:: RCOMP_1 semantic presentation
scheme
:: RCOMP_1:sch 1
s1{ P
1
[
set
,
set
] } :
ex b
1
being
Function
of
NAT
,
REAL
st
for b
2
being
Nat
holds P
1
[b
2
,b
1
.
b
2
]
provided
E1
: for b
1
being
Nat
holds
ex b
2
being
real
number
st P
1
[b
1
,b
2
]
proof
end;
theorem
Th1
:
:: RCOMP_1:1
for b
1
, b
2
being
Subset
of
REAL
holds
( ( for b
3
being
real
number
holds
( b
3
in
b
1
implies b
3
in
b
2
) ) implies b
1
c=
b
2
)
proof
end;
theorem
Th2
:
:: RCOMP_1:2
canceled;
theorem
Th3
:
:: RCOMP_1:3
for b
1
, b
2
being
Subset
of
REAL
holds
( b
1
c=
b
2
& b
2
is
bounded_below
implies b
1
is
bounded_below
)
proof
end;
theorem
Th4
:
:: RCOMP_1:4
for b
1
, b
2
being
Subset
of
REAL
holds
( b
1
c=
b
2
& b
2
is
bounded_above
implies b
1
is
bounded_above
)
proof
end;
theorem
Th5
:
:: RCOMP_1:5
for b
1
, b
2
being
Subset
of
REAL
holds
( b
1
c=
b
2
& b
2
is
bounded
implies b
1
is
bounded
)
proof
end;
definition
let
c
1
, c
2
be
real
number
;
func
[.
c
1
,c
2
.]
->
Subset
of
REAL
equals
:: RCOMP_1:def 1
{
b
1
where B is
Real
: ( a
1
<=
b
1
& b
1
<=
a
2
)
}
;
coherence
{
b
1
where B is
Real
: ( c
1
<=
b
1
& b
1
<=
c
2
)
}
is
Subset
of
REAL
proof
end;
end;
::
deftheorem
Def1
defines
[.
RCOMP_1:def 1 :
for b
1
, b
2
being
real
number
holds
[.
b
1
,b
2
.]
=
{
b
3
where B is
Real
: ( b
1
<=
b
3
& b
3
<=
b
2
)
}
;
definition
let
c
1
, c
2
be
real
number
;
func
].
c
1
,c
2
.[
->
Subset
of
REAL
equals
:: RCOMP_1:def 2
{
b
1
where B is
Real
: ( a
1
<
b
1
& b
1
<
a
2
)
}
;
coherence
{
b
1
where B is
Real
: ( c
1
<
b
1
& b
1
<
c
2
)
}
is
Subset
of
REAL
proof
end;
end;
::
deftheorem
Def2
defines
].
RCOMP_1:def 2 :
for b
1
, b
2
being
real
number
holds
].
b
1
,b
2
.[
=
{
b
3
where B is
Real
: ( b
1
<
b
3
& b
3
<
b
2
)
}
;
theorem
Th6
:
:: RCOMP_1:6
canceled;
theorem
Th7
:
:: RCOMP_1:7
canceled;
theorem
Th8
:
:: RCOMP_1:8
for b
1
, b
2
, b
3
being
real
number
holds
( b
1
in
].
(
b
2
-
b
3
)
,
(
b
2
+
b
3
)
.[
iff
abs
(
b
1
-
b
2
)
<
b
3
)
proof
end;
theorem
Th9
:
:: RCOMP_1:9
for b
1
, b
2
, b
3
being
real
number
holds
( b
1
in
[.
b
2
,b
3
.]
iff
abs
(
(
b
2
+
b
3
)
-
(
2
*
b
1
)
)
<=
b
3
-
b
2
)
proof
end;
theorem
Th10
:
:: RCOMP_1:10
for b
1
, b
2
, b
3
being
real
number
holds
( b
1
in
].
b
2
,b
3
.[
iff
abs
(
(
b
2
+
b
3
)
-
(
2
*
b
1
)
)
<
b
3
-
b
2
)
proof
end;
theorem
Th11
:
:: RCOMP_1:11
for b
1
, b
2
being
real
number
holds
( b
1
<=
b
2
implies
[.
b
1
,b
2
.]
