dblseq_3.miz
begin
Lm1: for X be
set, x be
object holds ((X
-->
0. )
. x)
=
0
proof
let X be
set, x be
object;
set M = (X
-->
0. );
per cases ;
suppose x
in (
dom M);
then x
in X;
hence ((X
-->
0. )
. x)
=
0 by
FUNCOP_1: 7;
end;
suppose not x
in (
dom M);
then (M
. x)
=
0 by
FUNCT_1:def 2;
hence ((X
-->
0. )
. x)
=
0 ;
end;
end;
registration
let X be non
empty
set;
cluster
nonnegative
nonpositive for
Function of X,
REAL ;
existence
proof
reconsider M = (X
-->
0 ) as
Function of X,
REAL by
FUNCOP_1: 45,
XREAL_0:def 1;
take M;
A1: M is
Function of X,
ExtREAL by
FUNCOP_1: 45,
XXREAL_0:def 1;
for x be
object holds
0
<= (M
. x);
hence M is
nonnegative by
A1,
SUPINF_2: 51;
for x be
object holds (M
. x)
<=
0 by
Lm1;
hence M is
nonpositive by
A1,
MESFUNC5: 8;
end;
end
registration
let X be non
empty
set;
cluster
without-infty
without+infty
nonnegative
nonpositive for
Function of X,
ExtREAL ;
existence
proof
reconsider M = (X
-->
0. ) as
Function of X,
ExtREAL ;
take M;
for x be
object holds
-infty
< (M
. x);
hence M is
without-infty by
MESFUNC5:def 5;
for x be
object holds (M
. x)
<
+infty by
Lm1;
hence M is
without+infty by
MESFUNC5:def 6;
for x be
object holds
0
<= (M
. x);
hence M is
nonnegative by
SUPINF_2: 51;
for x be
object holds (M
. x)
<=
0 by
Lm1;
hence M is
nonpositive by
MESFUNC5: 8;
end;
end
Lm2: for X be non
empty
set holds (X
-->
0. ) is
without-infty
Function of X,
ExtREAL & (X
-->
0. ) is
without+infty
Function of X,
ExtREAL
proof
let X be non
empty
set;
reconsider P = (X
-->
0. ) as
Function of X,
ExtREAL ;
for x be
object holds
-infty
< (P
. x);
hence (X
-->
0. ) is
without-infty
Function of X,
ExtREAL by
MESFUNC5:def 5;
reconsider M = (X
-->
0. ) as
Function of X,
ExtREAL ;
for x be
object holds (M
. x)
<
+infty by
Lm1;
hence (X
-->
0. ) is
without+infty
Function of X,
ExtREAL by
MESFUNC5:def 6;
end;
registration
let X be non
empty
set;
cluster
nonnegative ->
without-infty for
Function of X,
ExtREAL ;
correctness
proof
now
let f be
nonnegative
Function of X,
ExtREAL ;
for x be
object holds (f
. x)
>
-infty by
SUPINF_2: 51;
hence f is
without-infty by
MESFUNC5:def 5;
end;
hence thesis;
end;
cluster
nonpositive ->
without+infty for
Function of X,
ExtREAL ;
correctness
proof
now
let f be
nonpositive
Function of X,
ExtREAL ;
for x be
object holds (f
. x)
<
+infty by
MESFUNC5: 8;
hence f is
without+infty by
MESFUNC5:def 6;
end;
hence thesis;
end;
cluster
without-infty for
without+infty
Function of X,
ExtREAL ;
existence
proof
reconsider f = (X
-->
0. ) as
without+infty
Function of X,
ExtREAL by
Lm2;
take f;
thus f is
without-infty by
Lm2;
end;
end
definition
let X be non
empty
set, f be
Function of X,
ExtREAL ;
:: original:
-
redefine
func
- f ->
Function of X,
ExtREAL ;
correctness
proof
(
dom (
- f))
= (
dom f) by
MESFUNC1:def 7;
then
A1: (
dom (
- f))
= X by
FUNCT_2:def 1;
now
let x be
object;
assume x
in X;
then
reconsider x1 = x as
Element of X;
((
- f)
. x)
= (
- (f
. x1)) by
A1,
MESFUNC1:def 7;
hence ((
- f)
. x)
in
ExtREAL ;
end;
hence (
- f) is
Function of X,
ExtREAL by
A1,
FUNCT_2: 3;
end;
end
registration
let X be non
empty
set, f be
without-infty
Function of X,
ExtREAL ;
cluster (
- f) ->
without+infty;
correctness
proof
now
let x be
object;
per cases ;
suppose
A1: x
in (
dom (
- f));
then
reconsider x1 = x as
Element of X;
((
- f)
. x)
= (
- (f
. x1)) by
A1,
MESFUNC1:def 7;
hence ((
- f)
. x)
<
+infty by
XXREAL_3: 5,
XXREAL_3: 38,
MESFUNC5:def 5;
end;
suppose not x
in (
dom (
- f));
hence ((
- f)
. x)
<
+infty by
FUNCT_1:def 2;
end;
end;
hence (
- f) is
without+infty by
MESFUNC5:def 6;
end;
end
registration
let X be non
empty
set, f be
without+infty
Function of X,
ExtREAL ;
cluster (
- f) ->
without-infty;
correctness
proof
now
let x be
object;
per cases ;
suppose
A1: x
in (
dom (
- f));
then
reconsider x1 = x as
Element of X;
((
- f)
. x)
= (
- (f
. x1)) by
A1,
MESFUNC1:def 7;
hence ((
- f)
. x)
>
-infty by
XXREAL_3: 6,
XXREAL_3: 38,
MESFUNC5:def 6;
end;
suppose not x
in (
dom (
- f));
hence ((
- f)
. x)
>
-infty by
FUNCT_1:def 2;
end;
end;
hence (
- f) is
without-infty by
MESFUNC5:def 5;
end;
end
registration
let X be non
empty
set, f be
nonnegative
Function of X,
ExtREAL ;
cluster (
- f) ->
nonpositive;
correctness
proof
now
let x be
object;
per cases ;
suppose
A1: x
in (
dom (
- f));
then
reconsider x1 = x as
Element of X;
A2: ((
- f)
. x)
= (
- (f
. x1)) by
A1,
MESFUNC1:def 7;
(f
. x1)
>=
0 by
SUPINF_2: 51;
hence ((
- f)
. x)
<=
0 by
A2;
end;
suppose not x
in (
dom (
- f));
hence ((
- f)
. x)
<=
0 by
FUNCT_1:def 2;
end;
end;
hence (
- f) is
nonpositive by
MESFUNC5: 8;
end;
end
registration
let X be non
empty
set, f be
nonpositive
Function of X,
ExtREAL ;
cluster (
- f) ->
nonnegative;
correctness
proof
now
let x be
object;
per cases ;
suppose
A1: x
in (
dom (
- f));
then
reconsider x1 = x as
Element of X;
A2: ((
- f)
. x)
= (
- (f
. x1)) by
A1,
MESFUNC1:def 7;
(f
. x1)
<=
0 by
MESFUNC5: 8;
hence ((
- f)
. x)
>=
0 by
A2;
end;
suppose not x
in (
dom (
- f));
hence ((
- f)
. x)
>=
0 by
FUNCT_1:def 2;
end;
end;
hence (
- f) is
nonnegative by
SUPINF_2: 51;
end;
end
registration
let A,B be non
empty
set;
let f be
without-infty
Function of
[:A, B:],
ExtREAL ;
cluster (
~ f) ->
without-infty;
correctness
proof
now
let x be
object;
per cases ;
suppose x
in (
dom (
~ f));
then
consider b,a be
object such that
A1: b
in B & a
in A & x
=
[b, a] by
ZFMISC_1:def 2;
reconsider a as
Element of A by
A1;
reconsider b as
Element of B by
A1;
(f
. (a,b))
>
-infty by
MESFUNC5:def 5;
then ((
~ f)
. (b,a))
>
-infty by
FUNCT_4:def 8;
hence ((
~ f)
. x)
>
-infty by
A1;
end;
suppose not x
in (
dom (
~ f));
hence ((
~ f)
. x)
>
-infty by
FUNCT_1:def 2;
end;
end;
hence thesis by
MESFUNC5:def 5;
end;
end
registration
let A,B be non
empty
set;
let f be
without+infty
Function of
[:A, B:],
ExtREAL ;
cluster (
~ f) ->
without+infty;
correctness
proof
now
let x be
object;
per cases ;
suppose x
in (
dom (
~ f));
then
consider b,a be
object such that
A1: b
in B & a
in A & x
=
[b, a] by
ZFMISC_1:def 2;
reconsider a as
Element of A by
A1;
reconsider b as
Element of B by
A1;
(f
. (a,b))
<
+infty by
MESFUNC5:def 6;
then ((
~ f)
. (b,a))
<
+infty by
FUNCT_4:def 8;
hence ((
~ f)
. x)
<
+infty by
A1;
end;
suppose not x
in (
dom (
~ f));
hence ((
~ f)
. x)
<
+infty by
FUNCT_1:def 2;
end;
end;
hence thesis by
MESFUNC5:def 6;
end;
end
registration
let A,B be non
empty
set;
let f be
nonnegative
Function of
[:A, B:],
ExtREAL ;
cluster (
~ f) ->
nonnegative;
correctness
proof
now
let x be
object;
assume x
in (
dom (
~ f));
then
consider b,a be
object such that
A1: b
in B & a
in A & x
=
[b, a] by
ZFMISC_1:def 2;
reconsider a as
Element of A by
A1;
reconsider b as
Element of B by
A1;
(f
. (a,b))
>=
0 by
SUPINF_2: 51;
then ((
~ f)
. (b,a))
>=
0 by
FUNCT_4:def 8;
hence ((
~ f)
. x)
>=
0 by
A1;
end;
hence thesis by
SUPINF_2: 52;
end;
end
registration
let A,B be non
empty
set;
let f be
nonpositive
Function of
[:A, B:],
ExtREAL ;
cluster (
~ f) ->
nonpositive;
correctness
proof
now
let x be
object;
per cases ;
suppose not x
in (
dom (
~ f));
hence ((
~ f)
. x)
<=
0. by
FUNCT_1:def 2;
end;
suppose x
in (
dom (
~ f));
then
consider b,a be
object such that
A1: b
in B & a
in A & x
=
[b, a] by
ZFMISC_1:def 2;
reconsider a as
Element of A by
A1;
reconsider b as
Element of B by
A1;
(f
. (a,b))
<=
0 by
MESFUNC5: 8;
then ((
~ f)
. (b,a))
<=
0 by
FUNCT_4:def 8;
hence ((
~ f)
. x)
<=
0. by
A1;
end;
end;
hence (
~ f) is
nonpositive by
MESFUNC5: 8;
end;
end
theorem ::
DBLSEQ_3:1
Th1: for seq be
ExtREAL_sequence holds (
Partial_Sums (
- seq))
= (
- (
Partial_Sums seq))
proof
let seq be
ExtREAL_sequence;
A1: (
dom (
- seq))
=
NAT & (
dom (
- (
Partial_Sums seq)))
=
NAT by
FUNCT_2:def 1;
defpred
Q[
Nat] means ((
- (
Partial_Sums seq))
. $1)
= (
- ((
Partial_Sums seq)
. $1));
A3:
Q[
0 ] by
A1,
MESFUNC1:def 7;
A4: for n be
Nat st
Q[n] holds
Q[(n
+ 1)] by
A1,
MESFUNC1:def 7;
A5: for n be
Nat holds
Q[n] from
NAT_1:sch 2(
A3,
A4);
defpred
P[
Nat] means ((
Partial_Sums (
- seq))
. $1)
= ((
- (
Partial_Sums seq))
. $1);
((
Partial_Sums (
- seq))
.
0 )
= ((
- seq)
.
0 ) by
MESFUNC9:def 1;
then
A6: ((
Partial_Sums (
- seq))
.
0 )
= (
- (seq
.
0 )) by
A1,
MESFUNC1:def 7;
((
Partial_Sums seq)
.
0 )
= (seq
.
0 ) by
MESFUNC9:def 1;
then
A7:
P[
0 ] by
A1,
A6,
MESFUNC1:def 7;
A8: for n be
Nat st
P[n] holds
P[(n
+ 1)]
proof
let n be
Nat;
assume
A9:
P[n];
((
Partial_Sums (
- seq))
. (n
+ 1))
= (((
- (
Partial_Sums seq))
. n)
+ ((
- seq)
. (n
+ 1))) by
A9,
MESFUNC9:def 1
.= (((
- (
Partial_Sums seq))
. n)
+ (
- (seq
. (n
+ 1)))) by
A1,
MESFUNC1:def 7
.= ((
- ((
Partial_Sums seq)
. n))
- (seq
. (n
+ 1))) by
A5
.= (
- (((
Partial_Sums seq)
. n)
+ (seq
. (n
+ 1)))) by
XXREAL_3: 25
.= (
- ((
Partial_Sums seq)
. (n
+ 1))) by
MESFUNC9:def 1;
hence
P[(n
+ 1)] by
A1,
MESFUNC1:def 7;
end;
for n be
Nat holds
P[n] from
NAT_1:sch 2(
A7,
A8);
then for n be
Element of
NAT holds ((
Partial_Sums (
- seq))
. n)
= ((
- (
Partial_Sums seq))
. n);
hence thesis by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:2
Th2: for X be non
empty
set, f be
PartFunc of X,
ExtREAL holds (
- (
- f))
= f
proof
let X be non
empty
set, f be
PartFunc of X,
ExtREAL ;
A1: (
dom f)
= (
dom (
- f)) by
MESFUNC1:def 7;
then
A2: (
dom f)
= (
dom (
- (
- f))) by
MESFUNC1:def 7;
now
let x be
object;
assume
A3: x
in (
dom f);
then ((
- f)
. x)
= (
- (f
. x)) by
A1,
MESFUNC1:def 7;
then ((
- (
- f))
. x)
= (
- (
- (f
. x))) by
A2,
A3,
MESFUNC1:def 7;
hence ((
- (
- f))
. x)
= (f
. x);
end;
hence thesis by
A2,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:3
for X,Y be non
empty
set, f be
Function of
[:X, Y:],
ExtREAL holds (
~ (
- f))
= (
- (
~ f))
proof
let X,Y be non
empty
set, f be
Function of
[:X, Y:],
ExtREAL ;
now
let z be
Element of
[:Y, X:];
consider y,x be
object such that
A1: y
in Y & x
in X & z
=
[y, x] by
ZFMISC_1:def 2;
A2: (
dom (
- f))
=
[:X, Y:] & (
dom (
- (
~ f)))
=
[:Y, X:] by
FUNCT_2:def 1;
reconsider y as
Element of Y by
A1;
reconsider x as
Element of X by
A1;
reconsider z1 =
[x, y] as
Element of
[:X, Y:] by
ZFMISC_1: 87;
((
~ (
- f))
. z)
= ((
~ (
- f))
. (y,x)) by
A1;
then ((
~ (
- f))
. z)
= ((
- f)
. (x,y)) by
FUNCT_4:def 8;
then ((
~ (
- f))
. z)
= (
- (f
. z1)) by
A2,
MESFUNC1:def 7;
then ((
~ (
- f))
. z)
= (
- (f
. (x,y)));
then ((
~ (
- f))
. z)
= (
- ((
~ f)
. (y,x))) by
FUNCT_4:def 8;
hence ((
~ (
- f))
. z)
= ((
- (
~ f))
. z) by
A1,
A2,
MESFUNC1:def 7;
end;
hence thesis by
FUNCT_2:def 8;
end;
registration
let seq be
nonnegative
ExtREAL_sequence;
cluster (
Partial_Sums seq) ->
nonnegative;
correctness by
MESFUNC9: 16;
end
registration
let seq be
nonpositive
ExtREAL_sequence;
cluster (
Partial_Sums seq) ->
nonpositive;
correctness
proof
set f = (
- seq);
(
Partial_Sums f) is
nonnegative;
then (
- (
Partial_Sums seq)) is
nonnegative by
Th1;
then (
- (
- (
Partial_Sums seq))) is
nonpositive;
hence (
Partial_Sums seq) is
nonpositive by
Th2;
end;
end
theorem ::
DBLSEQ_3:4
Th4: for seq be
nonnegative
ExtREAL_sequence, m be
Nat holds (seq
. m)
<= ((
Partial_Sums seq)
. m)
proof
let seq be
nonnegative
ExtREAL_sequence, m be
Nat;
defpred
P[
Nat] means (seq
. $1)
<= ((
Partial_Sums seq)
. $1);
A1:
P[
0 ] by
MESFUNC9:def 1;
A2: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
P[k];
((
Partial_Sums seq)
. (k
+ 1))
= (((
Partial_Sums seq)
. k)
+ (seq
. (k
+ 1))) by
MESFUNC9:def 1;
hence
P[(k
+ 1)] by
XXREAL_3: 39,
SUPINF_2: 51;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A1,
A2);
hence (seq
. m)
<= ((
Partial_Sums seq)
. m);
end;
theorem ::
DBLSEQ_3:5
for seq be
nonpositive
ExtREAL_sequence, m be
Nat holds (seq
. m)
>= ((
Partial_Sums seq)
. m)
proof
let seq be
nonpositive
ExtREAL_sequence, m be
Nat;
reconsider f = (
- seq) as
nonnegative
ExtREAL_sequence;
A1: (
dom f)
=
NAT & (
dom seq)
=
NAT & (
dom (
- f))
=
NAT & (
dom (
Partial_Sums seq))
=
NAT & (
dom (
- (
Partial_Sums seq)))
=
NAT & (
dom (
- (
- (
Partial_Sums seq))))
=
NAT by
FUNCT_2:def 1;
A2: m
in
NAT by
ORDINAL1:def 12;
(f
. m)
<= ((
Partial_Sums f)
. m) by
Th4;
then (f
. m)
<= ((
- (
Partial_Sums seq))
. m) by
Th1;
then (
- ((
- seq)
. m))
>= (
- ((
- (
Partial_Sums seq))
. m)) by
XXREAL_3: 38;
then ((
- (
- seq))
. m)
>= (
- ((
- (
Partial_Sums seq))
. m)) by
A1,
A2,
MESFUNC1:def 7;
then (seq
. m)
>= (
- ((
- (
Partial_Sums seq))
. m)) by
Th2;
then (seq
. m)
>= (
- (
- ((
Partial_Sums seq)
. m))) by
A1,
A2,
MESFUNC1:def 7;
hence thesis;
end;
theorem ::
DBLSEQ_3:6
for X be non
empty
set, f be
without-infty
without+infty
Function of X,
ExtREAL holds f is
Function of X,
REAL
proof
let X be non
empty
set, f be
without-infty
without+infty
Function of X,
ExtREAL ;
A1: (
dom f)
= X by
FUNCT_2:def 1;
now
let x be
object;
assume x
in X;
then (f
. x)
in
ExtREAL & (f
. x)
>
-infty & (f
. x)
<
+infty by
FUNCT_2: 5,
MESFUNC5:def 5,
MESFUNC5:def 6;
hence (f
. x)
in
REAL by
XXREAL_0: 14;
end;
hence thesis by
A1,
FUNCT_2: 3;
end;
definition
let X be non
empty
set;
let f1,f2 be
without-infty
Function of X,
ExtREAL ;
:: original:
+
redefine
func f1
+ f2 ->
without-infty
Function of X,
ExtREAL ;
correctness
proof
A1: (
dom f1)
= X & (
dom f2)
= X by
FUNCT_2:def 1;
{
-infty }
misses (
rng f1) &
{
-infty }
misses (
rng f2) by
ZFMISC_1: 50,
MESFUNC5:def 3;
then (f1
"
{
-infty })
=
{} & (f2
"
{
-infty })
=
{} by
RELAT_1: 138;
then (((
dom f1)
/\ (
dom f2))
\ (((f1
"
{
-infty })
/\ (f2
"
{
+infty }))
\/ ((f1
"
{
+infty })
/\ (f2
"
{
-infty }))))
= X by
A1;
then
A2: (
dom (f1
+ f2))
= X by
MESFUNC1:def 3;
now
assume
-infty
in (
rng (f1
+ f2));
then
consider x be
object such that
A3: x
in (
dom (f1
+ f2)) & ((f1
+ f2)
. x)
=
-infty by
FUNCT_1:def 3;
reconsider x as
Element of X by
A3;
((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x)) by
A3,
MESFUNC1:def 3;
then (f1
. x)
=
-infty or (f2
. x)
=
-infty by
A3,
XXREAL_3: 17;
hence contradiction by
MESFUNC5:def 5;
end;
hence thesis by
A2,
FUNCT_2:def 1,
MESFUNC5:def 3;
end;
end
definition
let X be non
empty
set;
let f1,f2 be
without+infty
Function of X,
ExtREAL ;
:: original:
+
redefine
func f1
+ f2 ->
without+infty
Function of X,
ExtREAL ;
correctness
proof
A1: (
dom f1)
= X & (
dom f2)
= X by
FUNCT_2:def 1;
{
+infty }
misses (
rng f1) &
{
+infty }
misses (
rng f2) by
ZFMISC_1: 50,
MESFUNC5:def 4;
then (f1
"
{
+infty })
=
{} & (f2
"
{
+infty })
=
{} by
RELAT_1: 138;
then (((
dom f1)
/\ (
dom f2))
\ (((f1
"
{
-infty })
/\ (f2
"
{
+infty }))
\/ ((f1
"
{
+infty })
/\ (f2
"
{
-infty }))))
= X by
A1;
then
A2: (
dom (f1
+ f2))
= X by
MESFUNC1:def 3;
now
assume
+infty
in (
rng (f1
+ f2));
then
consider x be
object such that
A3: x
in (
dom (f1
+ f2)) & ((f1
+ f2)
. x)
=
+infty by
FUNCT_1:def 3;
reconsider x as
Element of X by
A3;
((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x)) by
A3,
MESFUNC1:def 3;
then (f1
. x)
=
+infty or (f2
. x)
=
+infty by
A3,
XXREAL_3: 16;
hence contradiction by
MESFUNC5:def 6;
end;
hence thesis by
A2,
FUNCT_2:def 1,
MESFUNC5:def 4;
end;
end
definition
let X be non
empty
set;
let f1 be
without-infty
Function of X,
ExtREAL ;
let f2 be
without+infty
Function of X,
ExtREAL ;
:: original:
-
redefine
func f1
- f2 ->
without-infty
Function of X,
ExtREAL ;
correctness
proof
A1: (
dom f1)
= X & (
dom f2)
= X by
FUNCT_2:def 1;
{
-infty }
misses (
rng f1) &
{
+infty }
misses (
rng f2) by
ZFMISC_1: 50,
MESFUNC5:def 3,
MESFUNC5:def 4;
then (f1
"
{
-infty })
=
{} & (f2
"
{
+infty })
=
{} by
RELAT_1: 138;
then (((
dom f1)
/\ (
dom f2))
\ (((f1
"
{
+infty })
/\ (f2
"
{
+infty }))
\/ ((f1
"
{
-infty })
/\ (f2
"
{
-infty }))))
= X by
A1;
then
A2: (
dom (f1
- f2))
= X by
MESFUNC1:def 4;
now
assume
-infty
in (
rng (f1
- f2));
then
consider x be
object such that
A3: x
in (
dom (f1
- f2)) & ((f1
- f2)
. x)
=
-infty by
FUNCT_1:def 3;
reconsider x as
Element of X by
A3;
((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x)) by
A3,
MESFUNC1:def 4;
then (f1
. x)
=
-infty or (f2
. x)
=
+infty by
A3,
XXREAL_3: 19;
hence contradiction by
MESFUNC5:def 5,
MESFUNC5:def 6;
end;
hence thesis by
A2,
FUNCT_2:def 1,
MESFUNC5:def 3;
end;
end
definition
let X be non
empty
set;
let f1 be
without+infty
Function of X,
ExtREAL ;
let f2 be
without-infty
Function of X,
ExtREAL ;
:: original:
-
redefine
func f1
- f2 ->
without+infty
Function of X,
ExtREAL ;
correctness
proof
A1: (
dom f1)
= X & (
dom f2)
= X by
FUNCT_2:def 1;
{
-infty }
misses (
rng f2) &
{
+infty }
misses (
rng f1) by
ZFMISC_1: 50,
MESFUNC5:def 3,
MESFUNC5:def 4;
then (f2
"
{
-infty })
=
{} & (f1
"
{
+infty })
=
{} by
RELAT_1: 138;
then (((
dom f1)
/\ (
dom f2))
\ (((f1
"
{
+infty })
/\ (f2
"
{
+infty }))
\/ ((f1
"
{
-infty })
/\ (f2
"
{
-infty }))))
= X by
A1;
then
A2: (
dom (f1
- f2))
= X by
MESFUNC1:def 4;
now
assume
+infty
in (
rng (f1
- f2));
then
consider x be
object such that
A3: x
in (
dom (f1
- f2)) & ((f1
- f2)
. x)
=
+infty by
FUNCT_1:def 3;
reconsider x as
Element of X by
A3;
((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x)) by
A3,
MESFUNC1:def 4;
then (f2
. x)
=
-infty or (f1
. x)
=
+infty by
A3,
XXREAL_3: 18;
hence contradiction by
MESFUNC5:def 5,
MESFUNC5:def 6;
end;
hence thesis by
A2,
FUNCT_2:def 1,
MESFUNC5:def 4;
end;
end
theorem ::
DBLSEQ_3:7
Th7: for X be non
empty
set, x be
Element of X, f1,f2 be
Function of X,
ExtREAL holds (f1 is
without-infty & f2 is
without-infty implies ((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x))) & (f1 is
without+infty & f2 is
without+infty implies ((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x))) & (f1 is
without-infty & f2 is
without+infty implies ((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x))) & (f1 is
without+infty & f2 is
without-infty implies ((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x)))
proof
let X be non
empty
set, x be
Element of X, f1,f2 be
Function of X,
ExtREAL ;
hereby
assume f1 is
without-infty & f2 is
without-infty;
then
reconsider F1 = f1, F2 = f2 as
without-infty
Function of X,
ExtREAL ;
(
dom (F1
+ F2))
= X by
