:: ALI2 semantic presentation
definition
let
c
1
be non
empty
MetrSpace
;
mode
contraction
of c
1
->
Function
of a
1
,a
1
means
:
Def1
:
:: ALI2:def 1
ex b
1
being
Real
st
( 0
<
b
1
& b
1
<
1 & ( for b
2
, b
3
being
Point
of a
1
holds
dist
(
a
2
.
b
2
)
,
(
a
2
.
b
3
)
<=
b
1
*
(
dist
b
2
,b
3
)
) );
existence
ex b
1
being
Function
of c
1
,c
1
ex b
2
being
Real
st
( 0
<
b
2
& b
2
<
1 & ( for b
3
, b
4
being
Point
of c
1
holds
dist
(
b
1
.
b
3
)
,
(
b
1
.
b
4
)
<=
b
2
*
(
dist
b
3
,b
4
)
) )
proof
end;
end;
::
deftheorem
Def1
defines
contraction
ALI2:def 1 :
for b
1
being non
empty
MetrSpace
for b
2
being
Function
of b
1
,b
1
holds
( b
2
is
contraction
of b
1
iff ex b
3
being
Real
st
( 0
<
b
3
& b
3
<
1 & ( for b
4
, b
5
being
Point
of b
1
holds
dist
(
b
2
.
b
4
)
,
(
b
2
.
b
5
)
<=
b
3
*
(
dist
b
4
,b
5
)
) ) );
theorem
Th1
:
:: ALI2:1
canceled;
theorem
Th2
:
:: ALI2:2
for b
1
being non
empty
MetrSpace
for b
2
being
contraction
of b
1
holds
not (
TopSpaceMetr
b
1
is
compact
& ( for b
3
being
Point
of b
1
holds
not ( b
2
.
b
3
=
b
3
& ( for b
4
being
Point
of b
1
holds
( b
2
.
b
4
=
b
4
implies b
4
=
b
3
) ) ) ) )
proof
end;