=
].
b
1
,b
2
.[
\/
{
b
1
,b
2
}
)
proof
end;
theorem
Th12
:
:: RCOMP_1:12
for b
1
, b
2
being
real
number
holds
( b
1
<=
b
2
implies
].
b
2
,b
1
.[
=
{}
)
proof
end;
theorem
Th13
:
:: RCOMP_1:13
for b
1
, b
2
being
real
number
holds
( b
1
<
b
2
implies
[.
b
2
,b
1
.]
=
{}
)
proof
end;
theorem
Th14
:
:: RCOMP_1:14
for b
1
being
real
number
holds
[.
b
1
,b
1
.]
=
{
b
1
}
proof
end;
theorem
Th15
:
:: RCOMP_1:15
for b
1
, b
2
being
real
number
holds
( not ( b
1
<
b
2
& not
].
b
1
,b
2
.[
<>
{}
) & ( b
1
<=
b
2
implies ( b
1
in
[.
b
1
,b
2
.]
& b
2
in
[.
b
1
,b
2
.]
) ) &
].
b
1
,b
2
.[
c=
[.
b
1
,b
2
.]
)
proof
end;
theorem
Th16
:
:: RCOMP_1:16
for b
1
, b
2
, b
3
, b
4
being
real
number
holds
( b
1
in
[.
b
2
,b
3
.]
& b
4
in
[.
b
2
,b
3
.]
implies
[.
b
1
,b
4
.]
c=
[.
b
2
,b
3
.]
)
proof
end;
theorem
Th17
:
:: RCOMP_1:17
for b
1
, b
2
, b
3
, b
4
being
real
number
holds
( b
1
in
].
b
2
,b
3
.[
& b
4
in
].
b
2
,b
3
.[
implies
[.
b
1
,b
4
.]
c=
].
b
2
,b
3
.[
)
proof
end;
theorem
Th18
:
:: RCOMP_1:18
for b
1
, b
2
being
real
number
holds
( b
1
<=
b
2
implies
[.
b
1
,b
2
.]
=
[.
b
1
,b
2
.]
\/
[.
b
2
,b
1
.]
)
proof
end;
definition
let
c
1
be
Subset
of
REAL
;
attr
a
1
is
compact
means
:
Def3
:
:: RCOMP_1:def 3
for b
1
being
Real_Sequence
holds
not (
rng
b
1
c=
a
1
& ( for b
2
being
Real_Sequence
holds
not ( b
2
is
subsequence
of b
1
& b
2
is
convergent
&
lim
b
2
in
a
1
) ) );
end;
::
deftheorem
Def3
defines
compact
RCOMP_1:def 3 :
for b
1
being
Subset
of
REAL
holds
( b
1
is
compact
iff for b
2
being
Real_Sequence
holds
not (
rng
b
2
c=
b
1
& ( for b
3
being
Real_Sequence
holds
not ( b
3
is
subsequence
of b
2
& b
3
is
convergent
&
lim
b
3
in
b
1
) ) ) );
definition
let
c
1
be
Subset
of
REAL
;
attr
a
1
is
closed
means
:
Def4
:
:: RCOMP_1:def 4
for b
1
being
Real_Sequence
holds
(
rng
b
1
c=
a
1
& b
1
is
convergent
implies
lim
b
1
in
a
1
);
end;
::
deftheorem
Def4
defines
closed
RCOMP_1:def 4 :
for b
1
being
Subset
of
REAL
holds
( b
1
is
closed
iff for b
2
being
Real_Sequence
holds
(
rng
b
2
c=
b
1
& b
2
is
convergent
implies
lim
b
2
in
b
1
) );
definition
let
c
1
be
Subset
of
REAL
;
attr
a
1
is
open
means
:
Def5
:
:: RCOMP_1:def 5
a
1
`
is
closed
;
end;
::
deftheorem
Def5
defines
open
RCOMP_1:def 5 :
for b
1
being
Subset
of
REAL
holds
( b
1
is
open
iff b
1
`
is
closed
);
theorem
Th19
:
:: RCOMP_1:19
canceled;
theorem
Th20
:
:: RCOMP_1:20
canceled;
theorem
Th21
:
:: RCOMP_1:21
canceled;
theorem
Th22
:
:: RCOMP_1:22
for b
1
, b
2
being
real
number
for b
3
being
Real_Sequence
holds
(
rng
b
3
c=
[.