FUNCT_2:def 1;
hence ((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x)) by
MESFUNC1:def 3;
end;
hereby
assume f1 is
without+infty & f2 is
without+infty;
then
reconsider F1 = f1, F2 = f2 as
without+infty
Function of X,
ExtREAL ;
(
dom (F1
+ F2))
= X by
FUNCT_2:def 1;
hence ((f1
+ f2)
. x)
= ((f1
. x)
+ (f2
. x)) by
MESFUNC1:def 3;
end;
hereby
assume
A1: f1 is
without-infty & f2 is
without+infty;
then
reconsider F1 = f1 as
without-infty
Function of X,
ExtREAL ;
reconsider F2 = f2 as
without+infty
Function of X,
ExtREAL by
A1;
(
dom (F1
- F2))
= X by
FUNCT_2:def 1;
hence ((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x)) by
MESFUNC1:def 4;
end;
hereby
assume
A1: f1 is
without+infty & f2 is
without-infty;
then
reconsider F1 = f1 as
without+infty
Function of X,
ExtREAL ;
reconsider F2 = f2 as
without-infty
Function of X,
ExtREAL by
A1;
(
dom (F1
- F2))
= X by
FUNCT_2:def 1;
hence ((f1
- f2)
. x)
= ((f1
. x)
- (f2
. x)) by
MESFUNC1:def 4;
end;
end;
Lm3: for X be non
empty
set, f1,f2 be
without-infty
Function of X,
ExtREAL holds (f1
+ f2)
= (f1
- (
- f2))
proof
let X be non
empty
set, f1,f2 be
without-infty
Function of X,
ExtREAL ;
now
let x be
Element of X;
(
dom (
- f2))
= X by
FUNCT_2:def 1;
then ((
- f2)
. x)
= (
- (f2
. x)) by
MESFUNC1:def 7;
then ((f1
- (
- f2))
. x)
= ((f1
. x)
- (
- (f2
. x))) by
Th7;
hence ((f1
+ f2)
. x)
= ((f1
- (
- f2))
. x) by
Th7;
end;
hence thesis by
FUNCT_2:def 8;
end;
Lm4: for X be non
empty
set, f1,f2 be
without+infty
Function of X,
ExtREAL holds (f1
+ f2)
= (f1
- (
- f2))
proof
let X be non
empty
set, f1,f2 be
without+infty
Function of X,
ExtREAL ;
now
let x be
Element of X;
(
dom (
- f2))
= X by
FUNCT_2:def 1;
then ((
- f2)
. x)
= (
- (f2
. x)) by
MESFUNC1:def 7;
then ((f1
- (
- f2))
. x)
= ((f1
. x)
- (
- (f2
. x))) by
Th7;
hence ((f1
+ f2)
. x)
= ((f1
- (
- f2))
. x) by
Th7;
end;
hence thesis by
FUNCT_2:def 8;
end;
Lm5: for X be non
empty
set, f1 be
without-infty
Function of X,
ExtREAL , f2 be
without+infty
Function of X,
ExtREAL holds (f1
- f2)
= (f1
+ (
- f2)) & (f2
- f1)
= (f2
+ (
- f1))
proof
let X be non
empty
set, f1 be
without-infty
Function of X,
ExtREAL , f2 be
without+infty
Function of X,
ExtREAL ;
now
let x be
Element of X;
(
dom (
- f2))
= X by
FUNCT_2:def 1;
then ((
- f2)
. x)
= (
- (f2
. x)) by
MESFUNC1:def 7;
then ((f1
+ (
- f2))
. x)
= ((f1
. x)
- (f2
. x)) by
Th7;
hence ((f1
- f2)
. x)
= ((f1
+ (
- f2))
. x) by
Th7;
end;
hence (f1
- f2)
= (f1
+ (
- f2)) by
FUNCT_2:def 8;
now
let x be
Element of X;
(
dom (
- f1))
= X by
FUNCT_2:def 1;
then ((
- f1)
. x)
= (
- (f1
. x)) by
MESFUNC1:def 7;
then ((f2
+ (
- f1))
. x)
= ((f2
. x)
- (f1
. x)) by
Th7;
hence ((f2
- f1)
. x)
= ((f2
+ (
- f1))
. x) by
Th7;
end;
hence (f2
- f1)
= (f2
+ (
- f1)) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:8
Th8: for X be non
empty
set, f1,f2 be
without-infty
Function of X,
ExtREAL holds (f1
+ f2)
= (f1
- (
- f2)) & (
- (f1
+ f2))
= ((
- f1)
- f2)
proof
let X be non
empty
set, f1,f2 be
without-infty
Function of X,
ExtREAL ;
thus (f1
+ f2)
= (f1
- (
- f2)) by
Lm3;
A1: (
dom (
- f1))
= X by
FUNCT_2:def 1;
A2: (
dom (
- f2))
= X by
FUNCT_2:def 1;
A3: (
dom (
- (f1
+ f2)))
= X by
FUNCT_2:def 1;
now
let x be
Element of X;
((
- (f1
+ f2))
. x)
= (
- ((f1
+ f2)
. x)) by
A3,
MESFUNC1:def 7
.= (
- ((f1
. x)
+ (f2
. x))) by
Th7
.= ((
- (f1
. x))
- (f2
. x)) by
XXREAL_3: 25
.= (((
- f1)
. x)
+ (
- (f2
. x))) by
A1,
MESFUNC1:def 7
.= (((
- f1)
. x)
+ ((
- f2)
. x)) by
A2,
MESFUNC1:def 7
.= (((
- f1)
+ (
- f2))
. x) by
Th7;
hence ((
- (f1
+ f2))
. x)
= (((
- f1)
- f2)
. x) by
Lm5;
end;
hence thesis by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:9
Th9: for X be non
empty
set, f1,f2 be
without+infty
Function of X,
ExtREAL holds (f1
+ f2)
= (f1
- (
- f2)) & (
- (f1
+ f2))
= ((
- f1)
- f2)
proof
let X be non
empty
set, f1,f2 be
without+infty
Function of X,
ExtREAL ;
thus (f1
+ f2)
= (f1
- (
- f2)) by
Lm4;
A1: (
dom (
- f1))
= X by
FUNCT_2:def 1;
A2: (
dom (
- f2))
= X by
FUNCT_2:def 1;
A3: (
dom (
- (f1
+ f2)))
= X by
FUNCT_2:def 1;
now
let x be
Element of X;
((
- (f1
+ f2))
. x)
= (
- ((f1
+ f2)
. x)) by
A3,
MESFUNC1:def 7
.= (
- ((f1
. x)
+ (f2
. x))) by
Th7
.= ((
- (f1
. x))
- (f2
. x)) by
XXREAL_3: 25
.= (((
- f1)
. x)
+ (
- (f2
. x))) by
A1,
MESFUNC1:def 7
.= (((
- f1)
. x)
+ ((
- f2)
. x)) by
A2,
MESFUNC1:def 7
.= (((
- f1)
+ (
- f2))
. x) by
Th7;
hence ((
- (f1
+ f2))
. x)
= (((
- f1)
- f2)
. x) by
Lm5;
end;
hence thesis by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:10
Th10: for X be non
empty
set, f1 be
without-infty
Function of X,
ExtREAL , f2 be
without+infty
Function of X,
ExtREAL holds (f1
- f2)
= (f1
+ (
- f2)) & (f2
- f1)
= (f2
+ (
- f1)) & (
- (f1
- f2))
= ((
- f1)
+ f2) & (
- (f2
- f1))
= ((
- f2)
+ f1)
proof
let X be non
empty
set, f1 be
without-infty
Function of X,
ExtREAL , f2 be
without+infty
Function of X,
ExtREAL ;
thus
A1: (f1
- f2)
= (f1
+ (
- f2)) & (f2
- f1)
= (f2
+ (
- f1)) by
Lm5;
thus (
- (f1
- f2))
= ((
- f1)
- (
- f2)) by
A1,
Th8
.= ((
- f1)
+ (
- (
- f2))) by
Lm5
.= ((
- f1)
+ f2) by
Th2;
thus (
- (f2
- f1))
= ((
- f2)
- (
- f1)) by
A1,
Th9
.= ((
- f2)
+ (
- (
- f1))) by
Lm5
.= ((
- f2)
+ f1) by
Th2;
end;
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
:: original:
.
redefine
func f
. (n,m) ->
Element of
ExtREAL ;
coherence
proof
reconsider n, m as
Element of
NAT by
ORDINAL1:def 12;
(f
. (n,m))
in
ExtREAL ;
hence thesis;
end;
end
theorem ::
DBLSEQ_3:11
Th11: for f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((f1
+ f2)
. (n,m))
= ((f1
. (n,m))
+ (f2
. (n,m)))
proof
let f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
A1: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
reconsider z =
[n, m] as
Element of
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
[n, m]
in
[:
NAT ,
NAT :] by
A1,
ZFMISC_1: 87;
hence thesis by
Th7;
end;
theorem ::
DBLSEQ_3:12
for f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((f1
+ f2)
. (n,m))
= ((f1
. (n,m))
+ (f2
. (n,m)))
proof
let f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
A1: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
reconsider z =
[n, m] as
Element of
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
[n, m]
in
[:
NAT ,
NAT :] by
A1,
ZFMISC_1: 87;
hence thesis by
Th7;
end;
theorem ::
DBLSEQ_3:13
for f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((f1
- f2)
. (n,m))
= ((f1
. (n,m))
- (f2
. (n,m))) & ((f2
- f1)
. (n,m))
= ((f2
. (n,m))
- (f1
. (n,m)))
proof
let f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
A1: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
reconsider z =
[n, m] as
Element of
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
[n, m]
in
[:
NAT ,
NAT :] by
A1,
ZFMISC_1: 87;
hence thesis by
Th7;
end;
theorem ::
DBLSEQ_3:14
for X,Y be non
empty
set, f1,f2 be
without-infty
Function of
[:X, Y:],
ExtREAL holds (
~ (f1
+ f2))
= ((
~ f1)
+ (
~ f2))
proof
let X,Y be non
empty
set, f1,f2 be
without-infty
Function of
[:X, Y:],
ExtREAL ;
now
let z be
Element of
[:Y, X:];
consider y,x be
object such that
A1: y
in Y & x
in X & z
=
[y, x] by
ZFMISC_1:def 2;
reconsider y as
Element of Y by
A1;
reconsider x as
Element of X by
A1;
reconsider z1 =
[x, y] as
Element of
[:X, Y:] by
ZFMISC_1: 87;
((
~ (f1
+ f2))
. z)
= ((
~ (f1
+ f2))
. (y,x)) by
A1;
then ((
~ (f1
+ f2))
. z)
= ((f1
+ f2)
. (x,y)) by
FUNCT_4:def 8;
then
A2: ((
~ (f1
+ f2))
. z)
= ((f1
. z1)
+ (f2
. z1)) by
Th7;
(f1
. z1)
= (f1
. (x,y)) & (f2
. z1)
= (f2
. (x,y));
then (f1
. z1)
= ((
~ f1)
. (y,x)) & (f2
. z1)
= ((
~ f2)
. (y,x)) by
FUNCT_4:def 8;
hence ((
~ (f1
+ f2))
. z)
= (((
~ f1)
+ (
~ f2))
. z) by
A1,
A2,
Th7;
end;
hence thesis by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:15
for X,Y be non
empty
set, f1,f2 be
without+infty
Function of
[:X, Y:],
ExtREAL holds (
~ (f1
+ f2))
= ((
~ f1)
+ (
~ f2))
proof
let X,Y be non
empty
set, f1,f2 be
without+infty
Function of
[:X, Y:],
ExtREAL ;
now
let z be
Element of
[:Y, X:];
consider y,x be
object such that
A1: y
in Y & x
in X & z
=
[y, x] by
ZFMISC_1:def 2;
reconsider y as
Element of Y by
A1;
reconsider x as
Element of X by
A1;
reconsider z1 =
[x, y] as
Element of
[:X, Y:] by
ZFMISC_1: 87;
((
~ (f1
+ f2))
. z)
= ((
~ (f1
+ f2))
. (y,x)) by
A1;
then ((
~ (f1
+ f2))
. z)
= ((f1
+ f2)
. (x,y)) by
FUNCT_4:def 8;
then
A2: ((
~ (f1
+ f2))
. z)
= ((f1
. z1)
+ (f2
. z1)) by
Th7;
(f1
. z1)
= (f1
. (x,y)) & (f2
. z1)
= (f2
. (x,y));
then (f1
. z1)
= ((
~ f1)
. (y,x)) & (f2
. z1)
= ((
~ f2)
. (y,x)) by
FUNCT_4:def 8;
hence ((
~ (f1
+ f2))
. z)
= (((
~ f1)
+ (
~ f2))
. z) by
A1,
A2,
Th7;
end;
hence thesis by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:16
for X,Y be non
empty
set, f1 be
without-infty
Function of
[:X, Y:],
ExtREAL , f2 be
without+infty
Function of
[:X, Y:],
ExtREAL holds (
~ (f1
- f2))
= ((
~ f1)
- (
~ f2)) & (
~ (f2
- f1))
= ((
~ f2)
- (
~ f1))
proof
let X,Y be non
empty
set, f1 be
without-infty
Function of
[:X, Y:],
ExtREAL , f2 be
without+infty
Function of
[:X, Y:],
ExtREAL ;
now
let z be
Element of
[:Y, X:];
consider y,x be
object such that
A1: y
in Y & x
in X & z
=
[y, x] by
ZFMISC_1:def 2;
reconsider y as
Element of Y by
A1;
reconsider x as
Element of X by
A1;
reconsider z1 =
[x, y] as
Element of
[:X, Y:] by
ZFMISC_1: 87;
((
~ (f1
- f2))
. z)
= ((
~ (f1
- f2))
. (y,x)) by
A1;
then ((
~ (f1
- f2))
. z)
= ((f1
- f2)
. (x,y)) by
FUNCT_4:def 8;
then
A2: ((
~ (f1
- f2))
. z)
= ((f1
. z1)
- (f2
. z1)) by
Th7;
(f1
. z1)
= (f1
. (x,y)) & (f2
. z1)
= (f2
. (x,y));
then (f1
. z1)
= ((
~ f1)
. (y,x)) & (f2
. z1)
= ((
~ f2)
. (y,x)) by
FUNCT_4:def 8;
hence ((
~ (f1
- f2))
. z)
= (((
~ f1)
- (
~ f2))
. z) by
A1,
A2,
Th7;
end;
hence (
~ (f1
- f2))
= ((
~ f1)
- (
~ f2)) by
FUNCT_2:def 8;
now
let z be
Element of
[:Y, X:];
consider y,x be
object such that
A1: y
in Y & x
in X & z
=
[y, x] by
ZFMISC_1:def 2;
reconsider y as
Element of Y by
A1;
reconsider x as
Element of X by
A1;
reconsider z1 =
[x, y] as
Element of
[:X, Y:] by
ZFMISC_1: 87;
((
~ (f2
- f1))
. z)
= ((
~ (f2
- f1))
. (y,x)) by
A1;
then ((
~ (f2
- f1))
. z)
= ((f2
- f1)
. (x,y)) by
FUNCT_4:def 8;
then
A2: ((
~ (f2
- f1))
. z)
= ((f2
. z1)
- (f1
. z1)) by
Th7;
(f1
. z1)
= (f1
. (x,y)) & (f2
. z1)
= (f2
. (x,y));
then (f1
. z1)
= ((
~ f1)
. (y,x)) & (f2
. z1)
= ((
~ f2)
. (y,x)) by
FUNCT_4:def 8;
hence ((
~ (f2
- f1))
. z)
= (((
~ f2)
- (
~ f1))
. z) by
A1,
A2,
Th7;
end;
hence (
~ (f2
- f1))
= ((
~ f2)
- (
~ f1)) by
FUNCT_2:def 8;
end;
registration
cluster
convergent_to_+infty ->
convergent for
ExtREAL_sequence;
correctness by
MESFUNC5:def 11;
cluster
convergent_to_-infty ->
convergent for
ExtREAL_sequence;
correctness by
MESFUNC5:def 11;
cluster
convergent_to_finite_number ->
convergent for
ExtREAL_sequence;
correctness by
MESFUNC5:def 11;
end
registration
cluster
convergent for
ExtREAL_sequence;
existence
proof
reconsider z =
0 as
Element of
ExtREAL by
XXREAL_0:def 1;
reconsider M = (
NAT
--> z) as
ExtREAL_sequence;
take M;
for n be
Nat holds (M
. n)
=
0 by
Lm1;
then M is
convergent_to_finite_number by
MESFUNC5: 52;
hence M is
convergent;
end;
cluster
convergent for
without-infty
ExtREAL_sequence;
existence
proof
reconsider z =
0 as
Element of
ExtREAL by
XXREAL_0:def 1;
reconsider M = (
NAT
--> z) as
ExtREAL_sequence;
reconsider M as
without-infty
ExtREAL_sequence by
Lm2;
take M;
for n be
Nat holds (M
. n)
=
0 by
Lm1;
then M is
convergent_to_finite_number by
MESFUNC5: 52;
hence M is
convergent;
end;
cluster
convergent for
without+infty
ExtREAL_sequence;
existence
proof
reconsider z =
0 as
Element of
ExtREAL by
XXREAL_0:def 1;
reconsider M = (
NAT
--> z) as
ExtREAL_sequence;
reconsider M as
without+infty
ExtREAL_sequence by
Lm2;
take M;
for n be
Nat holds (M
. n)
=
0 by
Lm1;
then M is
convergent_to_finite_number by
MESFUNC5: 52;
hence M is
convergent;
end;
end
Lm6: for seq be
convergent
ExtREAL_sequence holds (seq is
convergent_to_finite_number implies (
- seq) is
convergent_to_finite_number) & (seq is
convergent_to_+infty implies (
- seq) is
convergent_to_-infty) & (seq is
convergent_to_-infty implies (
- seq) is
convergent_to_+infty) & (
- seq) is
convergent & (
lim (
- seq))
= (
- (
lim seq))
proof
let seq be
convergent
ExtREAL_sequence;
P0:
now
assume seq is
convergent_to_finite_number;
then
consider g be
Real such that
A1: (
lim seq)
= g & for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq
. m)
- (
lim seq)).|
< p by
MESFUNC9: 7;
reconsider G = (
- g) as
R_eal by
XXREAL_0:def 1;
A5:
now
let p be
Real;
assume
0
< p;
then
consider n be
Nat such that
A2: for m be
Nat st n
<= m holds
|.((seq
. m)
- (
lim seq)).|
< p by
A1;
take n;
hereby
let m be
Nat;
assume n
<= m;
then
A3:
|.((seq
. m)
- g).|
< p by
A1,
A2;
m
in
NAT by
ORDINAL1:def 12;
then m
in (
dom (
- seq)) by
FUNCT_2:def 1;
then (
- (((
- seq)
. m)
- G))
= (
- ((
- (seq
. m))
- G)) by
MESFUNC1:def 7
.= (
- (
- ((seq
. m)
- g))) by
XXREAL_3: 26
.= ((seq
. m)
- g);
hence
|.(((
- seq)
. m)
- G).|
< p by
A3,
EXTREAL1: 29;
end;
end;
hence
A4: (
- seq) is
convergent_to_finite_number by
MESFUNC5:def 8;
hence (
- seq) is
convergent;
(
lim (
- seq))
= (
- g) by
A4,
A5,
MESFUNC5:def 12;
hence (
lim (
- seq))
= (
- (
lim seq)) by
A1,
XXREAL_3:def 3;
end;
thus seq is
convergent_to_finite_number implies (
- seq) is
convergent_to_finite_number by
P0;
P1:
now
assume
A6: seq is
convergent_to_+infty;
A10:
now
let g be
Real;
assume
A7: g
<
0 ;
reconsider p = (
- g) as
ExtReal;
consider n be
Nat such that
A8: for m be
Nat st n
<= m holds p
<= (seq
. m) by
A6,
A7,
MESFUNC5:def 9;
take n;
hereby
let m be
Nat;
assume n
<= m;
then (
- p)
>= (
- (seq
. m)) by
A8,
XXREAL_3: 38;
then
A9: (
- (
- g))
>= (
- (seq
. m)) by
XXREAL_3:def 3;
m
in
NAT by
ORDINAL1:def 12;
then m
in (
dom (
- seq)) by
FUNCT_2:def 1;
hence ((
- seq)
. m)
<= g by
A9,
MESFUNC1:def 7;
end;
end;
hence (
- seq) is
convergent_to_-infty by
MESFUNC5:def 10;
hence (
- seq) is
convergent;
(
lim (
- seq))
=
-infty & (
lim seq)
=
+infty by
A6,
A10,
MESFUNC9: 7,
MESFUNC5:def 10;
hence (
lim (
- seq))
= (
- (
lim seq)) by
XXREAL_3: 6;
end;
thus seq is
convergent_to_+infty implies (
- seq) is
convergent_to_-infty by
P1;
P2:
now
assume
A11: seq is
convergent_to_-infty;
A15:
now
let g be
Real;
assume
A12: g
>
0 ;
reconsider p = (
- g) as
ExtReal;
consider n be
Nat such that
A13: for m be
Nat st n
<= m holds (seq
. m)
<= p by
A11,
A12,
MESFUNC5:def 10;
take n;
hereby
let m be
Nat;
assume n
<= m;
then (
- (seq
. m))
>= (
- p) by
A13,
XXREAL_3: 38;
then
A14: (
- (seq
. m))
>= (
- (
- g)) by
XXREAL_3:def 3;
m
in
NAT by
ORDINAL1:def 12;
then m
in (
dom (
- seq)) by
FUNCT_2:def 1;
hence ((
- seq)
. m)
>= g by
A14,
MESFUNC1:def 7;
end;
end;
hence (
- seq) is
convergent_to_+infty by
MESFUNC5:def 9;
hence (
- seq) is
convergent;
(
lim (
- seq))
=
+infty & (
lim seq)
=
-infty by
A11,
A15,
MESFUNC9: 7,
MESFUNC5:def 9;
hence (
lim (
- seq))
= (
- (
lim seq)) by
XXREAL_3: 5;
end;
thus seq is
convergent_to_-infty implies (
- seq) is
convergent_to_+infty by
P2;
thus thesis by
P0,
P1,
P2,
MESFUNC5:def 11;
end;
theorem ::
DBLSEQ_3:17
Th17: for seq be
convergent
ExtREAL_sequence holds (seq is
convergent_to_finite_number iff (
- seq) is
convergent_to_finite_number) & (seq is
convergent_to_+infty iff (
- seq) is
convergent_to_-infty) & (seq is
convergent_to_-infty iff (
- seq) is
convergent_to_+infty) & (
- seq) is
convergent & (
lim (
- seq))
= (
- (
lim seq))
proof
let seq be
convergent
ExtREAL_sequence;
now
assume (
- seq) is
convergent_to_finite_number;
then (
- (
- seq)) is
convergent_to_finite_number by
Lm6;
hence seq is
convergent_to_finite_number by
Th2;
end;
hence seq is
convergent_to_finite_number iff (
- seq) is
convergent_to_finite_number by
Lm6;
now
assume (
- seq) is
convergent_to_-infty;
then (
- (
- seq)) is
convergent_to_+infty by
Lm6;
hence seq is
convergent_to_+infty by
Th2;
end;
hence seq is
convergent_to_+infty iff (
- seq) is
convergent_to_-infty by
Lm6;
now
assume (
- seq) is
convergent_to_+infty;
then (
- (
- seq)) is
convergent_to_-infty by
Lm6;
hence seq is
convergent_to_-infty by
Th2;
end;
hence seq is
convergent_to_-infty iff (
- seq) is
convergent_to_+infty by
Lm6;
thus (
- seq) is
convergent & (
lim (
- seq))
= (
- (
lim seq)) by
Lm6;
end;
theorem ::
DBLSEQ_3:18
Th18: for seq1,seq2 be
without-infty
ExtREAL_sequence st seq1 is
convergent_to_+infty & seq2 is
convergent_to_+infty holds (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty
proof
let seq1,seq2 be
without-infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_+infty & seq2 is
convergent_to_+infty;
now
let g be
Real;
assume
A2:
0
< g;
then
consider n1 be
Nat such that
A3: for m be
Nat st n1
<= m holds (g
/ 2)
<= (seq1
. m) by
A1,
MESFUNC5:def 9;
consider n2 be
Nat such that
A4: for m be
Nat st n2
<= m holds (g
/ 2)
<= (seq2
. m) by
A1,
A2,
MESFUNC5:def 9;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A5: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A5,
XXREAL_0: 2;
then (g
/ 2)
<= (seq1
. m) & (g
/ 2)
<= (seq2
. m) by
A3,
A4;
then
A6: ((g
/ 2)
+ (g
/ 2))
<= ((seq1
. m)
+ (seq2
. m)) by
XXREAL_3: 36;
m is
Element of
NAT by
ORDINAL1:def 12;
hence g
<= ((seq1
+ seq2)
. m) by
A6,
Th7;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds g
<= ((seq1
+ seq2)
. m);
end;
hence
A7: (seq1
+ seq2) is
convergent_to_+infty by
MESFUNC5:def 9;
hence (seq1
+ seq2) is
convergent;
thus (
lim (seq1
+ seq2))
=
+infty by
A7,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:19
Th19: for seq1,seq2 be
without-infty
ExtREAL_sequence st seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number holds (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty
proof
let seq1,seq2 be
without-infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number;
then
consider S2 be
Real such that
A2: for g be
Real st
0
< g holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
MESFUNC5:def 8;
now
let g be
Real;
assume
A3:
0
< g;
set G = (
max (1,((2
* g)
- S2)));
A4: 1
<= G & ((2
* g)
- S2)
<= G by
