

dom

A

|

ρ

0

x

=

A

0

}

and

B

=

h

f

i{

x

X

0

dom

A

|

ρ

0

x

=

A

1

}

for families

A

0

A

1

of forms. Then for every

a

A

we have

a

=

fx

where

ρ

0

x

=

A

0

. Thus

ρ

1

a

= Υ

A

0

and

ρ

1

b

= Υ

A

1

;

ρ

1

a

ρ

1

b

;

a

b

. So

S

is a disjoint set.

Theorem 21.

Φ

is a bijection

im

f

im

g

.

Proof.

From the lemma.

4