b
1
,b
2
.]
implies b
3
is
bounded
)
proof
end;
theorem
Th23
:
:: RCOMP_1:23
for b
1
, b
2
being
real
number
holds
[.
b
1
,b
2
.]
is
closed
proof
end;
theorem
Th24
:
:: RCOMP_1:24
for b
1
, b
2
being
real
number
holds
[.
b
1
,b
2
.]
is
compact
proof
end;
theorem
Th25
:
:: RCOMP_1:25
for b
1
, b
2
being
real
number
holds
].
b
1
,b
2
.[
is
open
proof
end;
registration
let
c
1
, c
2
be
real
number
;
cluster
].
a
1
,a
2
.[
->
open
;
coherence
].
c
1
,c
2
.[
is
open
by
Th25
;
cluster
[.
a
1
,a
2
.]
->
closed
;
coherence
[.
c
1
,c
2
.]
is
closed
by
Th23
;
end;
theorem
Th26
:
:: RCOMP_1:26
for b
1
being
Subset
of
REAL
holds
( b
1
is
compact
implies b
1
is
closed
)
proof
end;
theorem
Th27
:
:: RCOMP_1:27
for b
1
being
Real_Sequence
for b
2
being
Subset
of
REAL
holds
( ( for b
3
being
real
number
holds
not ( b
3
in
b
2
& ( for b
4
being
real
number
for b
5
being
Nat
holds
not ( 0
<
b
4
& ( for b
6
being
Nat
holds
not ( b
5
<
b
6
& not b
4
<
abs
(
(
b
1
.
b
6
)
-
b
3
)
) ) ) ) ) ) implies for b
3
being
Real_Sequence
holds
not ( b
3
is
subsequence
of b
1
& b
3
is
convergent
&
lim
b
3
in
b
2
) )
proof
end;
theorem
Th28
:
:: RCOMP_1:28
for b
1
being
Subset
of
REAL
holds
( b
1
is
compact
implies b
1
is
bounded
)
proof
end;
theorem
Th29
:
:: RCOMP_1:29
for b
1
being
Subset
of
REAL
holds
( b
1
is
bounded
& b
1
is
closed
implies b
1
is
compact
)
proof
end;
theorem
Th30
:
:: RCOMP_1:30
for b
1
being
Subset
of
REAL
holds
( b
1
<>
{}
& b
1
is
closed
& b
1
is
bounded_above
implies
upper_bound
b
1
in
b
1
)
proof
end;
theorem
Th31
:
:: RCOMP_1:31
for b
1
being
Subset
of
REAL
holds
( b
1
<>
{}
& b
1
is
closed
& b
1
is
bounded_below
implies
lower_bound
b
1
in
b
1
)
proof
end;
theorem
Th32
:
:: RCOMP_1:32
for b
1
being
Subset
of
REAL
holds
( b
1
<>
{}
& b
1
is
compact
implies (
upper_bound
b
1
in
b
1
&
lower_bound
b
1
in
b
1
) )
proof
end;
theorem
Th33
:
:: RCOMP_1:33
for b
1
being
Subset
of
REAL
holds
not ( b
1
is
compact
& ( for b
2
, b
3
being
real
number
holds
( b
2
in
b
1
& b
3
in
b
1
implies
[.
b
2
,b
3
.]
c=
b
1
) ) & ( for b
2
, b
3
being
real
number
holds
not b
1
=
[.
b
2
,b
3
.]
) )
proof
end;
registration
cluster
open
Element
of
K18
(
REAL
);
existence
ex b
1
being
Subset
of
REAL
st b
1
is
open
proof
end;
end;
definition
let
c
1
be
real
number
;
canceled;
mode
Neighbourhood
of c
1
->
Subset
of
REAL
means
:
Def7
:
:: RCOMP_1:def 7
ex b
1
being
real
number
st
( 0
<
b
1
& a
2
=
].
(
a
1
-
b
1
)
,
(
a
1
+
b
1
)
.[
);
existence
ex b
1
being
Subset
of
REAL
ex b
2
being
real
number
st
( 0
<
b
2
& b
1
=
].