XXREAL_0: 25;
then
consider n1 be
Nat such that
A5: for m be
Nat st n1
<= m holds G
<= (seq1
. m) by
A1,
MESFUNC5:def 9;
consider n2 be
Nat such that
A6: for m be
Nat st n2
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
A2,
A3;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A7: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A7,
XXREAL_0: 2;
then
A8: G
<= (seq1
. m) &
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
A5,
A6;
reconsider g1 = g as
R_eal by
XXREAL_0:def 1;
(
- g1)
< ((seq2
. m)
- S2) qua
ExtReal by
A8,
EXTREAL1: 21;
then ((
- g1)
+ S2 qua
ExtReal)
< (seq2
. m) by
XXREAL_3: 53;
then
A9: (G
+ ((
- g1)
+ S2 qua
ExtReal))
<= ((seq1
. m)
+ (seq2
. m)) by
A8,
XXREAL_3: 36;
(
- g1)
= (
- g) by
XXREAL_3:def 3;
then ((
- g1)
+ S2 qua
ExtReal)
= ((
- g)
+ S2) by
XXREAL_3:def 2;
then (((2
* g)
- S2)
+ ((
- g1)
+ S2 qua
ExtReal))
= (((2
* g)
- S2)
+ ((
- g)
+ S2)) by
XXREAL_3:def 2;
then g
<= (G
+ ((
- g1)
+ S2 qua
ExtReal)) by
A4,
XXREAL_3: 36;
then
A10: g
<= ((seq1
. m)
+ (seq2
. m)) by
A9,
XXREAL_0: 2;
m is
Element of
NAT by
ORDINAL1:def 12;
hence g
<= ((seq1
+ seq2)
. m) by
A10,
Th7;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds g
<= ((seq1
+ seq2)
. m);
end;
hence
A11: (seq1
+ seq2) is
convergent_to_+infty by
MESFUNC5:def 9;
hence (seq1
+ seq2) is
convergent;
thus (
lim (seq1
+ seq2))
=
+infty by
A11,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:20
for seq1,seq2 be
without+infty
ExtREAL_sequence st seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number holds (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty
proof
let seq1,seq2 be
without+infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number;
then
consider S2 be
Real such that
A2: for g be
Real st
0
< g holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
MESFUNC5:def 8;
now
let g be
Real;
assume
A3:
0
< g;
set G = (
max (1,((2
* g)
- S2)));
A4: 1
<= G & ((2
* g)
- S2)
<= G by
XXREAL_0: 25;
then
consider n1 be
Nat such that
A5: for m be
Nat st n1
<= m holds G
<= (seq1
. m) by
A1,
MESFUNC5:def 9;
consider n2 be
Nat such that
A6: for m be
Nat st n2
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
A2,
A3;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A7: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A7,
XXREAL_0: 2;
then
A8: G
<= (seq1
. m) &
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
A5,
A6;
reconsider g1 = g as
R_eal by
XXREAL_0:def 1;
(
- g1)
< ((seq2
. m)
- S2) qua
ExtReal by
A8,
EXTREAL1: 21;
then ((
- g1)
+ S2 qua
ExtReal)
< (seq2
. m) by
XXREAL_3: 53;
then
A9: (G
+ ((
- g1)
+ S2 qua
ExtReal))
<= ((seq1
. m)
+ (seq2
. m)) by
A8,
XXREAL_3: 36;
(
- g1)
= (
- g) by
XXREAL_3:def 3;
then ((
- g1)
+ S2 qua
ExtReal)
= ((
- g)
+ S2) by
XXREAL_3:def 2;
then (((2
* g)
- S2)
+ ((
- g1)
+ S2 qua
ExtReal))
= (((2
* g)
- S2)
+ ((
- g)
+ S2)) by
XXREAL_3:def 2;
then g
<= (G
+ ((
- g1)
+ S2 qua
ExtReal)) by
A4,
XXREAL_3: 36;
then
A10: g
<= ((seq1
. m)
+ (seq2
. m)) by
A9,
XXREAL_0: 2;
m is
Element of
NAT by
ORDINAL1:def 12;
hence g
<= ((seq1
+ seq2)
. m) by
A10,
Th7;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds g
<= ((seq1
+ seq2)
. m);
end;
hence
A11: (seq1
+ seq2) is
convergent_to_+infty by
MESFUNC5:def 9;
hence (seq1
+ seq2) is
convergent;
thus (
lim (seq1
+ seq2))
=
+infty by
A11,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:21
Th21: for seq1,seq2 be
without-infty
ExtREAL_sequence st seq1 is
convergent_to_-infty & seq2 is
convergent_to_-infty holds (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty
proof
let seq1,seq2 be
without-infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_-infty & seq2 is
convergent_to_-infty;
now
let g be
Real;
assume
A2: g
<
0 ;
then
consider n1 be
Nat such that
A3: for m be
Nat st n1
<= m holds (seq1
. m)
<= (g
/ 2) by
A1,
MESFUNC5:def 10;
consider n2 be
Nat such that
A4: for m be
Nat st n2
<= m holds (seq2
. m)
<= (g
/ 2) by
A1,
A2,
MESFUNC5:def 10;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A5: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A5,
XXREAL_0: 2;
then (seq1
. m)
<= (g
/ 2) & (seq2
. m)
<= (g
/ 2) by
A3,
A4;
then
A6: ((seq1
. m)
+ (seq2
. m))
<= ((g
/ 2)
+ (g
/ 2)) by
XXREAL_3: 36;
m is
Element of
NAT by
ORDINAL1:def 12;
hence ((seq1
+ seq2)
. m)
<= g by
A6,
Th7;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds ((seq1
+ seq2)
. m)
<= g;
end;
hence
A7: (seq1
+ seq2) is
convergent_to_-infty by
MESFUNC5:def 10;
hence (seq1
+ seq2) is
convergent;
thus (
lim (seq1
+ seq2))
=
-infty by
A7,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:22
Th22: for seq1,seq2 be
without-infty
ExtREAL_sequence st seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number holds (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty
proof
let seq1,seq2 be
without-infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number;
then
consider S2 be
Real such that
A2: for g be
Real st
0
< g holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< g by
MESFUNC5:def 8;
now
let g be
Real;
assume
A3: g
<
0 ;
set G = (
min ((
- 1),((2
* g)
- S2)));
A4: G
<= (
- 1) & G
<= ((2
* g)
- S2) by
XXREAL_0: 17;
then
consider n1 be
Nat such that
A5: for m be
Nat st n1
<= m holds (seq1
. m)
<= G by
A1,
MESFUNC5:def 10;
consider n2 be
Nat such that
A6: for m be
Nat st n2
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< (
- g) by
A2,
A3;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A7: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A7,
XXREAL_0: 2;
then
A8: (seq1
. m)
<= G &
|.((seq2
. m)
- S2) qua
ExtReal.|
< (
- g) by
A5,
A6;
reconsider g1 = g as
R_eal by
XXREAL_0:def 1;
B1: (
- g1)
= (
- g) by
XXREAL_3:def 3;
then ((seq2
. m)
- S2) qua
ExtReal
< (
- g1) by
A8,
EXTREAL1: 21;
then (seq2
. m)
< ((
- g1)
+ S2 qua
ExtReal) by
XXREAL_3: 54;
then
A9: ((seq1
. m)
+ (seq2
. m))
<= (G
+ ((
- g1)
+ S2 qua
ExtReal)) by
A8,
XXREAL_3: 36;
((
- g1)
+ S2 qua
ExtReal)
= ((
- g)
+ S2) by
B1,
XXREAL_3:def 2;
then (((2
* g)
- S2)
+ ((
- g1)
+ S2 qua
ExtReal))
= (((2
* g)
- S2)
+ ((
- g)
+ S2)) by
XXREAL_3:def 2;
then (G
+ ((
- g1)
+ S2 qua
ExtReal))
<= g by
A4,
XXREAL_3: 36;
then
A10: ((seq1
. m)
+ (seq2
. m))
<= g by
A9,
XXREAL_0: 2;
m is
Element of
NAT by
ORDINAL1:def 12;
hence ((seq1
+ seq2)
. m)
<= g by
A10,
Th7;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds ((seq1
+ seq2)
. m)
<= g;
end;
hence
A11: (seq1
+ seq2) is
convergent_to_-infty by
MESFUNC5:def 10;
hence (seq1
+ seq2) is
convergent;
thus (
lim (seq1
+ seq2))
=
-infty by
A11,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:23
Th23: for seq1,seq2 be
without-infty
ExtREAL_sequence st seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number holds (seq1
+ seq2) is
convergent_to_finite_number & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
= ((
lim seq1)
+ (
lim seq2))
proof
let seq1,seq2 be
without-infty
ExtREAL_sequence;
assume
A1: seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number;
B2: not seq1 is
convergent_to_-infty & not seq1 is
convergent_to_+infty & not seq2 is
convergent_to_-infty & not seq2 is
convergent_to_+infty by
A1,
MESFUNC5: 50,
MESFUNC5: 51;
consider S1 be
Real such that
A2: (
lim seq1)
= S1 & (for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq1
. m)
- (
lim seq1)).|
< p) & seq1 is
convergent_to_finite_number by
A1,
B2,
MESFUNC5:def 12;
consider S2 be
Real such that
A3: (
lim seq2)
= S2 & (for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- (
lim seq2)).|
< p) & seq2 is
convergent_to_finite_number by
A1,
B2,
MESFUNC5:def 12;
B3:
now
let p be
Real;
assume
A4:
0
< p;
then
consider n1 be
Nat such that
A5: for m be
Nat st n1
<= m holds
|.((seq1
. m)
- S1) qua
ExtReal.|
< (p
/ 2) by
A2;
consider n2 be
Nat such that
A6: for m be
Nat st n2
<= m holds
|.((seq2
. m)
- S2) qua
ExtReal.|
< (p
/ 2) by
A3,
A4;
reconsider N1 = n1, N2 = n2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider n = (
max (N1,N2)) as
Nat;
A7: n1
<= n & n2
<= n by
XXREAL_0: 25;
now
let m be
Nat;
assume n
<= m;
then n1
<= m & n2
<= m by
A7,
XXREAL_0: 2;
then
A8:
|.((seq1
. m)
- S1) qua
ExtReal.|
< (p
/ 2) &
|.((seq2
. m)
- S2) qua
ExtReal.|
< (p
/ 2) by
A5,
A6;
then
A9: (
|.((seq1
. m)
- S1) qua
ExtReal.|
+
|.((seq2
. m)
- S2) qua
ExtReal.|)
< ((p
/ 2)
+ (p
/ 2)) by
XXREAL_3: 64;
|.((seq1
. m)
- S1) qua
ExtReal.|
<
+infty by
A8,
XXREAL_0: 2,
XXREAL_0: 4;
then
A10: ((seq1
. m)
- S1) qua
ExtReal
in
REAL by
EXTREAL1: 41;
A12: (seq1
. m)
<>
-infty & (seq2
. m)
<>
-infty & ((seq1
+ seq2)
. m)
<>
-infty by
MESFUNC5:def 5;
A13: m is
Element of
NAT by
ORDINAL1:def 12;
(((seq1
. m)
- S1) qua
ExtReal
+ ((seq2
. m)
- S2) qua
ExtReal)
= ((((seq1
. m)
- S1) qua
ExtReal
+ (seq2
. m))
- S2) qua
ExtReal by
A10,
XXREAL_3: 30
.= (((
- S1) qua
ExtReal
+ ((seq1
. m)
+ (seq2
. m)))
- S2) qua
ExtReal by
A12,
XXREAL_3: 29
.= ((((seq1
+ seq2)
. m)
- S1) qua
ExtReal
- S2) qua
ExtReal by
A13,
Th7
.= (((seq1
+ seq2)
. m)
- (S1 qua
ExtReal
+ S2) qua
ExtReal) by
XXREAL_3: 31;
then
|.(((seq1
+ seq2)
. m)
- (S1
+ S2)) qua
ExtReal.|
<= (
|.((seq1
. m)
- S1) qua
ExtReal.|
+
|.((seq2
. m)
- S2) qua
ExtReal.|) by
EXTREAL1: 24;
hence
|.(((seq1
+ seq2)
. m)
- (S1
+ S2)) qua
ExtReal.|
< p by
A9,
XXREAL_0: 2;
end;
hence ex n be
Nat st for m be
Nat st n
<= m holds
|.(((seq1
+ seq2)
. m)
- (S1
+ S2)) qua
ExtReal.|
< p;
end;
hence
A14: (seq1
+ seq2) is
convergent_to_finite_number by
MESFUNC5:def 8;
hence (seq1
+ seq2) is
convergent;
for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.(((seq1
+ seq2)
. m)
- ((
lim seq1)
+ (
lim seq2))).|
< p
proof
let p be
Real;
assume
0
< p;
then
consider n be
Nat such that
A17: for m be
Nat st n
<= m holds
|.(((seq1
+ seq2)
. m)
- (S1
+ S2)) qua
ExtReal.|
< p by
B3;
take n;
hereby
let m be
Nat;
assume n
<= m;
then
|.(((seq1
+ seq2)
. m)
- (S1
+ S2)) qua
ExtReal.|
< p by
A17;
hence
|.(((seq1
+ seq2)
. m)
- ((
lim seq1)
+ (
lim seq2))).|
< p by
A2,
A3,
XXREAL_3:def 2;
end;
end;
hence (
lim (seq1
+ seq2))
= ((
lim seq1)
+ (
lim seq2)) by
A2,
A3,
A14,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:24
Th24: for seq1,seq2 be
without+infty
ExtREAL_sequence holds (seq1 is
convergent_to_+infty & seq2 is
convergent_to_+infty implies (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty ) & (seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number implies (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty ) & (seq1 is
convergent_to_-infty & seq2 is
convergent_to_-infty implies (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty ) & (seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number implies (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty ) & (seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number implies (seq1
+ seq2) is
convergent_to_finite_number & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
= ((
lim seq1)
+ (
lim seq2)))
proof
let seq1,seq2 be
without+infty
ExtREAL_sequence;
reconsider f1 = (
- seq1), f2 = (
- seq2) as
without-infty
ExtREAL_sequence;
hereby
assume seq1 is
convergent_to_+infty & seq2 is
convergent_to_+infty;
then
A2: f1 is
convergent_to_-infty & f2 is
convergent_to_-infty by
Th17;
then
reconsider F = (f1
+ f2) as
convergent
ExtREAL_sequence by
Th21;
A3: (f1
+ f2) is
convergent_to_-infty & (f1
+ f2) is
convergent & (
lim (f1
+ f2))
=
-infty by
A2,
Th21;
(f1
+ f2)
= (f1
- seq2) by
Th10
.= (
- (seq1
+ seq2)) by
Th9;
then (seq1
+ seq2)
= (
- F) by
Th2;
hence (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty by
A3,
Th17,
XXREAL_3: 5;
end;
hereby
assume seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_-infty & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F = (f1
+ f2) as
convergent
ExtREAL_sequence by
Th22;
A3: (f1
+ f2) is
convergent_to_-infty & (f1
+ f2) is
convergent & (
lim (f1
+ f2))
=
-infty by
A2,
Th22;
(f1
+ f2)
= (f1
- seq2) by
Th10
.= (
- (seq1
+ seq2)) by
Th9;
then (seq1
+ seq2)
= (
- F) by
Th2;
hence (seq1
+ seq2) is
convergent_to_+infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
+infty by
A3,
Th17,
XXREAL_3: 5;
end;
hereby
assume seq1 is
convergent_to_-infty & seq2 is
convergent_to_-infty;
then
A2: f1 is
convergent_to_+infty & f2 is
convergent_to_+infty by
Th17;
then
reconsider F = (f1
+ f2) as
convergent
ExtREAL_sequence by
Th18;
A3: (f1
+ f2) is
convergent_to_+infty & (f1
+ f2) is
convergent & (
lim (f1
+ f2))
=
+infty by
A2,
Th18;
(f1
+ f2)
= (f1
- seq2) by
Th10
.= (
- (seq1
+ seq2)) by
Th9;
then (seq1
+ seq2)
= (
- F) by
Th2;
hence (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty by
A3,
Th17,
XXREAL_3: 6;
end;
hereby
assume seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_+infty & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F = (f1
+ f2) as
convergent
ExtREAL_sequence by
Th19;
A3: (f1
+ f2) is
convergent_to_+infty & (f1
+ f2) is
convergent & (
lim (f1
+ f2))
=
+infty by
A2,
Th19;
(f1
+ f2)
= (f1
- seq2) by
Th10
.= (
- (seq1
+ seq2)) by
Th9;
then (seq1
+ seq2)
= (
- F) by
Th2;
hence (seq1
+ seq2) is
convergent_to_-infty & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
=
-infty by
A3,
Th17,
XXREAL_3: 6;
end;
hereby
assume seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_finite_number & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F = (f1
+ f2) as
convergent
ExtREAL_sequence by
Th23;
A3: (f1
+ f2) is
convergent_to_finite_number & (f1
+ f2) is
convergent & (
lim (f1
+ f2))
= ((
lim f1)
+ (
lim f2)) by
A2,
Th23;
(f1
+ f2)
= (f1
- seq2) by
Th10
.= (
- (seq1
+ seq2)) by
Th9;
then
A4: (seq1
+ seq2)
= (
- F) by
Th2;
then
A5: (
lim (seq1
+ seq2))
= (
- ((
lim f1)
+ (
lim f2))) by
A3,
Th17;
seq1
= (
- f1) & seq2
= (
- f2) by
Th2;
then (
lim seq1)
= (
- (
lim f1)) & (
lim seq2)
= (
- (
lim f2)) by
A2,
Th17;
hence (seq1
+ seq2) is
convergent_to_finite_number & (seq1
+ seq2) is
convergent & (
lim (seq1
+ seq2))
= ((
lim seq1)
+ (
lim seq2)) by
A3,
A4,
A5,
Th17,
XXREAL_3: 9;
end;
end;
theorem ::
DBLSEQ_3:25
for seq1 be
without-infty
ExtREAL_sequence, seq2 be
without+infty
ExtREAL_sequence holds (seq1 is
convergent_to_+infty & seq2 is
convergent_to_-infty implies (seq1
- seq2) is
convergent_to_+infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_-infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
+infty & (
lim (seq2
- seq1))
=
-infty ) & (seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number implies (seq1
- seq2) is
convergent_to_+infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_-infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
+infty & (
lim (seq2
- seq1))
=
-infty ) & (seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number implies (seq1
- seq2) is
convergent_to_-infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_+infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
-infty & (
lim (seq2
- seq1))
=
+infty ) & (seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number implies (seq1
- seq2) is
convergent_to_finite_number & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_finite_number & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
= ((
lim seq1)
- (
lim seq2)) & (
lim (seq2
- seq1))
= ((
lim seq2)
- (
lim seq1)))
proof
let seq1 be
without-infty
ExtREAL_sequence, seq2 be
without+infty
ExtREAL_sequence;
reconsider f1 = (
- seq1) as
without+infty
ExtREAL_sequence;
reconsider f2 = (
- seq2) as
without-infty
ExtREAL_sequence;
hereby
assume
A1: seq1 is
convergent_to_+infty & seq2 is
convergent_to_-infty;
then
A2: f1 is
convergent_to_-infty & f2 is
convergent_to_+infty by
Th17;
then
reconsider F1 = (f1
+ seq2), F2 = (seq1
+ f2) as
convergent
ExtREAL_sequence by
A1,
Th24,
Th18;
A3: (f1
+ seq2) is
convergent_to_-infty & (f1
+ seq2) is
convergent & (
lim (f1
+ seq2))
=
-infty & (seq1
+ f2) is
convergent_to_+infty & (seq1
+ f2) is
convergent & (
lim (seq1
+ f2))
=
+infty by
A1,
A2,
Th24,
Th18;
A4: (seq1
- seq2)
= (seq1
+ (
- seq2)) by
Th10
.= (
- (seq2
- seq1)) by
Th10
.= (
- ((
- seq1)
+ seq2)) by
Th10;
(seq2
- seq1)
= (seq2
+ (
- seq1)) by
Th10
.= (
- (seq1
- seq2)) by
Th10
.= (
- ((
- seq2)
+ seq1)) by
Th10;
hence (seq1
- seq2) is
convergent_to_+infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_-infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
+infty & (
lim (seq2
- seq1))
=
-infty by
A3,
A4,
Th17,
XXREAL_3: 5,
XXREAL_3: 6;
end;
hereby
assume
A1: seq1 is
convergent_to_+infty & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_-infty & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F1 = (f1
+ seq2), F2 = (seq1
+ f2) as
convergent
ExtREAL_sequence by
A1,
Th24,
Th19;
A3: (f1
+ seq2) is
convergent_to_-infty & (f1
+ seq2) is
convergent & (
lim (f1
+ seq2))
=
-infty & (seq1
+ f2) is
convergent_to_+infty & (seq1
+ f2) is
convergent & (
lim (seq1
+ f2))
=
+infty by
A1,
A2,
Th24,
Th19;
A4: (seq1
- seq2)
= (seq1
+ (
- seq2)) by
Th10
.= (
- (seq2
- seq1)) by
Th10
.= (
- ((
- seq1)
+ seq2)) by
Th10;
(seq2
- seq1)
= (seq2
+ (
- seq1)) by
Th10
.= (
- (seq1
- seq2)) by
Th10
.= (
- ((
- seq2)
+ seq1)) by
Th10;
hence (seq1
- seq2) is
convergent_to_+infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_-infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
+infty & (
lim (seq2
- seq1))
=
-infty by
A3,
A4,
Th17,
XXREAL_3: 5,
XXREAL_3: 6;
end;
hereby
assume
A1: seq1 is
convergent_to_-infty & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_+infty & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F1 = (f1
+ seq2), F2 = (seq1
+ f2) as
convergent
ExtREAL_sequence by
A1,
Th24,
Th22;
A3: (f1
+ seq2) is
convergent_to_+infty & (f1
+ seq2) is
convergent & (
lim (f1
+ seq2))
=
+infty & (seq1
+ f2) is
convergent_to_-infty & (seq1
+ f2) is
convergent & (
lim (seq1
+ f2))
=
-infty by
A1,
A2,
Th24,
Th22;
A4: (seq1
- seq2)
= (seq1
+ (
- seq2)) by
Th10
.= (
- (seq2
- seq1)) by
Th10
.= (
- ((
- seq1)
+ seq2)) by
Th10;
(seq2
- seq1)
= (seq2
+ (
- seq1)) by
Th10
.= (
- (seq1
- seq2)) by
Th10
.= (
- ((
- seq2)
+ seq1)) by
Th10;
hence (seq1
- seq2) is
convergent_to_-infty & (seq1
- seq2) is
convergent & (seq2
- seq1) is
convergent_to_+infty & (seq2