(
c
1
-
b
2
)
,
(
c
1
+
b
2
)
.[
)
proof
end;
end;
::
deftheorem
Def6
RCOMP_1:def 6 :
canceled;
::
deftheorem
Def7
defines
Neighbourhood
RCOMP_1:def 7 :
for b
1
being
real
number
for b
2
being
Subset
of
REAL
holds
( b
2
is
Neighbourhood
of b
1
iff ex b
3
being
real
number
st
( 0
<
b
3
& b
2
=
].
(
b
1
-
b
3
)
,
(
b
1
+
b
3
)
.[
) );
registration
let
c
1
be
real
number
;
cluster
->
open
Neighbourhood
of a
1
;
coherence
for b
1
being
Neighbourhood
of c
1
holds b
1
is
open
proof
end;
end;
theorem
Th34
:
:: RCOMP_1:34
canceled;
theorem
Th35
:
:: RCOMP_1:35
canceled;
theorem
Th36
:
:: RCOMP_1:36
canceled;
theorem
Th37
:
:: RCOMP_1:37
for b
1
being
real
number
for b
2
being
Neighbourhood
of b
1
holds b
1
in
b
2
proof
end;
theorem
Th38
:
:: RCOMP_1:38
for b
1
being
real
number
for b
2
, b
3
being
Neighbourhood
of b
1
holds
ex b
4
being
Neighbourhood
of b
1
st
( b
4
c=
b
2
& b
4
c=
b
3
)
proof
end;
theorem
Th39
:
:: RCOMP_1:39
for b
1
being
open
Subset
of
REAL
for b
2
being
real
number
holds
not ( b
2
in
b
1
& ( for b
3
being
Neighbourhood
of b
2
holds
not b
3
c=
b
1
) )
proof
end;
theorem
Th40
:
:: RCOMP_1:40
for b
1
being
open
Subset
of
REAL
for b
2
being
real
number
holds
not ( b
2
in
b
1
& ( for b
3
being
real
number
holds
not ( 0
<
b
3
&
].
(
b
2
-
b
3
)
,
(
b
2
+
b
3
)
.[
c=
b
1
) ) )
proof
end;
theorem
Th41
:
:: RCOMP_1:41
for b
1
being
Subset
of
REAL
holds
( ( for b
2
being
real
number
holds
not ( b
2
in
b
1
& ( for b
3
being
Neighbourhood
of b
2
holds
not b
3
c=
b
1
) ) ) implies b
1
is
open
)
proof
end;
theorem
Th42
:
:: RCOMP_1:42
for b
1
being
Subset
of
REAL
holds
( ( for b
2
being
real
number
holds
not ( b
2
in
b
1
& ( for b
3
being
Neighbourhood
of b
2
holds
not b
3
c=
b
1
) ) ) iff b
1
is
open
)
by
Th39
,
Th41
;
theorem
Th43
:
:: RCOMP_1:43
for b
1
being
Subset
of
REAL
holds
not ( b
1
is
open
& b
1
is
bounded_above
&
upper_bound
b
1
in
b
1
)
proof
end;
theorem
Th44
:
:: RCOMP_1:44
for b
1
being
Subset
of
REAL
holds
not ( b
1
is
open
& b
1
is
bounded_below
&
lower_bound
b
1
in
b
1
)
proof
end;
theorem
Th45
:
:: RCOMP_1:45
for b
1
being
Subset
of
REAL
holds
not ( b
1
is
open
& b
1
is
bounded
& ( for b
2
, b
3
being
real
number
holds
( b
2
in
b
1
& b
3
in
b
1
implies
[.
b
2
,b
3
.]
c=
b
1
) ) & ( for b
2
, b
3
being
real
number
holds
not b
1
=
].
b
2
,b
3
.[
) )
proof
end;
theorem
Th46
:
:: RCOMP_1:46
for b
1
, b
2
being
real
number
holds
].
b
1
,b
2
.[
misses
{
b
1
,b
2
}
proof
end;
theorem
Th47
:
:: RCOMP_1:47
for b
1
, b
2
, b
3
being
real
number
holds
( b
3
in
].
b
1
,b
2
.[
iff ( b
1
<
b
3
& b
3
<
b
2
) )
proof
end;
theorem
Th48
:
:: RCOMP_1:48
for b
1
, b
2
, b
3
being
real
number
holds
( b
3
in
[.
b
1
,b
2
.]
iff ( b
1
<=
b
3
& b
3
<=
b
2
) )
proof
end;