- seq1) is
convergent & (
lim (seq1
- seq2))
=
-infty & (
lim (seq2
- seq1))
=
+infty by
A3,
A4,
Th17,
XXREAL_3: 5,
XXREAL_3: 6;
end;
assume
A1: seq1 is
convergent_to_finite_number & seq2 is
convergent_to_finite_number;
then
A2: f1 is
convergent_to_finite_number & f2 is
convergent_to_finite_number by
Th17;
then
reconsider F1 = (f1
+ seq2), F2 = (seq1
+ f2) as
convergent
ExtREAL_sequence by
A1,
Th24,
Th23;
A3: (f1
+ seq2) is
convergent_to_finite_number & (f1
+ seq2) is
convergent & (seq1
+ f2) is
convergent_to_finite_number & (seq1
+ f2) is
convergent & (
lim (f1
+ seq2))
= ((
lim f1)
+ (
lim seq2)) & (
lim (seq1
+ f2))
= ((
lim seq1)
+ (
lim f2)) by
A1,
A2,
Th24,
Th23;
A4: (seq1
- seq2)
= (seq1
+ (
- seq2)) by
Th10
.= (
- (seq2
- seq1)) by
Th10
.= (
- ((
- seq1)
+ seq2)) by
Th10;
then
A5: (
lim (seq1
- seq2))
= (
- ((
lim f1)
+ (
lim seq2))) by
A3,
Th17;
A6: (seq2
- seq1)
= (seq2
+ (
- seq1)) by
Th10
.= (
- (seq1
- seq2)) by
Th10
.= (
- ((
- seq2)
+ seq1)) by
Th10;
then
A7: (
lim (seq2
- seq1))
= (
- ((
lim f2)
+ (
lim seq1))) by
A3,
Th17;
seq1
= (
- f1) & seq2
= (
- f2) by
Th2;
then (
lim seq1)
= (
- (
lim f1)) & (
lim seq2)
= (
- (
lim f2)) by
A2,
Th17;
hence thesis by
A3,
A4,
A5,
A6,
A7,
Th17,
XXREAL_3: 9;
end;
begin
theorem ::
DBLSEQ_3:26
for seq1,seq2 be
ExtREAL_sequence st seq2 is
subsequence of seq1 & seq1 is
convergent_to_finite_number holds seq2 is
convergent_to_finite_number & (
lim seq1)
= (
lim seq2)
proof
let seq1,seq2 be
ExtREAL_sequence;
assume that
A1: seq2 is
subsequence of seq1 and
A2: seq1 is
convergent_to_finite_number;
not seq1 is
convergent_to_+infty & not seq1 is
convergent_to_-infty by
A2,
MESFUNC5: 50,
MESFUNC5: 51;
then
consider g be
Real such that
B3: (
lim seq1)
= g & (for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq1
. m)
- (
lim seq1)).|
< p) & seq1 is
convergent_to_finite_number by
A2,
MESFUNC5:def 12;
reconsider LIM2 = (
lim seq1) as
R_eal;
ex g be
Real st for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- g) qua
ExtReal.|
< p
proof
take g;
hereby
let p be
Real;
assume
0
< p;
then
consider n1 be
Nat such that
A4: for m be
Nat st n1
<= m holds
|.((seq1
. m)
- g) qua
ExtReal.|
< p by
B3;
take n = n1;
let m be
Nat such that
A5: n
<= m;
consider Nseq be
increasing
sequence of
NAT such that
A6: seq2
= (seq1
* Nseq) by
A1,
VALUED_0:def 17;
m
<= (Nseq
. m) by
SEQM_3: 14;
then
A7: n
<= (Nseq
. m) by
A5,
XXREAL_0: 2;
(seq2
. m)
= (seq1
. (Nseq
. m)) by
A6,
FUNCT_2: 15,
ORDINAL1:def 12;
hence
|.((seq2
. m)
- g) qua
ExtReal.|
< p by
A4,
A7;
end;
end;
hence
A8: seq2 is
convergent_to_finite_number by
MESFUNC5:def 8;
for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.((seq2
. m)
- LIM2).|
< p
proof
let p be
Real;
assume
0
< p;
then
consider n1 be
Nat such that
A10: for m be
Nat st n1
<= m holds
|.((seq1
. m)
- (
lim seq1)).|
< p by
B3;
take n = n1;
let m be
Nat such that
A11: n
<= m;
consider Nseq be
increasing
sequence of
NAT such that
A12: seq2
= (seq1
* Nseq) by
A1,
VALUED_0:def 17;
m
<= (Nseq
. m) by
SEQM_3: 14;
then
A13: n
<= (Nseq
. m) by
A11,
XXREAL_0: 2;
(seq2
. m)
= (seq1
. (Nseq
. m)) by
A12,
FUNCT_2: 15,
ORDINAL1:def 12;
hence
|.((seq2
. m)
- LIM2).|
< p by
A10,
A13;
end;
hence thesis by
A8,
B3,
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:27
for seq1,seq2 be
ExtREAL_sequence st seq2 is
subsequence of seq1 & seq1 is
convergent_to_+infty holds seq2 is
convergent_to_+infty & (
lim seq2)
=
+infty
proof
let seq1,seq2 be
ExtREAL_sequence;
assume that
A1: seq2 is
subsequence of seq1 and
A2: seq1 is
convergent_to_+infty;
now
let g be
Real;
assume
0
< g;
then
consider n1 be
Nat such that
A4: for m be
Nat st n1
<= m holds g
<= (seq1
. m) by
A2,
MESFUNC5:def 9;
take n = n1;
consider Nseq be
increasing
sequence of
NAT such that
A5: seq2
= (seq1
* Nseq) by
A1,
VALUED_0:def 17;
let m be
Nat;
assume
A6: n
<= m;
m
<= (Nseq
. m) by
SEQM_3: 14;
then
A7: n
<= (Nseq
. m) by
A6,
XXREAL_0: 2;
(seq2
. m)
= (seq1
. (Nseq
. m)) by
A5,
FUNCT_2: 15,
ORDINAL1:def 12;
hence g
<= (seq2
. m) by
A4,
A7;
end;
hence seq2 is
convergent_to_+infty by
MESFUNC5:def 9;
hence thesis by
MESFUNC5:def 12;
end;
theorem ::
DBLSEQ_3:28
for seq1,seq2 be
ExtREAL_sequence st seq2 is
subsequence of seq1 & seq1 is
convergent_to_-infty holds seq2 is
convergent_to_-infty & (
lim seq2)
=
-infty
proof
let seq1,seq2 be
ExtREAL_sequence;
assume that
A1: seq2 is
subsequence of seq1 and
A2: seq1 is
convergent_to_-infty;
now
let g be
Real;
assume g
<
0 ;
then
consider n1 be
Nat such that
A4: for m be
Nat st n1
<= m holds (seq1
. m)
<= g by
A2,
MESFUNC5:def 10;
take n = n1;
consider Nseq be
increasing
sequence of
NAT such that
A5: seq2
= (seq1
* Nseq) by
A1,
VALUED_0:def 17;
let m be
Nat;
assume
A6: n
<= m;
m
<= (Nseq
. m) by
SEQM_3: 14;
then
A7: n
<= (Nseq
. m) by
A6,
XXREAL_0: 2;
(seq2
. m)
= (seq1
. (Nseq
. m)) by
A5,
FUNCT_2: 15,
ORDINAL1:def 12;
hence (seq2
. m)
<= g by
A4,
A7;
end;
hence seq2 is
convergent_to_-infty by
MESFUNC5:def 10;
hence thesis by
MESFUNC5:def 12;
end;
begin
theorem ::
DBLSEQ_3:29
for Rseq be
Function of
[:
NAT ,
NAT :],
REAL st (
lim_in_cod1 Rseq) is
convergent holds (
cod1_major_iterated_lim Rseq)
= (
lim (
lim_in_cod1 Rseq))
proof
let Rseq be
Function of
[:
NAT ,
NAT :],
REAL ;
assume
A1: (
lim_in_cod1 Rseq) is
convergent;
then
consider g be
Real such that
A2: for p be
Real st
0
< p holds ex M be
Nat st for m be
Nat st M
<= m holds
|.(((
lim_in_cod1 Rseq)
. m)
- g) qua
Complex.|
< p by
SEQ_2:def 6;
g
= (
lim (
lim_in_cod1 Rseq)) by
A1,
A2,
SEQ_2:def 7;
hence thesis by
A1,
A2,
DBLSEQ_1:def 7;
end;
theorem ::
DBLSEQ_3:30
for Rseq be
Function of
[:
NAT ,
NAT :],
REAL st (
lim_in_cod2 Rseq) is
convergent holds (
cod2_major_iterated_lim Rseq)
= (
lim (
lim_in_cod2 Rseq))
proof
let Rseq be
Function of
[:
NAT ,
NAT :],
REAL ;
assume
A1: (
lim_in_cod2 Rseq) is
convergent;
then
consider g be
Real such that
A2: for p be
Real st
0
< p holds ex M be
Nat st for m be
Nat st M
<= m holds
|.(((
lim_in_cod2 Rseq)
. m)
- g) qua
Complex.|
< p by
SEQ_2:def 6;
g
= (
lim (
lim_in_cod2 Rseq)) by
A1,
A2,
SEQ_2:def 7;
hence thesis by
A1,
A2,
DBLSEQ_1:def 8;
end;
definition
let Eseq be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def1
attr Eseq is
P-convergent_to_finite_number means ex p be
Real st for e be
Real st
0
< e holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((Eseq
. (n,m))
- p qua
ExtReal).|
< e;
::
DBLSEQ_3:def2
attr Eseq is
P-convergent_to_+infty means for g be
Real st
0
< g holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds g
<= (Eseq
. (n,m));
::
DBLSEQ_3:def3
attr Eseq is
P-convergent_to_-infty means for g be
Real st g
<
0 holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds (Eseq
. (n,m))
<= g;
end
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def4
attr f is
convergent_in_cod1_to_+infty means for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
convergent_to_+infty;
::
DBLSEQ_3:def5
attr f is
convergent_in_cod1_to_-infty means for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
convergent_to_-infty;
::
DBLSEQ_3:def6
attr f is
convergent_in_cod1_to_finite means for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
convergent_to_finite_number;
::
DBLSEQ_3:def7
attr f is
convergent_in_cod1 means for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
convergent;
::
DBLSEQ_3:def8
attr f is
convergent_in_cod2_to_+infty means for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
convergent_to_+infty;
::
DBLSEQ_3:def9
attr f is
convergent_in_cod2_to_-infty means for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
convergent_to_-infty;
::
DBLSEQ_3:def10
attr f is
convergent_in_cod2_to_finite means for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
convergent_to_finite_number;
::
DBLSEQ_3:def11
attr f is
convergent_in_cod2 means for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
convergent;
end
theorem ::
DBLSEQ_3:31
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (f is
convergent_in_cod1_to_+infty or f is
convergent_in_cod1_to_-infty or f is
convergent_in_cod1_to_finite implies f is
convergent_in_cod1) & (f is
convergent_in_cod2_to_+infty or f is
convergent_in_cod2_to_-infty or f is
convergent_in_cod2_to_finite implies f is
convergent_in_cod2)
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
hereby
assume
A1: f is
convergent_in_cod1_to_+infty or f is
convergent_in_cod1_to_-infty or f is
convergent_in_cod1_to_finite;
per cases by
A1;
suppose
A2: f is
convergent_in_cod1_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m)) is
convergent_to_+infty by
A2;
hence (
ProjMap2 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod1;
end;
suppose
A3: f is
convergent_in_cod1_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m)) is
convergent_to_-infty by
A3;
hence (
ProjMap2 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod1;
end;
suppose
A4: f is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m)) is
convergent_to_finite_number by
A4;
hence (
ProjMap2 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod1;
end;
end;
assume
A5: f is
convergent_in_cod2_to_+infty or f is
convergent_in_cod2_to_-infty or f is
convergent_in_cod2_to_finite;
per cases by
A5;
suppose
A6: f is
convergent_in_cod2_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m)) is
convergent_to_+infty by
A6;
hence (
ProjMap1 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod2;
end;
suppose
A7: f is
convergent_in_cod2_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m)) is
convergent_to_-infty by
A7;
hence (
ProjMap1 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod2;
end;
suppose
A8: f is
convergent_in_cod2_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m)) is
convergent_to_finite_number by
A8;
hence (
ProjMap1 (f,m)) is
convergent;
end;
hence f is
convergent_in_cod2;
end;
end;
theorem ::
DBLSEQ_3:32
Th32: for X,Y,Z be non
empty
set, F be
Function of
[:X, Y:], Z, x be
Element of X holds (
ProjMap1 (F,x))
= (
ProjMap2 ((
~ F),x))
proof
let X,Y,Z be non
empty
set;
let F be
Function of
[:X, Y:], Z;
let x be
Element of X;
now
let y be
Element of Y;
((
ProjMap1 (F,x))
. y)
= (F
. (x,y)) by
MESFUNC9:def 6
.= ((
~ F)
. (y,x)) by
FUNCT_4:def 8;
hence ((
ProjMap1 (F,x))
. y)
= ((
ProjMap2 ((
~ F),x))
. y) by
MESFUNC9:def 7;
end;
hence (
ProjMap1 (F,x))
= (
ProjMap2 ((
~ F),x)) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:33
Th33: for X,Y,Z be non
empty
set, F be
Function of
[:X, Y:], Z, y be
Element of Y holds (
ProjMap2 (F,y))
= (
ProjMap1 ((
~ F),y))
proof
let X,Y,Z be non
empty
set;
let F be
Function of
[:X, Y:], Z;
let y be
Element of Y;
now
let x be
Element of X;
((
ProjMap2 (F,y))
. x)
= (F
. (x,y)) by
MESFUNC9:def 7
.= ((
~ F)
. (y,x)) by
FUNCT_4:def 8;
hence ((
ProjMap2 (F,y))
. x)
= ((
ProjMap1 ((
~ F),y))
. x) by
MESFUNC9:def 6;
end;
hence (
ProjMap2 (F,y))
= (
ProjMap1 ((
~ F),y)) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:34
Th34: for X,Y be non
empty
set, F be
Function of
[:X, Y:],
ExtREAL , x be
Element of X holds (
ProjMap1 ((
- F),x))
= (
- (
ProjMap1 (F,x)))
proof
let X,Y be non
empty
set, F be
Function of
[:X, Y:],
ExtREAL , x be
Element of X;
now
let y be
Element of Y;
[x, y]
in
[:X, Y:] by
ZFMISC_1: 87;
then
A1:
[x, y]
in (
dom (
- F)) by
FUNCT_2:def 1;
A2: (
dom (
- (
ProjMap1 (F,x))))
= Y by
FUNCT_2:def 1;
((
ProjMap1 ((
- F),x))
. y)
= ((
- F)
. (x,y)) by
MESFUNC9:def 6;
then ((
ProjMap1 ((
- F),x))
. y)
= (
- (F
. (x,y))) by
A1,
MESFUNC1:def 7;
then ((
ProjMap1 ((
- F),x))
. y)
= (
- ((
ProjMap1 (F,x))
. y)) by
MESFUNC9:def 6;
hence ((
ProjMap1 ((
- F),x))
. y)
= ((
- (
ProjMap1 (F,x)))
. y) by
A2,
MESFUNC1:def 7;
end;
hence (
ProjMap1 ((
- F),x))
= (
- (
ProjMap1 (F,x))) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:35
Th35: for X,Y be non
empty
set, F be
Function of
[:X, Y:],
ExtREAL , y be
Element of Y holds (
ProjMap2 ((
- F),y))
= (
- (
ProjMap2 (F,y)))
proof
let X,Y be non
empty
set, F be
Function of
[:X, Y:],
ExtREAL , y be
Element of Y;
now
let x be
Element of X;
[x, y]
in
[:X, Y:] by
ZFMISC_1: 87;
then
A1:
[x, y]
in (
dom (
- F)) by
FUNCT_2:def 1;
A2: (
dom (
- (
ProjMap2 (F,y))))
= X by
FUNCT_2:def 1;
((
ProjMap2 ((
- F),y))
. x)
= ((
- F)
. (x,y)) by
MESFUNC9:def 7;
then ((
ProjMap2 ((
- F),y))
. x)
= (
- (F
. (x,y))) by
A1,
MESFUNC1:def 7;
then ((
ProjMap2 ((
- F),y))
. x)
= (
- ((
ProjMap2 (F,y))
. x)) by
MESFUNC9:def 7;
hence ((
ProjMap2 ((
- F),y))
. x)
= ((
- (
ProjMap2 (F,y)))
. x) by
A2,
MESFUNC1:def 7;
end;
hence (
ProjMap2 ((
- F),y))
= (
- (
ProjMap2 (F,y))) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:36
Th36: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (f is
convergent_in_cod1_to_+infty iff (
~ f) is
convergent_in_cod2_to_+infty) & (f is
convergent_in_cod2_to_+infty iff (
~ f) is
convergent_in_cod1_to_+infty) & (f is
convergent_in_cod1_to_-infty iff (
~ f) is
convergent_in_cod2_to_-infty) & (f is
convergent_in_cod2_to_-infty iff (
~ f) is
convergent_in_cod1_to_-infty) & (f is
convergent_in_cod1_to_finite iff (
~ f) is
convergent_in_cod2_to_finite) & (f is
convergent_in_cod2_to_finite iff (
~ f) is
convergent_in_cod1_to_finite)
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
now
hereby
assume
A1: f is
convergent_in_cod1_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap1 ((
~ f),m)) is
convergent_to_+infty by
A1;
end;
hence (
~ f) is
convergent_in_cod2_to_+infty;
end;
assume
A2: (
~ f) is
convergent_in_cod2_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap2 (f,m)) is
convergent_to_+infty by
A2;
end;
hence f is
convergent_in_cod1_to_+infty;
end;
hence f is
convergent_in_cod1_to_+infty iff (
~ f) is
convergent_in_cod2_to_+infty;
now
hereby
assume
A3: f is
convergent_in_cod2_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap2 ((
~ f),m)) is
convergent_to_+infty by
A3;
end;
hence (
~ f) is
convergent_in_cod1_to_+infty;
end;
assume
A4: (
~ f) is
convergent_in_cod1_to_+infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap1 (f,m)) is
convergent_to_+infty by
A4;
end;
hence f is
convergent_in_cod2_to_+infty;
end;
hence f is
convergent_in_cod2_to_+infty iff (
~ f) is
convergent_in_cod1_to_+infty;
now
hereby
assume
A5: f is
convergent_in_cod1_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap1 ((
~ f),m)) is
convergent_to_-infty by
A5;
end;
hence (
~ f) is
convergent_in_cod2_to_-infty;
end;
assume
A6: (
~ f) is
convergent_in_cod2_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap2 (f,m)) is
convergent_to_-infty by
A6;
end;
hence f is
convergent_in_cod1_to_-infty;
end;
hence f is
convergent_in_cod1_to_-infty iff (
~ f) is
convergent_in_cod2_to_-infty;
now
hereby
assume
A7: f is
convergent_in_cod2_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap2 ((
~ f),m)) is
convergent_to_-infty by
A7;
end;
hence (
~ f) is
convergent_in_cod1_to_-infty;
end;
assume
A8: (
~ f) is
convergent_in_cod1_to_-infty;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap1 (f,m)) is
convergent_to_-infty by
A8;
end;
hence f is
convergent_in_cod2_to_-infty;
end;
hence f is
convergent_in_cod2_to_-infty iff (
~ f) is
convergent_in_cod1_to_-infty;
now
hereby
assume
A9: f is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap1 ((
~ f),m)) is
convergent_to_finite_number by
A9;
end;
hence (
~ f) is
convergent_in_cod2_to_finite;
end;
assume
A10: (
~ f) is
convergent_in_cod2_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m))
= (
ProjMap1 ((
~ f),m)) by
Th33;
hence (
ProjMap2 (f,m)) is
convergent_to_finite_number by
A10;
end;
hence f is
convergent_in_cod1_to_finite;
end;
hence f is
convergent_in_cod1_to_finite iff (
~ f) is
convergent_in_cod2_to_finite;
now
hereby
assume
A11: f is
convergent_in_cod2_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap2 ((
~ f),m)) is
convergent_to_finite_number by
A11;
end;
hence (
~ f) is
convergent_in_cod1_to_finite;
end;
assume
A12: (
~ f) is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (
ProjMap1 (f,m)) is
convergent_to_finite_number by
A12;
end;
hence f is
convergent_in_cod2_to_finite;
end;
hence f is
convergent_in_cod2_to_finite iff (
~ f) is
convergent_in_cod1_to_finite;
end;
theorem ::
DBLSEQ_3:37
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (f is
convergent_in_cod1_to_+infty iff (
- f) is
convergent_in_cod1_to_-infty) & (f is
convergent_in_cod1_to_-infty iff (
- f) is
convergent_in_cod1_to_+infty) & (f is
convergent_in_cod1_to_finite iff (
- f) is
convergent_in_cod1_to_finite) & (f is
convergent_in_cod2_to_+infty iff (
- f) is
convergent_in_cod2_to_-infty) & (f is
convergent_in_cod2_to_-infty iff (
- f) is
convergent_in_cod2_to_+infty) & (f is
convergent_in_cod2_to_finite iff (
- f) is
convergent_in_cod2_to_finite)
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
now
hereby
assume
A1: f is
convergent_in_cod1_to_+infty;
now
let m be
Element of
NAT ;
A2: (
ProjMap2 (f,m)) is
convergent_to_+infty by
A1;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
hence (
ProjMap2 ((
- f),m)) is
convergent_to_-infty by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod1_to_-infty;
end;
assume
A3: (
- f) is
convergent_in_cod1_to_-infty;
now
let m be
Element of
NAT ;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
then (
- (
ProjMap2 (f,m))) is
convergent_to_-infty by
A3;
then (
- (
- (
ProjMap2 (f,m)))) is
convergent_to_+infty by
Th17;
hence (
ProjMap2 (f,m)) is
convergent_to_+infty by
Th2;
end;
hence f is
convergent_in_cod1_to_+infty;
end;
hence f is
convergent_in_cod1_to_+infty iff (
- f) is
convergent_in_cod1_to_-infty;
now
hereby
assume
A1: f is
convergent_in_cod1_to_-infty;
now
let m be
Element of
NAT ;
A2: (
ProjMap2 (f,m)) is
convergent_to_-infty by
A1;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
hence (
ProjMap2 ((
- f),m)) is
convergent_to_+infty by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod1_to_+infty;
end;
assume
A3: (
- f) is
convergent_in_cod1_to_+infty;
now
let m be
Element of
NAT ;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
then (
- (
ProjMap2 (f,m))) is
convergent_to_+infty by
A3;
then (
- (
- (
ProjMap2 (f,m)))) is
convergent_to_-infty by
Th17;
hence (
ProjMap2 (f,m)) is
convergent_to_-infty by
Th2;
end;
hence f is
convergent_in_cod1_to_-infty;
end;
hence f is
convergent_in_cod1_to_-infty iff (
- f) is
convergent_in_cod1_to_+infty;
now
hereby
assume
A1: f is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
A2: (
ProjMap2 (f,m)) is
convergent_to_finite_number by
A1;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
hence (
ProjMap2 ((
- f),m)) is
convergent_to_finite_number by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod1_to_finite;
end;
assume
A3: (
- f) is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
(
- (
ProjMap2 (f,m)))
= (
ProjMap2 ((
- f),m)) by
Th35;
then (
- (
ProjMap2 (f,m))) is
convergent_to_finite_number by
A3;
then (
- (
- (
ProjMap2 (f,m)))) is
convergent_to_finite_number by
Th17;
hence (
ProjMap2 (f,m)) is
convergent_to_finite_number by
Th2;
end;
hence f is
convergent_in_cod1_to_finite;
end;
hence f is
convergent_in_cod1_to_finite iff (
- f) is
convergent_in_cod1_to_finite;
now
hereby
assume
A1: f is
convergent_in_cod2_to_+infty;
now
let m be
Element of
NAT ;
A2: (
ProjMap1 (f,m)) is
convergent_to_+infty by
A1;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
hence (
ProjMap1 ((
- f),m)) is
convergent_to_-infty by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod2_to_-infty;
end;
assume
A3: (
- f) is
convergent_in_cod2_to_-infty;
now
let m be
Element of
NAT ;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
then (
- (
ProjMap1 (f,m))) is
convergent_to_-infty by
A3;
then (
- (
- (
ProjMap1 (f,m)))) is
convergent_to_+infty by
Th17;
hence (
ProjMap1 (f,m)) is
convergent_to_+infty by
Th2;
end;
hence f is
convergent_in_cod2_to_+infty;
end;
hence f is
convergent_in_cod2_to_+infty iff (
- f) is
convergent_in_cod2_to_-infty;
now
hereby
assume
A1: f is
convergent_in_cod2_to_-infty;
now
let m be
Element of
NAT ;
A2: (
ProjMap1 (f,m)) is
convergent_to_-infty by
A1;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
hence (
ProjMap1 ((
- f),m)) is
convergent_to_+infty by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod2_to_+infty;
end;
assume
A3: (
- f) is
convergent_in_cod2_to_+infty;
now
let m be
Element of
NAT ;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
then (
- (
ProjMap1 (f,m))) is
convergent_to_+infty by
A3;
then (
- (
- (
ProjMap1 (f,m)))) is
convergent_to_-infty by
Th17;
hence (
ProjMap1 (f,m)) is
convergent_to_-infty by
Th2;
end;
hence f is
convergent_in_cod2_to_-infty;
end;
hence f is
convergent_in_cod2_to_-infty iff (
- f) is
convergent_in_cod2_to_+infty;
now
hereby
assume
A1: f is
convergent_in_cod2_to_finite;
now
let m be
Element of
NAT ;
A2: (
ProjMap1 (f,m)) is
convergent_to_finite_number by
A1;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
hence (
ProjMap1 ((
- f),m)) is
convergent_to_finite_number by
A2,
Th17;
end;
hence (
- f) is
convergent_in_cod2_to_finite;
end;
assume
A3: (
- f) is
convergent_in_cod2_to_finite;
now
let m be
Element of
NAT ;
(
- (
ProjMap1 (f,m)))
= (
ProjMap1 ((
- f),m)) by
Th34;
then (
- (
ProjMap1 (f,m))) is
convergent_to_finite_number by
A3;
then (
- (
- (
ProjMap1 (f,m)))) is
convergent_to_finite_number by
Th17;
hence (
ProjMap1 (f,m)) is
convergent_to_finite_number by
Th2;
end;
hence f is
convergent_in_cod2_to_finite;
end;
hence f is
convergent_in_cod2_to_finite iff (
- f) is
convergent_in_cod2_to_finite;
end;
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def12
func
lim_in_cod1 f ->
ExtREAL_sequence means
:
D1DEF5: for m be
Element of
NAT holds (it
. m)
= (
lim (
ProjMap2 (f,m)));
existence
proof
defpred
P[
Element of
NAT ,
set] means $2
= (
lim (
ProjMap2 (f,$1)));
A1: for m be
Element of
NAT holds ex n be
Element of
ExtREAL st
P[m, n];
consider IT be
Function of
NAT ,
ExtREAL such that
A2: for m be
Element of
NAT holds
P[m, (IT
. m)] from
FUNCT_2:sch 3(
A1);
take IT;
thus thesis by
A2;
end;
uniqueness
proof
let f1,f2 be
Function of
NAT ,
ExtREAL ;
assume that
A3: for m be
Element of
NAT holds (f1
. m)
= (
lim (
ProjMap2 (f,m))) and
A4: for m be
Element of
NAT holds (f2
. m)
= (
lim (
ProjMap2 (f,m)));
now
let m be
Element of
NAT ;
thus (f1
. m)
= (
lim (
ProjMap2 (f,m))) by
A3
.= (f2
. m) by
A4;
end;
hence f1
= f2 by
FUNCT_2: 63;
end;
::
DBLSEQ_3:def13
func
lim_in_cod2 f ->
ExtREAL_sequence means
:
D1DEF6: for n be
Element of
NAT holds (it
. n)
= (
lim (
ProjMap1 (f,n)));
existence
proof
defpred
P[
Element of
NAT ,
set] means $2
= (
lim (
ProjMap1 (f,$1)));
A1: for m be
Element of
NAT holds ex n be
Element of
ExtREAL st
P[m, n];
consider IT be
Function of
NAT ,
ExtREAL such that
A2: for m be
Element of
NAT holds
P[m, (IT
. m)] from
FUNCT_2:sch 3(
A1);
take IT;
thus thesis by
A2;
end;
uniqueness
proof
let f1,f2 be
Function of
NAT ,
ExtREAL ;
assume that
A3: for n be
Element of
NAT holds (f1
. n)
= (
lim (
ProjMap1 (f,n))) and
A4: for n be
Element of
NAT holds (f2
. n)
= (
lim (
ProjMap1 (f,n)));
now
let n be
Element of
NAT ;
thus (f1
. n)
= (
lim (
ProjMap1 (f,n))) by
A3
.= (f2
. n) by
A4;
end;
hence f1
= f2 by
FUNCT_2: 63;
end;
end
theorem ::
DBLSEQ_3:38
Th38: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
lim_in_cod1 f)
= (
lim_in_cod2 (
~ f)) & (
lim_in_cod2 f)
= (
lim_in_cod1 (
~ f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
now
let n be
Element of
NAT ;
((
lim_in_cod1 f)
. n)
= (
lim (
ProjMap2 (f,n))) by
D1DEF5
.= (
lim (
ProjMap1 ((
~ f),n))) by
Th33;
hence ((
lim_in_cod1 f)
. n)
= ((
lim_in_cod2 (
~ f))
. n) by
D1DEF6;
end;
hence (
lim_in_cod1 f)
= (
lim_in_cod2 (
~ f)) by
FUNCT_2:def 8;
now
let n be
Element of
NAT ;
((
lim_in_cod2 f)
. n)
= (
lim (
ProjMap1 (f,n))) by
D1DEF6
.= (
lim (
ProjMap2 ((
~ f),n))) by
Th32;
hence ((
lim_in_cod2 f)
. n)
= ((
lim_in_cod1 (
~ f))
. n) by
D1DEF5;
end;
hence (
lim_in_cod2 f)
= (
lim_in_cod1 (
~ f)) by
FUNCT_2:def 8;
end;
registration
let X,Y be non
empty
set, F be
without+infty
Function of
[:X, Y:],
ExtREAL , x be
Element of X;
cluster (
ProjMap1 (F,x)) ->
without+infty;
correctness
proof
now
let y be
object;
per cases ;
suppose not y
in (
dom (
ProjMap1 (F,x)));
hence ((
ProjMap1 (F,x))
. y)
<
+infty by
FUNCT_1:def 2;
end;
suppose y
in (
dom (
ProjMap1 (F,x)));
then
reconsider y1 = y as
Element of Y;
((
ProjMap1 (F,x))
. y)
= (F
. (x,y1)) by
MESFUNC9:def 6;
hence ((
ProjMap1 (F,x))
. y)
<
+infty by
MESFUNC5:def 6;
end;
end;
hence thesis by
MESFUNC5:def 6;
end;
end
registration
let X,Y be non
empty
set, F be
without+infty
Function of
[:X, Y:],
ExtREAL , y be
Element of Y;
cluster (
ProjMap2 (F,y)) ->
without+infty;
correctness
proof
now
let x be
object;
per cases ;
suppose not x
in (
dom (
ProjMap2 (F,y)));
hence ((
ProjMap2 (F,y))
. x)
<
+infty by
FUNCT_1:def 2;
end;
suppose x
in (
dom (
ProjMap2 (F,y)));
then
reconsider x1 = x as
Element of X;
((
ProjMap2 (F,y))
. x)
= (F
. (x1,y)) by
MESFUNC9:def 7;
hence ((
ProjMap2 (F,y))
. x)
<
+infty by
MESFUNC5:def 6;
end;
end;
hence thesis by
MESFUNC5:def 6;
end;
end
registration
let X,Y be non
empty
set, F be
without-infty
Function of
[:X, Y:],
ExtREAL , x be
Element of X;
cluster (
ProjMap1 (F,x)) ->
without-infty;
correctness
proof
now
let y be
object;
per cases ;
suppose not y
in (
dom (
ProjMap1 (F,x)));
hence
-infty
< ((
ProjMap1 (F,x))
. y) by
FUNCT_1:def 2;
end;
suppose y
in (
dom (
ProjMap1 (F,x)));
then
reconsider y1 = y as
Element of Y;
((
ProjMap1 (F,x))
. y)
= (F
. (x,y1)) by
MESFUNC9:def 6;
hence
-infty
< ((
ProjMap1 (F,x))
. y) by
MESFUNC5:def 5;
end;
end;
hence thesis by
MESFUNC5:def 5;
end;
end
registration
let X,Y be non
empty
set, F be
without-infty
Function of
[:X, Y:],
ExtREAL , y be
Element of Y;
cluster (
ProjMap2 (F,y)) ->
without-infty;
correctness
proof
now
let x be
object;
per cases ;
suppose not x
in (
dom (
ProjMap2 (F,y)));
hence
-infty
< ((
ProjMap2 (F,y))
. x) by
FUNCT_1:def 2;
end;
suppose x
in (
dom (
ProjMap2 (F,y)));
then
reconsider x1 = x as
Element of X;
((
ProjMap2 (F,y))
. x)
= (F
. (x1,y)) by
MESFUNC9:def 7;
hence
-infty
< ((
ProjMap2 (F,y))
. x) by
MESFUNC5:def 5;
end;
end;
hence thesis by
MESFUNC5:def 5;
end;
end
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def14
func
Partial_Sums_in_cod2 f ->
Function of
[:
NAT ,
NAT :],
ExtREAL means
:
DefCSM: for n,m be
Nat holds (it
. (n,
0 ))
= (f
. (n,
0 )) & (it
. (n,(m
+ 1)))
= ((it
. (n,m))
+ (f
. (n,(m
+ 1))));
existence
proof
deffunc
F0(
Element of
NAT ) = (f
. ($1,
0 ));
consider f0 be
Function of
NAT ,
ExtREAL such that
A1: for n be
Element of
NAT holds (f0
. n)
=
F0(n) from
FUNCT_2:sch 4;
deffunc
F(
Element of
ExtREAL ,
Nat,
Nat) = ($1
+ (f
. ($2,($3
+ 1))));
consider IT be
Function of
[:
NAT ,
NAT :],
ExtREAL such that
A2: for a be
Element of
NAT holds (IT
. (a,
0 ))
= (f0
. a) & for n be
Nat holds (IT
. (a,(n
+ 1)))
=
F(.,a,n) from
DBLSEQ_2:sch 1;
take IT;
hereby
let n,m be
Nat;
A3: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then (IT
. (n,
0 ))
= (f0
. n) by
A2;
hence (IT
. (n,
0 ))
= (f
. (n,
0 )) & (IT
. (n,(m
+ 1)))
= ((IT
. (n,m))
+ (f
. (n,(m
+ 1)))) by
A1,
A2,
A3;
end;
end;
uniqueness
proof
let f1,f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume that
A1: for n,m be
natural
number holds (f1
. (n,
0 ))
= (f
. (n,
0 )) & (f1
. (n,(m
+ 1)))
= ((f1
. (n,m))
+ (f
. (n,(m
+ 1)))) and
A2: for n,m be
natural
number holds (f2
. (n,
0 ))
= (f
. (n,
0 )) & (f2
. (n,(m
+ 1)))
= ((f2
. (n,m))
+ (f
. (n,(m
+ 1))));
A3: (
dom f1)
=
[:
NAT ,
NAT :] & (
dom f2)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
for n,m be
object st n
in
NAT & m
in
NAT holds (f1
. (n,m))
= (f2
. (n,m))
proof
let n,m be
object;
assume n
in
NAT & m
in
NAT ;
then
reconsider n1 = n, m1 = m as
Element of
NAT ;
defpred
P[
Nat] means (f1
. (n1,$1))
= (f2
. (n1,$1));
(f1
. (n1,
0 ))
= (f
. (n1,
0 )) by
A1;
then
A4:
P[
0 ] by
A2;
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
(f1
. (n1,(k
+ 1)))
= ((f1
. (n1,k))
+ (f
. (n1,(k
+ 1)))) by
A1;
hence (f1
. (n1,(k
+ 1)))
= (f2
. (n1,(k
+ 1))) by
A2,
A6;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A4,
A5);
then
P[m1];
hence (f1
. (n,m))
= (f2
. (n,m));
end;
hence thesis by
A3,
FUNCT_3: 6;
end;
end
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def15
func
Partial_Sums_in_cod1 f ->
Function of
[:
NAT ,
NAT :],
ExtREAL means
:
DefRSM: for n,m be
Nat holds (it
. (
0 ,m))
= (f
. (
0 ,m)) & (it
. ((n
+ 1),m))
= ((it
. (n,m))
+ (f
. ((n
+ 1),m)));
existence
proof
deffunc
F0(
Element of
NAT ) = (f
. (
0 ,$1));
consider f0 be
Function of
NAT ,
ExtREAL such that
A1: for n be
Element of
NAT holds (f0
. n)
=
F0(n) from
FUNCT_2:sch 4;
deffunc
F(
Element of
ExtREAL ,
Nat,
Nat) = ($1
+ (f
. (($3
+ 1),$2)));
consider IT be
Function of
[:
NAT ,
NAT :],
ExtREAL such that
A2: for a be
Element of
NAT holds (IT
. (
0 ,a))
= (f0
. a) & for n be
Nat holds (IT
. ((n
+ 1),a))
=
F(.,a,n) from
DBLSEQ_2:sch 3;
take IT;
hereby
let n,m be
Nat;
A3: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then (IT
. (
0 ,m))
= (f0
. m) by
A2;
hence (IT
. (
0 ,m))
= (f
. (
0 ,m)) & (IT
. ((n
+ 1),m))
= ((IT
. (n,m))
+ (f
. ((n
+ 1),m))) by
A1,
A2,
A3;
end;
end;
uniqueness
proof
let f1,f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume that
A1: for n,m be
natural
number holds (f1
. (
0 ,m))
= (f
. (
0 ,m)) & (f1
. ((n
+ 1),m))
= ((f1
. (n,m))
+ (f
. ((n
+ 1),m))) and
A2: for n,m be
natural
number holds (f2
. (
0 ,m))
= (f
. (
0 ,m)) & (f2
. ((n
+ 1),m))
= ((f2
. (n,m))
+ (f
. ((n
+ 1),m)));
A3: (
dom f1)
=
[:
NAT ,
NAT :] & (
dom f2)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
for n,m be
object st n
in
NAT & m
in
NAT holds (f1
. (n,m))
= (f2
. (n,m))
proof
let n,m be
object;
assume n
in
NAT & m
in
NAT ;
then
reconsider n1 = n, m1 = m as
Element of
NAT ;
defpred
P[
Nat] means (f1
. ($1,m1))
= (f2
. ($1,m1));
(f1
. (
0 ,m1))
= (f
. (
0 ,m1)) by
A1;
then
A4:
P[
0 ] by
A2;
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
(f1
. ((k
+ 1),m1))
= ((f1
. (k,m1))
+ (f
. ((k
+ 1),m1))) by
A1;
hence (f1
. ((k
+ 1),m1))
= (f2
. ((k
+ 1),m1)) by
A2,
A6;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A4,
A5);
then
P[n1];
hence (f1
. (n,m))
= (f2
. (n,m));
end;
hence thesis by
A3,
FUNCT_3: 6;
end;
end
registration
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod2 f) ->
without-infty;
correctness
proof
for x be
object holds
-infty
< ((
Partial_Sums_in_cod2 f)
. x)
proof
let x be
object;
per cases ;
suppose x
in (
dom (
Partial_Sums_in_cod2 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & x
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
>
-infty ;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then
A2:
P[
0 ] by
MESFUNC5:def 5;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. (n,(k
+ 1)))
>
-infty by
MESFUNC5:def 5;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence
P[(k
+ 1)] by
A4,
A5,
XXREAL_3: 17,
XXREAL_0: 6;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then
P[m];
hence
-infty
< ((
Partial_Sums_in_cod2 f)
. x) by
A1;
end;
suppose not x
in (
dom (
Partial_Sums_in_cod2 f));
hence
-infty
< ((
Partial_Sums_in_cod2 f)
. x) by
FUNCT_1:def 2;
end;
end;
hence (
Partial_Sums_in_cod2 f) is
without-infty by
MESFUNC5:def 5;
end;
end
registration
let f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod2 f) ->
without+infty;
correctness
proof
for x be
object holds
+infty
> ((
Partial_Sums_in_cod2 f)
. x)
proof
let x be
object;
per cases ;
suppose x
in (
dom (
Partial_Sums_in_cod2 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & x
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
<
+infty ;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then
A2:
P[
0 ] by
MESFUNC5:def 6;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. (n,(k
+ 1)))
<
+infty by
MESFUNC5:def 6;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence
P[(k
+ 1)] by
A4,
A5,
XXREAL_3: 16,
XXREAL_0: 4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then
P[m];
hence
+infty
> ((
Partial_Sums_in_cod2 f)
. x) by
A1;
end;
suppose not x
in (
dom (
Partial_Sums_in_cod2 f));
hence
+infty
> ((
Partial_Sums_in_cod2 f)
. x) by
FUNCT_1:def 2;
end;
end;
hence (
Partial_Sums_in_cod2 f) is
without+infty by
MESFUNC5:def 6;
end;
end
registration
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod2 f) ->
nonnegative;
correctness
proof
for z be
object st z
in (
dom (
Partial_Sums_in_cod2 f)) holds
0.
<= ((
Partial_Sums_in_cod2 f)
. z)
proof
let z be
object;
assume z
in (
dom (
Partial_Sums_in_cod2 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
>=
0 ;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then
A2:
P[
0 ] by
SUPINF_2: 51;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. (n,(k
+ 1)))
>=
0 by
SUPINF_2: 51;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
>=
0 by
A5,
A4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then ((
Partial_Sums_in_cod2 f)
. (n,m))
>=
0 ;
hence
0.
<= ((
Partial_Sums_in_cod2 f)
. z) by
A1;
end;
hence thesis by
SUPINF_2: 52;
end;
end
registration
let f be
nonpositive
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod2 f) ->
nonpositive;
correctness
proof
for z be
set st z
in (
dom (
Partial_Sums_in_cod2 f)) holds
0.
>= ((
Partial_Sums_in_cod2 f)
. z)
proof
let z be
set;
assume z
in (
dom (
Partial_Sums_in_cod2 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
<=
0 ;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then
A2:
P[
0 ] by
MESFUNC5: 8;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. (n,(k
+ 1)))
<=
0 by
MESFUNC5: 8;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
<=
0 by
A5,
A4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then ((
Partial_Sums_in_cod2 f)
. (n,m))
<=
0 ;
hence
0.
>= ((
Partial_Sums_in_cod2 f)
. z) by
A1;
end;
hence thesis by
MESFUNC5: 9;
end;
end
registration
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod1 f) ->
without-infty;
correctness
proof
for x be
object holds
-infty
< ((
Partial_Sums_in_cod1 f)
. x)
proof
let x be
object;
per cases ;
suppose x
in (
dom (
Partial_Sums_in_cod1 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & x
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,m))
>
-infty ;
((
Partial_Sums_in_cod1 f)
. (
0 ,m))
= (f
. (
0 ,m)) by
DefRSM;
then
A2:
P[
0 ] by
MESFUNC5:def 5;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. ((k
+ 1),m))
>
-infty by
MESFUNC5:def 5;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM;
hence
P[(k
+ 1)] by
A4,
A5,
XXREAL_3: 17,
XXREAL_0: 6;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then
P[n];
hence
-infty
< ((
Partial_Sums_in_cod1 f)
. x) by
A1;
end;
suppose not x
in (
dom (
Partial_Sums_in_cod1 f));
hence
-infty
< ((
Partial_Sums_in_cod1 f)
. x) by
FUNCT_1:def 2;
end;
end;
hence (
Partial_Sums_in_cod1 f) is
without-infty by
MESFUNC5:def 5;
end;
end
registration
let f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod1 f) ->
without+infty;
correctness
proof
for x be
object holds
+infty
> ((
Partial_Sums_in_cod1 f)
. x)
proof
let x be
object;
per cases ;
suppose x
in (
dom (
Partial_Sums_in_cod1 f));
then
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & x
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,m))
<
+infty ;
((
Partial_Sums_in_cod1 f)
. (
0 ,m))
= (f
. (
0 ,m)) by
DefRSM;
then
A2:
P[
0 ] by
MESFUNC5:def 6;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. ((k
+ 1),m))
<
+infty by
MESFUNC5:def 6;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM;
hence
P[(k
+ 1)] by
A4,
A5,
XXREAL_3: 16,
XXREAL_0: 4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then
P[n];
hence
+infty
> ((
Partial_Sums_in_cod1 f)
. x) by
A1;
end;
suppose not x
in (
dom (
Partial_Sums_in_cod1 f));
hence
+infty
> ((
Partial_Sums_in_cod1 f)
. x) by
FUNCT_1:def 2;
end;
end;
hence (
Partial_Sums_in_cod1 f) is
without+infty by
MESFUNC5:def 6;
end;
end
registration
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod1 f) ->
nonnegative;
correctness
proof
for z be
object st z
in (
dom (
Partial_Sums_in_cod1 f)) holds
0.
<= ((
Partial_Sums_in_cod1 f)
. z)
proof
let z be
object;
assume z
in (
dom (
Partial_Sums_in_cod1 f));
then
consider m,n be
object such that
A1: m
in
NAT & n
in
NAT & z
=
[m, n] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,n))
>=
0 ;
((
Partial_Sums_in_cod1 f)
. (
0 ,n))
= (f
. (
0 ,n)) by
DefRSM;
then
A2:
P[
0 ] by
SUPINF_2: 51;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. ((k
+ 1),n))
>=
0 by
SUPINF_2: 51;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),n))
= (((
Partial_Sums_in_cod1 f)
. (k,n))
+ (f
. ((k
+ 1),n))) by
DefRSM;
hence ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),n))
>=
0 by
A5,
A4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then ((
Partial_Sums_in_cod1 f)
. (m,n))
>=
0 ;
hence
0.
<= ((
Partial_Sums_in_cod1 f)
. z) by
A1;
end;
hence thesis by
SUPINF_2: 52;
end;
end
registration
let f be
nonpositive
Function of
[:
NAT ,
NAT :],
ExtREAL ;
cluster (
Partial_Sums_in_cod1 f) ->
nonpositive;
correctness
proof
for z be
set st z
in (
dom (
Partial_Sums_in_cod1 f)) holds
0.
>= ((
Partial_Sums_in_cod1 f)
. z)
proof
let z be
set;
assume z
in (
dom (
Partial_Sums_in_cod1 f));
then
consider m,n be
object such that
A1: m
in
NAT & n
in
NAT & z
=
[m, n] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,n))
<=
0 ;
((
Partial_Sums_in_cod1 f)
. (
0 ,n))
= (f
. (
0 ,n)) by
DefRSM;
then
A2:
P[
0 ] by
MESFUNC5: 8;
A3: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P[k];
A5: (f
. ((k
+ 1),n))
<=
0 by
MESFUNC5: 8;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),n))
= (((
Partial_Sums_in_cod1 f)
. (k,n))
+ (f
. ((k
+ 1),n))) by
DefRSM;
hence ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),n))
<=
0 by
A5,
A4;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A3);
then ((
Partial_Sums_in_cod1 f)
. (m,n))
<=
0 ;
hence
0.
>= ((
Partial_Sums_in_cod1 f)
. z) by
A1;
end;
hence thesis by
MESFUNC5: 9;
end;
end
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def16
func
Partial_Sums f ->
Function of
[:
NAT ,
NAT :],
ExtREAL equals (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f));
correctness ;
end
theorem ::
DBLSEQ_3:39
Th39: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((
Partial_Sums_in_cod1 f)
. (n,m))
= ((
Partial_Sums_in_cod2 (
~ f))
. (m,n)) & ((
Partial_Sums_in_cod2 f)
. (n,m))
= ((
Partial_Sums_in_cod1 (
~ f))
. (m,n))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let n,m be
Nat;
reconsider n1 = n, m1 = m as
Element of
NAT by
ORDINAL1:def 12;
defpred
P1[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,m))
= ((
Partial_Sums_in_cod2 (
~ f))
. (m,$1));
((
Partial_Sums_in_cod1 f)
. (
0 ,m))
= (f
. (
0 ,m)) by
DefRSM
.= ((
~ f)
. (m1,
0 )) by
FUNCT_4:def 8;
then
A2:
P1[
0 ] by
DefCSM;
A3: for k be
Nat st
P1[k] holds
P1[(k
+ 1)]
proof
let k be
Nat;
assume
A4:
P1[k];
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM
.= (((
Partial_Sums_in_cod2 (
~ f))
. (m,k))
+ ((
~ f)
. (m1,(k
+ 1)))) by
A4,
FUNCT_4:def 8;
hence
P1[(k
+ 1)] by
DefCSM;
end;
for k be
Nat holds
P1[k] from
NAT_1:sch 2(
A2,
A3);
hence ((
Partial_Sums_in_cod1 f)
. (n,m))
= ((
Partial_Sums_in_cod2 (
~ f))
. (m,n));
defpred
P2[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
= ((
Partial_Sums_in_cod1 (
~ f))
. ($1,n));
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM
.= ((
~ f)
. (
0 ,n1)) by
FUNCT_4:def 8;
then
A5:
P2[
0 ] by
DefRSM;
A6: for k be
Nat st
P2[k] holds
P2[(k
+ 1)]
proof
let k be
Nat;
assume
A7:
P2[k];
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM
.= (((
Partial_Sums_in_cod1 (
~ f))
. (k,n))
+ ((
~ f)
. ((k
+ 1),n1))) by
A7,
FUNCT_4:def 8;
hence
P2[(k
+ 1)] by
DefRSM;
end;
for k be
Nat holds
P2[k] from
NAT_1:sch 2(
A5,
A6);
hence ((
Partial_Sums_in_cod2 f)
. (n,m))
= ((
Partial_Sums_in_cod1 (
~ f))
. (m,n));
end;
theorem ::
DBLSEQ_3:40
Th40: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
~ (
Partial_Sums_in_cod1 f))
= (
Partial_Sums_in_cod2 (
~ f)) & (
~ (
Partial_Sums_in_cod2 f))
= (
Partial_Sums_in_cod1 (
~ f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
now
let z be
Element of
[:
NAT ,
NAT :];
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A1;
((
Partial_Sums_in_cod2 (
~ f))
. z)
= ((
Partial_Sums_in_cod2 (
~ f))
. (n,m)) by
A1
.= ((
Partial_Sums_in_cod1 f)
. (m,n)) by
Th39
.= ((
~ (
Partial_Sums_in_cod1 f))
. (n,m)) by
FUNCT_4:def 8;
hence ((
Partial_Sums_in_cod2 (
~ f))
. z)
= ((
~ (
Partial_Sums_in_cod1 f))
. z) by
A1;
end;
hence (
~ (
Partial_Sums_in_cod1 f))
= (
Partial_Sums_in_cod2 (
~ f)) by
FUNCT_2:def 8;
now
let z be
Element of
[:
NAT ,
NAT :];
consider n,m be
object such that
A2: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A2;
((
Partial_Sums_in_cod1 (
~ f))
. z)
= ((
Partial_Sums_in_cod1 (
~ f))
. (n,m)) by
A2
.= ((
Partial_Sums_in_cod2 f)
. (m,n)) by
Th39
.= ((
~ (
Partial_Sums_in_cod2 f))
. (n,m)) by
FUNCT_4:def 8;
hence ((
Partial_Sums_in_cod1 (
~ f))
. z)
= ((
~ (
Partial_Sums_in_cod2 f))
. z) by
A2;
end;
hence (
~ (
Partial_Sums_in_cod2 f))
= (
Partial_Sums_in_cod1 (
~ f)) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:41
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , g be
ext-real-valued
Function, n be
Nat st (for k be
Nat holds ((
Partial_Sums_in_cod1 f)
. (n,k))
= (g
. k)) holds (for k be
Nat holds ((
Partial_Sums f)
. (n,k))
= ((
Partial_Sums g)
. k)) & ((
lim_in_cod2 (
Partial_Sums f))
. n)
= (
Sum g)
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , g be
ext-real-valued
Function, n be
Nat;
assume
A1: for k be
Nat holds ((
Partial_Sums_in_cod1 f)
. (n,k))
= (g
. k);
A4:
now
let k be
Nat;
defpred
P[
Nat] means ((
Partial_Sums f)
. (n,$1))
= ((
Partial_Sums g)
. $1);
((
Partial_Sums f)
. (n,
0 ))
= ((
Partial_Sums_in_cod1 f)
. (n,
0 )) by
DefCSM
.= (g
.
0 ) by
A1;
then
A2:
P[
0 ] by
MESFUNC9:def 1;
A3: for m be
Nat st
P[m] holds
P[(m
+ 1)]
proof
let m be
Nat;
assume
A4:
P[m];
((
Partial_Sums f)
. (n,(m
+ 1)))
= (((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f))
. (n,m))
+ ((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))) by
DefCSM
.= (((
Partial_Sums g)
. m)
+ (g
. (m
+ 1))) by
A1,
A4;
hence
P[(m
+ 1)] by
MESFUNC9:def 1;
end;
for m be
Nat holds
P[m] from
NAT_1:sch 2(
A2,
A3);
hence ((
Partial_Sums f)
. (n,k))
= ((
Partial_Sums g)
. k);
end;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
now
let k be
Element of
NAT ;
((
ProjMap1 ((
Partial_Sums f),n1))
. k)
= ((
Partial_Sums f)
. (n,k)) by
MESFUNC9:def 6;
hence ((
ProjMap1 ((
Partial_Sums f),n1))
. k)
= ((
Partial_Sums g)
. k) by
A4;
end;
then (
ProjMap1 ((
Partial_Sums f),n1))
= (
Partial_Sums g) by
FUNCT_2:def 8;
then ((
lim_in_cod2 (
Partial_Sums f))
. n1)
= (
lim (
Partial_Sums g)) by
D1DEF6;
hence thesis by
A4,
MESFUNC9:def 3;
end;
theorem ::
DBLSEQ_3:42
Th42: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums_in_cod2 (
- f))
= (
- (
Partial_Sums_in_cod2 f)) & (
Partial_Sums_in_cod1 (
- f))
= (
- (
Partial_Sums_in_cod1 f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A1: (
dom (
- (
Partial_Sums_in_cod2 f)))
=
[:
NAT ,
NAT :] & (
dom (
- (
Partial_Sums_in_cod1 f)))
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
A2: (
dom (
- f))
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
for z be
Element of
[:
NAT ,
NAT :] holds ((
- (
Partial_Sums_in_cod2 f))
. z)
= ((
Partial_Sums_in_cod2 (
- f))
. z)
proof
let z be
Element of
[:
NAT ,
NAT :];
consider n,m be
object such that
A3: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A3;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 (
- f))
. (n,$1))
= (
- ((
Partial_Sums_in_cod2 f)
. (n,$1)));
reconsider z0 =
[n,
0 ] as
Element of
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
A4:
[n,
0 ]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
((
Partial_Sums_in_cod2 (
- f))
. (n,
0 ))
= ((
- f)
. (n,
0 )) by
DefCSM
.= (
- (f
. (n,
0 ))) by
A4,
A2,
MESFUNC1:def 7;
then
A5:
P[
0 ] by
DefCSM;
A6: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A7:
P[k];
A8:
[n, (k
+ 1)]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
thus ((
Partial_Sums_in_cod2 (
- f))
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 (
- f))
. (n,k))
+ ((
- f)
. (n,(k
+ 1)))) by
DefCSM
.= ((
- ((
Partial_Sums_in_cod2 f)
. (n,k)))
- (f
. (n,(k
+ 1)))) by
A7,
A8,
A2,
MESFUNC1:def 7
.= (
- (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1))))) by
XXREAL_3: 25
.= (
- ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))) by
DefCSM;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A5,
A6);
then ((
Partial_Sums_in_cod2 (
- f))
. (n,m))
= (
- ((
Partial_Sums_in_cod2 f)
. (n,m)));
hence thesis by
A3,
A1,
MESFUNC1:def 7;
end;
hence (
Partial_Sums_in_cod2 (
- f))
= (
- (
Partial_Sums_in_cod2 f)) by
FUNCT_2:def 8;
for z be
Element of
[:
NAT ,
NAT :] holds ((
- (
Partial_Sums_in_cod1 f))
. z)
= ((
Partial_Sums_in_cod1 (
- f))
. z)
proof
let z be
Element of
[:
NAT ,
NAT :];
consider n,m be
object such that
A3: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A3;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 (
- f))
. ($1,m))
= (
- ((
Partial_Sums_in_cod1 f)
. ($1,m)));
reconsider z0 =
[
0 , m] as
Element of
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
A4:
[
0 , m]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
((
Partial_Sums_in_cod1 (
- f))
. (
0 ,m))
= ((
- f)
. (
0 ,m)) by
DefRSM
.= (
- (f
. (
0 ,m))) by
A4,
A2,
MESFUNC1:def 7;
then
A5:
P[
0 ] by
DefRSM;
A6: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A7:
P[k];
A8:
[(k
+ 1), m]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
thus ((
Partial_Sums_in_cod1 (
- f))
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 (
- f))
. (k,m))
+ ((
- f)
. ((k
+ 1),m))) by
DefRSM
.= ((
- ((
Partial_Sums_in_cod1 f)
. (k,m)))
- (f
. ((k
+ 1),m))) by
A7,
A8,
A2,
MESFUNC1:def 7
.= (
- (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m)))) by
XXREAL_3: 25
.= (
- ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))) by
DefRSM;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A5,
A6);
then ((
Partial_Sums_in_cod1 (
- f))
. (n,m))
= (
- ((
Partial_Sums_in_cod1 f)
. (n,m)));
hence thesis by
A3,
A1,
MESFUNC1:def 7;
end;
hence (
Partial_Sums_in_cod1 (
- f))
= (
- (
Partial_Sums_in_cod1 f)) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:43
Th43: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , m,n be
Element of
NAT holds ((
Partial_Sums_in_cod1 f)
. (m,n))
= ((
Partial_Sums (
ProjMap2 (f,n)))
. m) & ((
Partial_Sums_in_cod2 f)
. (m,n))
= ((
Partial_Sums (
ProjMap1 (f,m)))
. n)
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let m,n be
Element of
NAT ;
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,n))
= ((
Partial_Sums (
ProjMap2 (f,n)))
. $1);
((
Partial_Sums (
ProjMap2 (f,n)))
.
0 )
= ((
ProjMap2 (f,n))
.
0 ) by
MESFUNC9:def 1
.= (f
. (
0 ,n)) by
MESFUNC9:def 7;
then
a1:
P[
0 ] by
DefRSM;
a2: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
P[k];
then ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),n))
= (((
Partial_Sums (
ProjMap2 (f,n)))
. k)
+ (f
. ((k
+ 1),n))) by
DefRSM
.= (((
Partial_Sums (
ProjMap2 (f,n)))
. k)
+ ((
ProjMap2 (f,n))
. (k
+ 1))) by
MESFUNC9:def 7;
hence
P[(k
+ 1)] by
MESFUNC9:def 1;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
a1,
a2);
hence ((
Partial_Sums_in_cod1 f)
. (m,n))
= ((
Partial_Sums (
ProjMap2 (f,n)))
. m);
defpred
Q[
Nat] means ((
Partial_Sums_in_cod2 f)
. (m,$1))
= ((
Partial_Sums (
ProjMap1 (f,m)))
. $1);
((
Partial_Sums (
ProjMap1 (f,m)))
.
0 )
= ((
ProjMap1 (f,m))
.
0 ) by
MESFUNC9:def 1
.= (f
. (m,
0 )) by
MESFUNC9:def 6;
then
a3:
Q[
0 ] by
DefCSM;
a4: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
Q[k];
then ((
Partial_Sums_in_cod2 f)
. (m,(k
+ 1)))
= (((
Partial_Sums (
ProjMap1 (f,m)))
. k)
+ (f
. (m,(k
+ 1)))) by
DefCSM
.= (((
Partial_Sums (
ProjMap1 (f,m)))
. k)
+ ((
ProjMap1 (f,m))
. (k
+ 1))) by
MESFUNC9:def 6;
hence
Q[(k
+ 1)] by
MESFUNC9:def 1;
end;
for k be
Nat holds
Q[k] from
NAT_1:sch 2(
a3,
a4);
hence thesis;
end;
theorem ::
DBLSEQ_3:44
Th44: for f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums_in_cod2 (f1
+ f2))
= ((
Partial_Sums_in_cod2 f1)
+ (
Partial_Sums_in_cod2 f2)) & (
Partial_Sums_in_cod1 (f1
+ f2))
= ((
Partial_Sums_in_cod1 f1)
+ (
Partial_Sums_in_cod1 f2))
proof
let f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
set CS1 = (
Partial_Sums_in_cod2 f1);
set CS2 = (
Partial_Sums_in_cod2 f2);
set CS12 = (
Partial_Sums_in_cod2 (f1
+ f2));
set RS1 = (
Partial_Sums_in_cod1 f1);
set RS2 = (
Partial_Sums_in_cod1 f2);
set RS12 = (
Partial_Sums_in_cod1 (f1
+ f2));
now
let n be
Element of
NAT , m be
Element of
NAT ;
defpred
C[
Nat] means (CS12
. (n,$1))
= ((CS1
. (n,$1))
+ (CS2
. (n,$1)));
(CS12
. (n,
0 ))
= ((f1
+ f2)
. (n,
0 )) by
DefCSM
.= ((f1
. (n,
0 ))
+ (f2
. (n,
0 ))) by
Th11
.= ((CS1
. (n,
0 ))
+ (f2
. (n,
0 ))) by
DefCSM;
then
a1:
C[
0 ] by
DefCSM;
a2: for k be
Nat st
C[k] holds
C[(k
+ 1)]
proof
let k be
Nat;
assume
a3:
C[k];
X1: (CS1
. (n,k))
<>
-infty & (CS2
. (n,k))
<>
-infty & (f1
. (n,(k
+ 1)))
<>
-infty & (f2
. (n,(k
+ 1)))
<>
-infty & ((f1
+ f2)
. (n,(k
+ 1)))
<>
-infty by
MESFUNC5:def 5;
then
X2: ((CS2
. (n,k))
+ (f2
. (n,(k
+ 1))))
<>
-infty by
XXREAL_3: 17;
(CS12
. (n,(k
+ 1)))
= ((CS12
. (n,k))
+ ((f1
+ f2)
. (n,(k
+ 1)))) by
DefCSM
.= ((CS1
. (n,k))
+ (((f1
+ f2)
. (n,(k
+ 1)))
+ (CS2
. (n,k)))) by
a3,
X1,
XXREAL_3: 29
.= ((CS1
. (n,k))
+ (((f1
. (n,(k
+ 1)))
+ (f2
. (n,(k
+ 1))))
+ (CS2
. (n,k)))) by
Th11
.= ((CS1
. (n,k))
+ ((f1
. (n,(k
+ 1)))
+ ((CS2
. (n,k))
+ (f2
. (n,(k
+ 1)))))) by
X1,
XXREAL_3: 29
.= (((CS1
. (n,k))
+ (f1
. (n,(k
+ 1))))
+ ((CS2
. (n,k))
+ (f2
. (n,(k
+ 1))))) by
X1,
X2,
XXREAL_3: 29
.= ((CS1
. (n,(k
+ 1)))
+ ((CS2
. (n,k))
+ (f2
. (n,(k
+ 1))))) by
DefCSM;
hence
C[(k
+ 1)] by
DefCSM;
end;
for k be
Nat holds
C[k] from
NAT_1:sch 2(
a1,
a2);
then
C[m];
hence (CS12
. (n,m))
= ((CS1
+ CS2)
. (n,m)) by
Th11;
end;
hence CS12
= (CS1
+ CS2) by
BINOP_1: 2;
now
let n,m be
Element of
NAT ;
defpred
R[
Nat] means (RS12
. ($1,m))
= ((RS1
. ($1,m))
+ (RS2
. ($1,m)));
(RS12
. (
0 ,m))
= ((f1
+ f2)
. (
0 ,m)) by
DefRSM
.= ((f1
. (
0 ,m))
+ (f2
. (
0 ,m))) by
Th11
.= ((RS1
. (
0 ,m))
+ (f2
. (
0 ,m))) by
DefRSM;
then
a4:
R[
0 ] by
DefRSM;
a5: for k be
Nat st
R[k] holds
R[(k
+ 1)]
proof
let k be
Nat;
assume
a6:
R[k];
X3: (RS1
. (k,m))
<>
-infty & (RS2
. (k,m))
<>
-infty & (f1
. ((k
+ 1),m))
<>
-infty & (f2
. ((k
+ 1),m))
<>
-infty & ((f1
+ f2)
. ((k
+ 1),m))
<>
-infty by
MESFUNC5:def 5;
then
X4: ((RS2
. (k,m))
+ (f2
. ((k
+ 1),m)))
<>
-infty by
XXREAL_3: 17;
(RS12
. ((k
+ 1),m))
= ((RS12
. (k,m))
+ ((f1
+ f2)
. ((k
+ 1),m))) by
DefRSM
.= ((RS1
. (k,m))
+ (((f1
+ f2)
. ((k
+ 1),m))
+ (RS2
. (k,m)))) by
a6,
X3,
XXREAL_3: 29
.= ((RS1
. (k,m))
+ (((f1
. ((k
+ 1),m))
+ (f2
. ((k
+ 1),m)))
+ (RS2
. (k,m)))) by
Th11
.= ((RS1
. (k,m))
+ ((f1
. ((k
+ 1),m))
+ ((RS2
. (k,m))
+ (f2
. ((k
+ 1),m))))) by
X3,
XXREAL_3: 29
.= (((RS1
. (k,m))
+ (f1
. ((k
+ 1),m)))
+ ((RS2
. (k,m))
+ (f2
. ((k
+ 1),m)))) by
X3,
X4,
XXREAL_3: 29
.= ((RS1
. ((k
+ 1),m))
+ ((RS2
. (k,m))
+ (f2
. ((k
+ 1),m)))) by
DefRSM;
hence
R[(k
+ 1)] by
DefRSM;
end;
for k be
Nat holds
R[k] from
NAT_1:sch 2(
a4,
a5);
then
R[n];
hence (RS12
. (n,m))
= ((RS1
+ RS2)
. (n,m)) by
Th11;
end;
hence RS12
= (RS1
+ RS2) by
BINOP_1: 2;
end;
theorem ::
DBLSEQ_3:45
Th45: for f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums_in_cod2 (f1
+ f2))
= ((
Partial_Sums_in_cod2 f1)
+ (
Partial_Sums_in_cod2 f2)) & (
Partial_Sums_in_cod1 (f1
+ f2))
= ((
Partial_Sums_in_cod1 f1)
+ (
Partial_Sums_in_cod1 f2))
proof
let f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
reconsider F1 = (
- f1), F2 = (
- f2) as
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
(F1
+ F2)
= ((
- f1)
- f2) by
Th10
.= (
- (f1
+ f2)) by
Th9;
then
A1: (
- (F1
+ F2))
= (f1
+ f2) by
Th2;
then (
Partial_Sums_in_cod2 (f1
+ f2))
= (
- (
Partial_Sums_in_cod2 (F1
+ F2))) by
Th42
.= (
- ((
Partial_Sums_in_cod2 F1)
+ (
Partial_Sums_in_cod2 F2))) by
Th44
.= (
- ((
- (
Partial_Sums_in_cod2 f1))
+ (
Partial_Sums_in_cod2 F2))) by
Th42
.= (
- ((
- (
Partial_Sums_in_cod2 f1))
+ (
- (
Partial_Sums_in_cod2 f2)))) by
Th42
.= ((
- (
- (
Partial_Sums_in_cod2 f1)))
- (
- (
Partial_Sums_in_cod2 f2))) by
Th8
.= ((
Partial_Sums_in_cod2 f1)
- (
- (
Partial_Sums_in_cod2 f2))) by
Th2
.= ((
Partial_Sums_in_cod2 f1)
+ (
- (
- (
Partial_Sums_in_cod2 f2)))) by
Th10;
hence (
Partial_Sums_in_cod2 (f1
+ f2))
= ((
Partial_Sums_in_cod2 f1)
+ (
Partial_Sums_in_cod2 f2)) by
Th2;
(
Partial_Sums_in_cod1 (f1
+ f2))
= (
- (
Partial_Sums_in_cod1 (F1
+ F2))) by
A1,
Th42
.= (
- ((
Partial_Sums_in_cod1 F1)
+ (
Partial_Sums_in_cod1 F2))) by
Th44
.= (
- ((
- (
Partial_Sums_in_cod1 f1))
+ (
Partial_Sums_in_cod1 F2))) by
Th42
.= (
- ((
- (
Partial_Sums_in_cod1 f1))
+ (
- (
Partial_Sums_in_cod1 f2)))) by
Th42
.= ((
- (
- (
Partial_Sums_in_cod1 f1)))
- (
- (
Partial_Sums_in_cod1 f2))) by
Th8
.= ((
Partial_Sums_in_cod1 f1)
- (
- (
Partial_Sums_in_cod1 f2))) by
Th2
.= ((
Partial_Sums_in_cod1 f1)
+ (
- (
- (
Partial_Sums_in_cod1 f2)))) by
Th10;
hence thesis by
Th2;
end;
theorem ::
DBLSEQ_3:46
Th46: for f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums_in_cod2 (f1
- f2))
= ((
Partial_Sums_in_cod2 f1)
- (
Partial_Sums_in_cod2 f2)) & (
Partial_Sums_in_cod1 (f1
- f2))
= ((
Partial_Sums_in_cod1 f1)
- (
Partial_Sums_in_cod1 f2)) & (
Partial_Sums_in_cod2 (f2
- f1))
= ((
Partial_Sums_in_cod2 f2)
- (
Partial_Sums_in_cod2 f1)) & (
Partial_Sums_in_cod1 (f2
- f1))
= ((
Partial_Sums_in_cod1 f2)
- (
Partial_Sums_in_cod1 f1))
proof
let f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
(
Partial_Sums_in_cod2 (f1
- f2))
= (
Partial_Sums_in_cod2 (f1
+ (
- f2))) by
Th10
.= ((
Partial_Sums_in_cod2 f1)
+ (
Partial_Sums_in_cod2 (
- f2))) by
Th44
.= ((
Partial_Sums_in_cod2 f1)
+ (
- (
Partial_Sums_in_cod2 f2))) by
Th42;
hence (
Partial_Sums_in_cod2 (f1
- f2))
= ((
Partial_Sums_in_cod2 f1)
- (
Partial_Sums_in_cod2 f2)) by
Th10;
(
Partial_Sums_in_cod1 (f1
- f2))
= (
Partial_Sums_in_cod1 (f1
+ (
- f2))) by
Th10
.= ((
Partial_Sums_in_cod1 f1)
+ (
Partial_Sums_in_cod1 (
- f2))) by
Th44
.= ((
Partial_Sums_in_cod1 f1)
+ (
- (
Partial_Sums_in_cod1 f2))) by
Th42;
hence (
Partial_Sums_in_cod1 (f1
- f2))
= ((
Partial_Sums_in_cod1 f1)
- (
Partial_Sums_in_cod1 f2)) by
Th10;
(
Partial_Sums_in_cod2 (f2
- f1))
= (
Partial_Sums_in_cod2 (f2
+ (
- f1))) by
Th10
.= ((
Partial_Sums_in_cod2 f2)
+ (
Partial_Sums_in_cod2 (
- f1))) by
Th45
.= ((
Partial_Sums_in_cod2 f2)
+ (
- (
Partial_Sums_in_cod2 f1))) by
Th42;
hence (
Partial_Sums_in_cod2 (f2
- f1))
= ((
Partial_Sums_in_cod2 f2)
- (
Partial_Sums_in_cod2 f1)) by
Th10;
(
Partial_Sums_in_cod1 (f2
- f1))
= (
Partial_Sums_in_cod1 (f2
+ (
- f1))) by
Th10
.= ((
Partial_Sums_in_cod1 f2)
+ (
Partial_Sums_in_cod1 (
- f1))) by
Th45
.= ((
Partial_Sums_in_cod1 f2)
+ (
- (
Partial_Sums_in_cod1 f1))) by
Th42;
hence (
Partial_Sums_in_cod1 (f2
- f1))
= ((
Partial_Sums_in_cod1 f2)
- (
Partial_Sums_in_cod1 f1)) by
Th10;
end;
theorem ::
DBLSEQ_3:47
Th47: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((
Partial_Sums f)
. ((n
+ 1),m))
= (((
Partial_Sums_in_cod2 f)
. ((n
+ 1),m))
+ ((
Partial_Sums f)
. (n,m))) & ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,(m
+ 1)))
= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,m)))
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let n,m be
Nat;
set RPS = (
Partial_Sums f);
set CPS = (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f));
set ROW = (
Partial_Sums_in_cod1 f);
set COL = (
Partial_Sums_in_cod2 f);
defpred
P[
Nat] means (RPS
. ((n
+ 1),$1))
= ((COL
. ((n
+ 1),$1))
+ (RPS
. (n,$1)));
a1: (RPS
. (n,
0 ))
= (ROW
. (n,
0 )) by
DefCSM;
(RPS
. ((n
+ 1),
0 ))
= (ROW
. ((n
+ 1),
0 )) by
DefCSM
.= ((ROW
. (n,
0 ))
+ (f
. ((n
+ 1),
0 ))) by
DefRSM;
then
a3:
P[
0 ] by
a1,
DefCSM;
a4: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A5:
P[k];
a6: (COL
. ((n
+ 1),(k
+ 1)))
= ((COL
. ((n
+ 1),k))
+ (f
. ((n
+ 1),(k
+ 1)))) by
DefCSM;
X1: (COL
. ((n
+ 1),k))
<>
-infty & (f
. ((n
+ 1),(k
+ 1)))
<>
-infty & (RPS
. (n,k))
<>
-infty & (ROW
. (n,(k
+ 1)))
<>
-infty & (RPS
. ((n
+ 1),k))
<>
-infty by
MESFUNC5:def 5;
then
X2: ((COL
. ((n
+ 1),k))
+ (f
. ((n
+ 1),(k
+ 1))))
<>
-infty by
XXREAL_3: 17;
(RPS
. (n,(k
+ 1)))
= ((RPS
. (n,k))
+ (ROW
. (n,(k
+ 1)))) by
DefCSM;
then ((COL
. ((n
+ 1),(k
+ 1)))
+ (RPS
. (n,(k
+ 1))))
= ((((COL
. ((n
+ 1),k))
+ (f
. ((n
+ 1),(k
+ 1))))
+ (RPS
. (n,k)))
+ (ROW
. (n,(k
+ 1)))) by
a6,
X1,
X2,
XXREAL_3: 29
.= ((((COL
. ((n
+ 1),k))
+ (RPS
. (n,k)))
+ (f
. ((n
+ 1),(k
+ 1))))
+ (ROW
. (n,(k
+ 1)))) by
X1,
XXREAL_3: 29
.= ((RPS
. ((n
+ 1),k))
+ ((f
. ((n
+ 1),(k
+ 1)))
+ (ROW
. (n,(k
+ 1))))) by
A5,
X1,
XXREAL_3: 29
.= ((RPS
. ((n
+ 1),k))
+ (ROW
. ((n
+ 1),(k
+ 1)))) by
DefRSM;
hence
P[(k
+ 1)] by
DefCSM;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
a3,
a4);
hence (RPS
. ((n
+ 1),m))
= ((COL
. ((n
+ 1),m))
+ (RPS
. (n,m)));
defpred
Q[
Nat] means (CPS
. ($1,(m
+ 1)))
= ((ROW
. ($1,(m
+ 1)))
+ (CPS
. ($1,m)));
b1: (CPS
. (
0 ,m))
= (COL
. (
0 ,m)) by
DefRSM;
(CPS
. (
0 ,(m
+ 1)))
= (COL
. (
0 ,(m
+ 1))) by
DefRSM
.= ((COL
. (
0 ,m))
+ (f
. (
0 ,(m
+ 1)))) by
DefCSM;
then
b3:
Q[
0 ] by
b1,
DefRSM;
b4: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
B5:
Q[k];
b6: (ROW
. ((k
+ 1),(m
+ 1)))
= ((ROW
. (k,(m
+ 1)))
+ (f
. ((k
+ 1),(m
+ 1)))) by
DefRSM;
X3: (ROW
. (k,(m
+ 1)))
<>
-infty & (f
. ((k
+ 1),(m
+ 1)))
<>
-infty & (CPS
. (k,m))
<>
-infty & (COL
. ((k
+ 1),m))
<>
-infty & (CPS
. (k,(m
+ 1)))
<>
-infty by
MESFUNC5:def 5;
then
X4: ((ROW
. (k,(m
+ 1)))
+ (f
. ((k
+ 1),(m
+ 1))))
<>
-infty by
XXREAL_3: 17;
(CPS
. ((k
+ 1),m))
= ((CPS
. (k,m))
+ (COL
. ((k
+ 1),m))) by
DefRSM;
then ((ROW
. ((k
+ 1),(m
+ 1)))
+ (CPS
. ((k
+ 1),m)))
= ((((ROW
. (k,(m
+ 1)))
+ (f
. ((k
+ 1),(m
+ 1))))
+ (CPS
. (k,m)))
+ (COL
. ((k
+ 1),m))) by
b6,
X3,
X4,
XXREAL_3: 29
.= ((((ROW
. (k,(m
+ 1)))
+ (CPS
. (k,m)))
+ (f
. ((k
+ 1),(m
+ 1))))
+ (COL
. ((k
+ 1),m))) by
X3,
XXREAL_3: 29
.= ((CPS
. (k,(m
+ 1)))
+ ((f
. ((k
+ 1),(m
+ 1)))
+ (COL
. ((k
+ 1),m)))) by
B5,
X3,
XXREAL_3: 29
.= ((CPS
. (k,(m
+ 1)))
+ (COL
. ((k
+ 1),(m
+ 1)))) by
DefCSM;
hence
Q[(k
+ 1)] by
DefRSM;
end;
for k be
Nat holds
Q[k] from
NAT_1:sch 2(
b3,
b4);
hence thesis;
end;
theorem ::
DBLSEQ_3:48
for f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((
Partial_Sums f)
. ((n
+ 1),m))
= (((
Partial_Sums_in_cod2 f)
. ((n
+ 1),m))
+ ((
Partial_Sums f)
. (n,m))) & ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,(m
+ 1)))
= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,m)))
proof
let f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let n,m be
Nat;
reconsider g = (
- f) as
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A2: (
dom (
- (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 g))))
=
[:
NAT ,
NAT :] & (
dom (
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))))
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
A4: (
dom (
- (
Partial_Sums_in_cod2 g)))
=
[:
NAT ,
NAT :] & (
dom (
- (
Partial_Sums_in_cod1 g)))
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
A5: (
dom (
- (
Partial_Sums g)))
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
A3:
[(n
+ 1), m]
in
[:
NAT ,
NAT :] &
[n, m]
in
[:
NAT ,
NAT :] &
[n, (m
+ 1)]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
A1: (
Partial_Sums f)
= (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 (
- g))) by
Th2
.= (
Partial_Sums_in_cod2 (
- (
Partial_Sums_in_cod1 g))) by
Th42
.= (
- (
Partial_Sums g)) by
Th42;
thus ((
Partial_Sums f)
. ((n
+ 1),m))
= (
- ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 g))
. ((n
+ 1),m))) by
A1,
A2,
A3,
MESFUNC1:def 7
.= (
- (((
Partial_Sums_in_cod2 g)
. ((n
+ 1),m))
+ ((
Partial_Sums g)
. (n,m)))) by
Th47
.= ((
- ((
Partial_Sums_in_cod2 g)
. ((n
+ 1),m)))
- ((
Partial_Sums g)
. (n,m))) by
XXREAL_3: 25
.= (((
- (
Partial_Sums_in_cod2 g))
. ((n
+ 1),m))
+ (
- ((
Partial_Sums g)
. (n,m)))) by
A3,
A4,
MESFUNC1:def 7
.= (((
Partial_Sums_in_cod2 (
- g))
. ((n
+ 1),m))
+ (
- ((
Partial_Sums g)
. (n,m)))) by
Th42
.= (((
Partial_Sums_in_cod2 f)
. ((n
+ 1),m))
+ (
- ((
Partial_Sums g)
. (n,m)))) by
Th2
.= (((
Partial_Sums_in_cod2 f)
. ((n
+ 1),m))
+ ((
Partial_Sums f)
. (n,m))) by
A1,
A3,
A5,
MESFUNC1:def 7;
thus ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,(m
+ 1)))
= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
- g)))
. (n,(m
+ 1))) by
Th2
.= ((
Partial_Sums_in_cod1 (
- (
Partial_Sums_in_cod2 g)))
. (n,(m
+ 1))) by
Th42
.= ((
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g)))
. (n,(m
+ 1))) by
Th42
.= (
- ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))
. (n,(m
+ 1)))) by
A3,
A2,
MESFUNC1:def 7
.= (
- (((
Partial_Sums_in_cod1 g)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))
. (n,m)))) by
Th47
.= ((
- ((
Partial_Sums_in_cod1 g)
. (n,(m
+ 1))))
- ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))
. (n,m))) by
XXREAL_3: 25
.= (((
- (
Partial_Sums_in_cod1 g))
. (n,(m
+ 1)))
- ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))
. (n,m))) by
A4,
A3,
MESFUNC1:def 7
.= (((
- (
Partial_Sums_in_cod1 g))
. (n,(m
+ 1)))
+ ((
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g)))
. (n,m))) by
A3,
A2,
MESFUNC1:def 7
.= (((
Partial_Sums_in_cod1 (
- g))
. (n,(m
+ 1)))
+ ((
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g)))
. (n,m))) by
Th42
.= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g)))
. (n,m))) by
Th2
.= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
- (
Partial_Sums_in_cod2 g)))
. (n,m))) by
Th42
.= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
- g)))
. (n,m))) by
Th42
.= (((
Partial_Sums_in_cod1 f)
. (n,(m
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,m))) by
Th2;
end;
Lm7: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , m,n be
Nat holds ((
Partial_Sums f)
. (m,n))
= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,n))
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , m,n be
Nat;
defpred
P1[
Nat] means for m be
Nat holds ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,$1))
= ((
Partial_Sums f)
. (m,$1));
defpred
P2[
Nat] means ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. ($1,
0 ))
= ((
Partial_Sums f)
. ($1,
0 ));
Y3: ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (
0 ,
0 ))
= ((
Partial_Sums_in_cod2 f)
. (
0 ,
0 )) by
DefRSM
.= (f
. (
0 ,
0 )) by
DefCSM;
((
Partial_Sums f)
. (
0 ,
0 ))
= ((
Partial_Sums_in_cod1 f)
. (
0 ,
0 )) by
DefCSM
.= (f
. (
0 ,
0 )) by
DefRSM;
then
Y1:
P2[
0 ] by
Y3;
Y2: for i be
Nat st
P2[i] holds
P2[(i
+ 1)]
proof
let i be
Nat;
assume
Y3:
P2[i];
Y4: ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. ((i
+ 1),
0 ))
= (((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,
0 ))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),
0 ))) by
DefRSM
.= (((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,
0 ))
+ (f
. ((i
+ 1),
0 ))) by
DefCSM;
((
Partial_Sums f)
. ((i
+ 1),
0 ))
= ((
Partial_Sums_in_cod1 f)
. ((i
+ 1),
0 )) by
DefCSM
.= (((
Partial_Sums_in_cod1 f)
. (i,
0 ))
+ (f
. ((i
+ 1),
0 ))) by
DefRSM
.= (((
Partial_Sums f)
. (i,
0 ))
+ (f
. ((i
+ 1),
0 ))) by
DefCSM;
hence
P2[(i
+ 1)] by
Y3,
Y4;
end;
for n be
Nat holds
P2[n] from
NAT_1:sch 2(
Y1,
Y2);
then
X1:
P1[
0 ];
X2: for j be
Nat st
P1[j] holds
P1[(j
+ 1)]
proof
let j be
Nat;
assume
Z3:
P1[j];
for m be
Nat holds ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,(j
+ 1)))
= ((
Partial_Sums f)
. (m,(j
+ 1)))
proof
let n be
Nat;
defpred
P3[
Nat] means ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. ($1,(j
+ 1)))
= ((
Partial_Sums f)
. ($1,(j
+ 1)));
Z4: ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (
0 ,(j
+ 1)))
= ((
Partial_Sums_in_cod2 f)
. (
0 ,(j
+ 1))) by
DefRSM
.= (((
Partial_Sums_in_cod2 f)
. (
0 ,j))
+ (f
. (
0 ,(j
+ 1)))) by
DefCSM
.= (((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (
0 ,j))
+ (f
. (
0 ,(j
+ 1)))) by
DefRSM;
((
Partial_Sums f)
. (
0 ,(j
+ 1)))
= (((
Partial_Sums f)
. (
0 ,j))
+ ((
Partial_Sums_in_cod1 f)
. (
0 ,(j
+ 1)))) by
DefCSM
.= (((
Partial_Sums f)
. (
0 ,j))
+ (f
. (
0 ,(j
+ 1)))) by
DefRSM
.= (((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (
0 ,j))
+ (f
. (
0 ,(j
+ 1)))) by
Z3;
then
Z1:
P3[
0 ] by
Z4;
Z2: for i be
Nat st
P3[i] holds
P3[(i
+ 1)]
proof
let i be
Nat;
assume
P3[i];
W1: ((
Partial_Sums f)
. (i,j))
<>
-infty & ((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1)))
<>
-infty & ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))
<>
-infty & (f
. ((i
+ 1),(j
+ 1)))
<>
-infty & ((
Partial_Sums_in_cod1 f)
. ((i
+ 1),(j
+ 1)))
<>
-infty & ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))
<>
-infty by
MESFUNC5:def 5;
then
W2: (((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1))))
<>
-infty by
XXREAL_3: 17;
Z6: ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,j))
= ((
Partial_Sums f)
. (i,j)) by
Z3;
((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. ((i
+ 1),(j
+ 1)))
= (((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,(j
+ 1)))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),(j
+ 1)))) by
DefRSM
.= ((((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1))))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),(j
+ 1)))) by
Z6,
Th47
.= ((((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1))))
+ (((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))
+ (f
. ((i
+ 1),(j
+ 1))))) by
DefCSM
.= (((((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1))))
+ (f
. ((i
+ 1),(j
+ 1))))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))) by
W1,
W2,
XXREAL_3: 29
.= ((((
Partial_Sums f)
. (i,j))
+ (((
Partial_Sums_in_cod1 f)
. (i,(j
+ 1)))
+ (f
. ((i
+ 1),(j
+ 1)))))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))) by
W1,
XXREAL_3: 29
.= ((((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod1 f)
. ((i
+ 1),(j
+ 1))))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j))) by
DefRSM
.= ((((
Partial_Sums f)
. (i,j))
+ ((
Partial_Sums_in_cod2 f)
. ((i
+ 1),j)))
+ ((
Partial_Sums_in_cod1 f)
. ((i
+ 1),(j
+ 1)))) by
W1,
XXREAL_3: 29
.= (((
Partial_Sums f)
. ((i
+ 1),j))
+ ((
Partial_Sums_in_cod1 f)
. ((i
+ 1),(j
+ 1)))) by
Th47;
hence thesis by
DefCSM;
end;
for n be
Nat holds
P3[n] from
NAT_1:sch 2(
Z1,
Z2);
hence thesis;
end;
hence
P1[(j
+ 1)];
end;
for m be
Nat holds
P1[m] from
NAT_1:sch 2(
X1,
X2);
hence thesis;
end;
Lm8: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums f)
= (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
now
let x be
Element of
[:
NAT ,
NAT :];
consider n,m be
object such that
A1: n
in
NAT & m
in
NAT & x
=
[n, m] by
ZFMISC_1:def 2;
reconsider n1 = n, m1 = m as
Nat by
A1;
((
Partial_Sums f)
. (n1,m1))
= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n1,m1)) by
Lm7;
hence ((
Partial_Sums f)
. x)
= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. x) by
A1;
end;
hence thesis by
FUNCT_2: 63;
end;
Lm9: for f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums f)
= (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
proof
let f be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
reconsider g = (
- f) as
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A1: (
Partial_Sums f)
= (
Partial_Sums (
- g)) by
Th2
.= (
Partial_Sums_in_cod2 (
- (
Partial_Sums_in_cod1 g))) by
Th42
.= (
- (
Partial_Sums g)) by
Th42;
(
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
= (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
- g))) by
Th2
.= (
Partial_Sums_in_cod1 (
- (
Partial_Sums_in_cod2 g))) by
Th42
.= (
- (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 g))) by
Th42;
hence thesis by
A1,
Lm8;
end;
theorem ::
DBLSEQ_3:49
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL st f is
without-infty or f is
without+infty holds (
Partial_Sums f)
= (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)) by
Lm8,
Lm9;
theorem ::
DBLSEQ_3:50
for f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums (f1
+ f2))
= ((
Partial_Sums f1)
+ (
Partial_Sums f2))
proof
let f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
(
Partial_Sums (f1
+ f2))
= (
Partial_Sums_in_cod2 ((
Partial_Sums_in_cod1 f1)
+ (
Partial_Sums_in_cod1 f2))) by
Th44;
hence thesis by
Th44;
end;
theorem ::
DBLSEQ_3:51
for f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums (f1
+ f2))
= ((
Partial_Sums f1)
+ (
Partial_Sums f2))
proof
let f1,f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
(
Partial_Sums (f1
+ f2))
= (
Partial_Sums_in_cod2 ((
Partial_Sums_in_cod1 f1)
+ (
Partial_Sums_in_cod1 f2))) by
Th45;
hence thesis by
Th45;
end;
theorem ::
DBLSEQ_3:52
for f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums (f1
- f2))
= ((
Partial_Sums f1)
- (
Partial_Sums f2)) & (
Partial_Sums (f2
- f1))
= ((
Partial_Sums f2)
- (
Partial_Sums f1))
proof
let f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
without+infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
(
Partial_Sums (f1
- f2))
= (
Partial_Sums_in_cod2 ((
Partial_Sums_in_cod1 f1)
- (
Partial_Sums_in_cod1 f2))) by
Th46;
hence (
Partial_Sums (f1
- f2))
= ((
Partial_Sums f1)
- (
Partial_Sums f2)) by
Th46;
(
Partial_Sums (f2
- f1))
= (
Partial_Sums_in_cod2 ((
Partial_Sums_in_cod1 f2)
- (
Partial_Sums_in_cod1 f1))) by
Th46;
hence (
Partial_Sums (f2
- f1))
= ((
Partial_Sums f2)
- (
Partial_Sums f1)) by
Th46;
end;
theorem ::
DBLSEQ_3:53
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , k be
Element of
NAT holds (
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
= (
Partial_Sums (
ProjMap2 (f,k))) & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
= (
Partial_Sums (
ProjMap1 (f,k)))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , k be
Element of
NAT ;
now
let n be
Element of
NAT ;
((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
= ((
Partial_Sums_in_cod1 f)
. (n,k)) by
MESFUNC9:def 7;
hence ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
= ((
Partial_Sums (
ProjMap2 (f,k)))
. n) by
Th43;
end;
hence (
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
= (
Partial_Sums (
ProjMap2 (f,k))) by
FUNCT_2:def 8;
now
let n be
Element of
NAT ;
((
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
. n)
= ((
Partial_Sums_in_cod2 f)
. (k,n)) by
MESFUNC9:def 6;
hence ((
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
. n)
= ((
Partial_Sums (
ProjMap1 (f,k)))
. n) by
Th43;
end;
hence (
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
= (
Partial_Sums (
ProjMap1 (f,k))) by
FUNCT_2:def 8;
end;
theorem ::
DBLSEQ_3:54
Th54: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL st f is
without-infty or f is
without+infty holds (
ProjMap1 ((
Partial_Sums f),
0 ))
= (
ProjMap1 ((
Partial_Sums_in_cod2 f),
0 )) & (
ProjMap2 ((
Partial_Sums f),
0 ))
= (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A0: f is
without-infty or f is
without+infty;
A1:
now
let m be
Element of
NAT ;
((
ProjMap1 ((
Partial_Sums f),
0 ))
. m)
= ((
Partial_Sums f)
. (
0 ,m)) by
MESFUNC9:def 6
.= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (
0 ,m)) by
A0,
Lm8,
Lm9
.= ((
Partial_Sums_in_cod2 f)
. (
0 ,m)) by
DefRSM;
hence ((
ProjMap1 ((
Partial_Sums f),
0 ))
. m)
= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),
0 ))
. m) by
MESFUNC9:def 6;
end;
now
let n be
Element of
NAT ;
((
ProjMap2 ((
Partial_Sums f),
0 ))
. n)
= ((
Partial_Sums f)
. (n,
0 )) by
MESFUNC9:def 7
.= ((
Partial_Sums_in_cod1 f)
. (n,
0 )) by
DefCSM;
hence ((
ProjMap2 ((
Partial_Sums f),
0 ))
. n)
= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))
. n) by
MESFUNC9:def 7;
end;
hence thesis by
A1,
FUNCT_2: 63;
end;
theorem ::
DBLSEQ_3:55
for C,D be non
empty
set, F1,F2 be
without-infty
Function of
[:C, D:],
ExtREAL , c be
Element of C holds (
ProjMap1 ((F1
+ F2),c))
= ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))
proof
let C,D be non
empty
set;
let F1,F2 be
without-infty
Function of
[:C, D:],
ExtREAL ;
let c be
Element of C;
A2: (
dom (
ProjMap1 ((F1
+ F2),c)))
= D & (
dom (
ProjMap1 (F1,c)))
= D & (
dom (
ProjMap1 (F2,c)))
= D by
FUNCT_2:def 1;
{
-infty }
misses (
rng (
ProjMap1 (F1,c))) &
{
-infty }
misses (
rng (
ProjMap1 (F2,c))) by
ZFMISC_1: 50,
MESFUNC5:def 3;
then ((
ProjMap1 (F1,c))
"
{
-infty })
=
{} & ((
ProjMap1 (F2,c))
"
{
-infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap1 ((F1
+ F2),c)))
= (((
dom (
ProjMap1 (F1,c)))
/\ (
dom (
ProjMap1 (F2,c))))
\ ((((
ProjMap1 (F1,c))
"
{
-infty })
/\ ((
ProjMap1 (F2,c))
"
{
+infty }))
\/ (((
ProjMap1 (F1,c))
"
{
+infty })
/\ ((
ProjMap1 (F2,c))
"
{
-infty })))) by
A2;
then
A5: (
dom (
ProjMap1 ((F1
+ F2),c)))
= (
dom ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))) by
MESFUNC1:def 3;
for d be
object st d
in (
dom (
ProjMap1 ((F1
+ F2),c))) holds ((
ProjMap1 ((F1
+ F2),c))
. d)
= (((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))
. d)
proof
let d be
object;
assume
A3: d
in (
dom (
ProjMap1 ((F1
+ F2),c)));
then
A4: ((
ProjMap1 ((F1
+ F2),c))
. d)
= ((F1
+ F2)
. (c,d)) & ((
ProjMap1 (F1,c))
. d)
= (F1
. (c,d)) & ((
ProjMap1 (F2,c))
. d)
= (F2
. (c,d)) by
MESFUNC9:def 6;
reconsider d1 = d as
Element of D by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap1 ((F1
+ F2),c))
. d)
= (((
ProjMap1 (F1,c))
. d)
+ ((
ProjMap1 (F2,c))
. d)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 3;
end;
hence (
ProjMap1 ((F1
+ F2),c))
= ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c))) by
A5,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:56
for C,D be non
empty
set, F1,F2 be
without-infty
Function of
[:C, D:],
ExtREAL , d be
Element of D holds (
ProjMap2 ((F1
+ F2),d))
= ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))
proof
let C,D be non
empty
set;
let F1,F2 be
without-infty
Function of
[:C, D:],
ExtREAL ;
let d be
Element of D;
A2: (
dom (
ProjMap2 ((F1
+ F2),d)))
= C & (
dom (
ProjMap2 (F1,d)))
= C & (
dom (
ProjMap2 (F2,d)))
= C by
FUNCT_2:def 1;
{
-infty }
misses (
rng (
ProjMap2 (F1,d))) &
{
-infty }
misses (
rng (
ProjMap2 (F2,d))) by
ZFMISC_1: 50,
MESFUNC5:def 3;
then ((
ProjMap2 (F1,d))
"
{
-infty })
=
{} & ((
ProjMap2 (F2,d))
"
{
-infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap2 ((F1
+ F2),d)))
= (((
dom (
ProjMap2 (F1,d)))
/\ (
dom (
ProjMap2 (F2,d))))
\ ((((
ProjMap2 (F1,d))
"
{
-infty })
/\ ((
ProjMap2 (F2,d))
"
{
+infty }))
\/ (((
ProjMap2 (F1,d))
"
{
+infty })
/\ ((
ProjMap2 (F2,d))
"
{
-infty })))) by
A2;
then
A5: (
dom (
ProjMap2 ((F1
+ F2),d)))
= (
dom ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))) by
MESFUNC1:def 3;
for c be
object st c
in (
dom (
ProjMap2 ((F1
+ F2),d))) holds ((
ProjMap2 ((F1
+ F2),d))
. c)
= (((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))
. c)
proof
let c be
object;
assume
A3: c
in (
dom (
ProjMap2 ((F1
+ F2),d)));
then
A4: ((
ProjMap2 ((F1
+ F2),d))
. c)
= ((F1
+ F2)
. (c,d)) & ((
ProjMap2 (F1,d))
. c)
= (F1
. (c,d)) & ((
ProjMap2 (F2,d))
. c)
= (F2
. (c,d)) by
MESFUNC9:def 7;
reconsider c1 = c as
Element of C by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap2 ((F1
+ F2),d))
. c)
= (((
ProjMap2 (F1,d))
. c)
+ ((
ProjMap2 (F2,d))
. c)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 3;
end;
hence (
ProjMap2 ((F1
+ F2),d))
= ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d))) by
A5,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:57
for C,D be non
empty
set, F1,F2 be
without+infty
Function of
[:C, D:],
ExtREAL , c be
Element of C holds (
ProjMap1 ((F1
+ F2),c))
= ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))
proof
let C,D be non
empty
set;
let F1,F2 be
without+infty
Function of
[:C, D:],
ExtREAL ;
let c be
Element of C;
A2: (
dom (
ProjMap1 ((F1
+ F2),c)))
= D & (
dom (
ProjMap1 (F1,c)))
= D & (
dom (
ProjMap1 (F2,c)))
= D by
FUNCT_2:def 1;
{
+infty }
misses (
rng (
ProjMap1 (F1,c))) &
{
+infty }
misses (
rng (
ProjMap1 (F2,c))) by
ZFMISC_1: 50,
MESFUNC5:def 4;
then ((
ProjMap1 (F1,c))
"
{
+infty })
=
{} & ((
ProjMap1 (F2,c))
"
{
+infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap1 ((F1
+ F2),c)))
= (((
dom (
ProjMap1 (F1,c)))
/\ (
dom (
ProjMap1 (F2,c))))
\ ((((
ProjMap1 (F1,c))
"
{
-infty })
/\ ((
ProjMap1 (F2,c))
"
{
+infty }))
\/ (((
ProjMap1 (F1,c))
"
{
+infty })
/\ ((
ProjMap1 (F2,c))
"
{
-infty })))) by
A2;
then
A5: (
dom (
ProjMap1 ((F1
+ F2),c)))
= (
dom ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))) by
MESFUNC1:def 3;
for d be
object st d
in (
dom (
ProjMap1 ((F1
+ F2),c))) holds ((
ProjMap1 ((F1
+ F2),c))
. d)
= (((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c)))
. d)
proof
let d be
object;
assume
A3: d
in (
dom (
ProjMap1 ((F1
+ F2),c)));
then
A4: ((
ProjMap1 ((F1
+ F2),c))
. d)
= ((F1
+ F2)
. (c,d)) & ((
ProjMap1 (F1,c))
. d)
= (F1
. (c,d)) & ((
ProjMap1 (F2,c))
. d)
= (F2
. (c,d)) by
MESFUNC9:def 6;
reconsider d1 = d as
Element of D by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap1 ((F1
+ F2),c))
. d)
= (((
ProjMap1 (F1,c))
. d)
+ ((
ProjMap1 (F2,c))
. d)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 3;
end;
hence (
ProjMap1 ((F1
+ F2),c))
= ((
ProjMap1 (F1,c))
+ (
ProjMap1 (F2,c))) by
A5,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:58
for C,D be non
empty
set, F1,F2 be
without+infty
Function of
[:C, D:],
ExtREAL , d be
Element of D holds (
ProjMap2 ((F1
+ F2),d))
= ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))
proof
let C,D be non
empty
set;
let F1,F2 be
without+infty
Function of
[:C, D:],
ExtREAL ;
let d be
Element of D;
A2: (
dom (
ProjMap2 ((F1
+ F2),d)))
= C & (
dom (
ProjMap2 (F1,d)))
= C & (
dom (
ProjMap2 (F2,d)))
= C by
FUNCT_2:def 1;
{
+infty }
misses (
rng (
ProjMap2 (F1,d))) &
{
+infty }
misses (
rng (
ProjMap2 (F2,d))) by
ZFMISC_1: 50,
MESFUNC5:def 4;
then ((
ProjMap2 (F1,d))
"
{
+infty })
=
{} & ((
ProjMap2 (F2,d))
"
{
+infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap2 ((F1
+ F2),d)))
= (((
dom (
ProjMap2 (F1,d)))
/\ (
dom (
ProjMap2 (F2,d))))
\ ((((
ProjMap2 (F1,d))
"
{
-infty })
/\ ((
ProjMap2 (F2,d))
"
{
+infty }))
\/ (((
ProjMap2 (F1,d))
"
{
+infty })
/\ ((
ProjMap2 (F2,d))
"
{
-infty })))) by
A2;
then
A5: (
dom (
ProjMap2 ((F1
+ F2),d)))
= (
dom ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))) by
MESFUNC1:def 3;
for c be
object st c
in (
dom (
ProjMap2 ((F1
+ F2),d))) holds ((
ProjMap2 ((F1
+ F2),d))
. c)
= (((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d)))
. c)
proof
let c be
object;
assume
A3: c
in (
dom (
ProjMap2 ((F1
+ F2),d)));
then
A4: ((
ProjMap2 ((F1
+ F2),d))
. c)
= ((F1
+ F2)
. (c,d)) & ((
ProjMap2 (F1,d))
. c)
= (F1
. (c,d)) & ((
ProjMap2 (F2,d))
. c)
= (F2
. (c,d)) by
MESFUNC9:def 7;
reconsider c1 = c as
Element of C by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap2 ((F1
+ F2),d))
. c)
= (((
ProjMap2 (F1,d))
. c)
+ ((
ProjMap2 (F2,d))
. c)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 3;
end;
hence (
ProjMap2 ((F1
+ F2),d))
= ((
ProjMap2 (F1,d))
+ (
ProjMap2 (F2,d))) by
A5,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:59
for C,D be non
empty
set, F1 be
without-infty
Function of
[:C, D:],
ExtREAL , F2 be
without+infty
Function of
[:C, D:],
ExtREAL , c be
Element of C holds (
ProjMap1 ((F1
- F2),c))
= ((
ProjMap1 (F1,c))
- (
ProjMap1 (F2,c))) & (
ProjMap1 ((F2
- F1),c))
= ((
ProjMap1 (F2,c))
- (
ProjMap1 (F1,c)))
proof
let C,D be non
empty
set;
let F1 be
without-infty
Function of
[:C, D:],
ExtREAL ;
let F2 be
without+infty
Function of
[:C, D:],
ExtREAL ;
let c be
Element of C;
A2: (
dom (
ProjMap1 ((F1
- F2),c)))
= D & (
dom (
ProjMap1 ((F2
- F1),c)))
= D & (
dom (
ProjMap1 (F1,c)))
= D & (
dom (
ProjMap1 (F2,c)))
= D by
FUNCT_2:def 1;
{
-infty }
misses (
rng (
ProjMap1 (F1,c))) &
{
+infty }
misses (
rng (
ProjMap1 (F2,c))) by
ZFMISC_1: 50,
MESFUNC5:def 3,
MESFUNC5:def 4;
then
B1: ((
ProjMap1 (F1,c))
"
{
-infty })
=
{} & ((
ProjMap1 (F2,c))
"
{
+infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap1 ((F1
- F2),c)))
= (((
dom (
ProjMap1 (F1,c)))
/\ (
dom (
ProjMap1 (F2,c))))
\ ((((
ProjMap1 (F1,c))
"
{
-infty })
/\ ((
ProjMap1 (F2,c))
"
{
-infty }))
\/ (((
ProjMap1 (F1,c))
"
{
+infty })
/\ ((
ProjMap1 (F2,c))
"
{
+infty })))) by
A2;
then
A5: (
dom (
ProjMap1 ((F1
- F2),c)))
= (
dom ((
ProjMap1 (F1,c))
- (
ProjMap1 (F2,c)))) by
MESFUNC1:def 4;
(
dom (
ProjMap1 ((F2
- F1),c)))
= (((
dom (
ProjMap1 (F2,c)))
/\ (
dom (
ProjMap1 (F1,c))))
\ ((((
ProjMap1 (F2,c))
"
{
+infty })
/\ ((
ProjMap1 (F1,c))
"
{
+infty }))
\/ (((
ProjMap1 (F2,c))
"
{
-infty })
/\ ((
ProjMap1 (F1,c))
"
{
-infty })))) by
A2,
B1;
then
A6: (
dom (
ProjMap1 ((F2
- F1),c)))
= (
dom ((
ProjMap1 (F2,c))
- (
ProjMap1 (F1,c)))) by
MESFUNC1:def 4;
for d be
object st d
in (
dom (
ProjMap1 ((F1
- F2),c))) holds ((
ProjMap1 ((F1
- F2),c))
. d)
= (((
ProjMap1 (F1,c))
- (
ProjMap1 (F2,c)))
. d)
proof
let d be
object;
assume
A3: d
in (
dom (
ProjMap1 ((F1
- F2),c)));
then
A4: ((
ProjMap1 ((F1
- F2),c))
. d)
= ((F1
- F2)
. (c,d)) & ((
ProjMap1 (F1,c))
. d)
= (F1
. (c,d)) & ((
ProjMap1 (F2,c))
. d)
= (F2
. (c,d)) by
MESFUNC9:def 6;
reconsider d1 = d as
Element of D by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap1 ((F1
- F2),c))
. d)
= (((
ProjMap1 (F1,c))
. d)
- ((
ProjMap1 (F2,c))
. d)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 4;
end;
hence (
ProjMap1 ((F1
- F2),c))
= ((
ProjMap1 (F1,c))
- (
ProjMap1 (F2,c))) by
A5,
FUNCT_1: 2;
for d be
object st d
in (
dom (
ProjMap1 ((F2
- F1),c))) holds ((
ProjMap1 ((F2
- F1),c))
. d)
= (((
ProjMap1 (F2,c))
- (
ProjMap1 (F1,c)))
. d)
proof
let d be
object;
assume
A3: d
in (
dom (
ProjMap1 ((F2
- F1),c)));
then
A4: ((
ProjMap1 ((F2
- F1),c))
. d)
= ((F2
- F1)
. (c,d)) & ((
ProjMap1 (F1,c))
. d)
= (F1
. (c,d)) & ((
ProjMap1 (F2,c))
. d)
= (F2
. (c,d)) by
MESFUNC9:def 6;
reconsider d1 = d as
Element of D by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap1 ((F2
- F1),c))
. d)
= (((
ProjMap1 (F2,c))
. d)
- ((
ProjMap1 (F1,c))
. d)) by
A4,
Th7;
hence thesis by
A3,
A6,
MESFUNC1:def 4;
end;
hence (
ProjMap1 ((F2
- F1),c))
= ((
ProjMap1 (F2,c))
- (
ProjMap1 (F1,c))) by
A6,
FUNCT_1: 2;
end;
theorem ::
DBLSEQ_3:60
for C,D be non
empty
set, F1 be
without-infty
Function of
[:C, D:],
ExtREAL , F2 be
without+infty
Function of
[:C, D:],
ExtREAL , d be
Element of D holds (
ProjMap2 ((F1
- F2),d))
= ((
ProjMap2 (F1,d))
- (
ProjMap2 (F2,d))) & (
ProjMap2 ((F2
- F1),d))
= ((
ProjMap2 (F2,d))
- (
ProjMap2 (F1,d)))
proof
let C,D be non
empty
set;
let F1 be
without-infty
Function of
[:C, D:],
ExtREAL ;
let F2 be
without+infty
Function of
[:C, D:],
ExtREAL ;
let d be
Element of D;
A2: (
dom (
ProjMap2 ((F1
- F2),d)))
= C & (
dom (
ProjMap2 ((F2
- F1),d)))
= C & (
dom (
ProjMap2 (F1,d)))
= C & (
dom (
ProjMap2 (F2,d)))
= C by
FUNCT_2:def 1;
{
-infty }
misses (
rng (
ProjMap2 (F1,d))) &
{
+infty }
misses (
rng (
ProjMap2 (F2,d))) by
ZFMISC_1: 50,
MESFUNC5:def 3,
MESFUNC5:def 4;
then
B1: ((
ProjMap2 (F1,d))
"
{
-infty })
=
{} & ((
ProjMap2 (F2,d))
"
{
+infty })
=
{} by
RELAT_1: 138;
then (
dom (
ProjMap2 ((F1
- F2),d)))
= (((
dom (
ProjMap2 (F1,d)))
/\ (
dom (
ProjMap2 (F2,d))))
\ ((((
ProjMap2 (F1,d))
"
{
-infty })
/\ ((
ProjMap2 (F2,d))
"
{
-infty }))
\/ (((
ProjMap2 (F1,d))
"
{
+infty })
/\ ((
ProjMap2 (F2,d))
"
{
+infty })))) by
A2;
then
A5: (
dom (
ProjMap2 ((F1
- F2),d)))
= (
dom ((
ProjMap2 (F1,d))
- (
ProjMap2 (F2,d)))) by
MESFUNC1:def 4;
(
dom (
ProjMap2 ((F2
- F1),d)))
= (((
dom (
ProjMap2 (F2,d)))
/\ (
dom (
ProjMap2 (F1,d))))
\ ((((
ProjMap2 (F2,d))
"
{
+infty })
/\ ((
ProjMap2 (F1,d))
"
{
+infty }))
\/ (((
ProjMap2 (F2,d))
"
{
-infty })
/\ ((
ProjMap2 (F1,d))
"
{
-infty })))) by
A2,
B1;
then
A6: (
dom (
ProjMap2 ((F2
- F1),d)))
= (
dom ((
ProjMap2 (F2,d))
- (
ProjMap2 (F1,d)))) by
MESFUNC1:def 4;
for c be
object st c
in (
dom (
ProjMap2 ((F1
- F2),d))) holds ((
ProjMap2 ((F1
- F2),d))
. c)
= (((
ProjMap2 (F1,d))
- (
ProjMap2 (F2,d)))
. c)
proof
let c be
object;
assume
A3: c
in (
dom (
ProjMap2 ((F1
- F2),d)));
then
A4: ((
ProjMap2 ((F1
- F2),d))
. c)
= ((F1
- F2)
. (c,d)) & ((
ProjMap2 (F1,d))
. c)
= (F1
. (c,d)) & ((
ProjMap2 (F2,d))
. c)
= (F2
. (c,d)) by
MESFUNC9:def 7;
reconsider c1 = c as
Element of C by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap2 ((F1
- F2),d))
. c)
= (((
ProjMap2 (F1,d))
. c)
- ((
ProjMap2 (F2,d))
. c)) by
A4,
Th7;
hence thesis by
A3,
A5,
MESFUNC1:def 4;
end;
hence (
ProjMap2 ((F1
- F2),d))
= ((
ProjMap2 (F1,d))
- (
ProjMap2 (F2,d))) by
A5,
FUNCT_1: 2;
for c be
object st c
in (
dom (
ProjMap2 ((F2
- F1),d))) holds ((
ProjMap2 ((F2
- F1),d))
. c)
= (((
ProjMap2 (F2,d))
- (
ProjMap2 (F1,d)))
. c)
proof
let c be
object;
assume
A3: c
in (
dom (
ProjMap2 ((F2
- F1),d)));
then
A4: ((
ProjMap2 ((F2
- F1),d))
. c)
= ((F2
- F1)
. (c,d)) & ((
ProjMap2 (F1,d))
. c)
= (F1
. (c,d)) & ((
ProjMap2 (F2,d))
. c)
= (F2
. (c,d)) by
MESFUNC9:def 7;
reconsider c1 = c as
Element of C by
A3;
[c, d]
in
[:C, D:] by
A3,
ZFMISC_1:def 2;
then ((
ProjMap2 ((F2
- F1),d))
. c)
= (((
ProjMap2 (F2,d))
. c)
- ((
ProjMap2 (F1,d))
. c)) by
A4,
Th7;
hence thesis by
A3,
A6,
MESFUNC1:def 4;
end;
hence (
ProjMap2 ((F2
- F1),d))
= ((
ProjMap2 (F2,d))
- (
ProjMap2 (F1,d))) by
A6,
FUNCT_1: 2;
end;
begin
theorem ::
DBLSEQ_3:61
for seq be
nonnegative
ExtREAL_sequence st not (
Partial_Sums seq) is
convergent_to_+infty holds for n be
Nat holds (seq
. n) is
Real
proof
let seq be
nonnegative
ExtREAL_sequence;
assume
A2: not (
Partial_Sums seq) is
convergent_to_+infty;
given N be
Nat such that
A3: not (seq
. N) is
Real;
not (seq
. N)
in
REAL by
A3;
then
A4: (seq
. N)
=
+infty or (seq
. N)
=
-infty by
XXREAL_0: 14;
A6: (
Partial_Sums seq) is
non-decreasing by
MESFUNC9: 16;
now
let g be
Real;
assume
0
< g;
take N;
hereby
let m be
Nat;
assume
A7: N
<= m;
per cases ;
suppose N
=
0 ;
then ((
Partial_Sums seq)
. N)
= (seq
. N) by
MESFUNC9:def 1;
then
A9: g
<= ((
Partial_Sums seq)
. N) by
A4,
SUPINF_2: 51,
XXREAL_0: 3;
((
Partial_Sums seq)
. N)
<= ((
Partial_Sums seq)
. m) by
A7,
MESFUNC9: 16,
RINFSUP2: 7;
hence g
<= ((
Partial_Sums seq)
. m) by
A9,
XXREAL_0: 2;
end;
suppose N
<>
0 ;
then
consider N1 be
Nat such that
A11: N
= (N1
+ 1) by
NAT_1: 6;
A12: ((
Partial_Sums seq)
. N1)
>=
0 by
SUPINF_2: 51;
((
Partial_Sums seq)
. N)
= (((
Partial_Sums seq)
. N1)
+ (seq
. N)) by
A11,
MESFUNC9:def 1
.=
+infty by
A4,
SUPINF_2: 51,
XXREAL_0: 4,
A12,
XXREAL_3: 39;
then ((
Partial_Sums seq)
. m)
=
+infty by
A6,
A7,
RINFSUP2: 7,
XXREAL_0: 4;
hence g
<= ((
Partial_Sums seq)
. m) by
XXREAL_0: 3;
end;
end;
end;
hence contradiction by
A2,
MESFUNC5:def 9;
end;
theorem ::
DBLSEQ_3:62
Th62: for seq be
nonnegative
ExtREAL_sequence st seq is
non-decreasing holds seq is
convergent_to_+infty or seq is
convergent_to_finite_number
proof
let seq be
nonnegative
ExtREAL_sequence;
assume
A1: seq is
non-decreasing;
now
assume seq is
convergent_to_-infty;
then
consider N be
Nat such that
A4: for n be
Nat st N
<= n holds (seq
. n)
<= (
- 1) by
MESFUNC5:def 10;
(seq
. N)
<= (
- 1) & (seq
. N)
>=
0 by
SUPINF_2: 51,
A4;
hence contradiction;
end;
hence thesis by
A1,
RINFSUP2: 37,
MESFUNC5:def 11;
end;
Lm10: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , n be
Element of
NAT st f is
nonnegative holds (
ProjMap1 (f,n)) is
nonnegative & (
ProjMap2 (f,n)) is
nonnegative
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , n be
Element of
NAT ;
assume
A1: f is
nonnegative;
now
let m be
object;
assume m
in (
dom (
ProjMap1 (f,n)));
then
reconsider m1 = m as
Element of
NAT ;
((
ProjMap1 (f,n))
. m1)
= (f
. (n,m1)) by
MESFUNC9:def 6;
hence ((
ProjMap1 (f,n))
. m)
>=
0 by
A1,
SUPINF_2: 51;
end;
hence (
ProjMap1 (f,n)) is
nonnegative by
SUPINF_2: 52;
now
let m be
object;
assume m
in (
dom (
ProjMap2 (f,n)));
then
reconsider m1 = m as
Element of
NAT ;
((
ProjMap2 (f,n))
. m1)
= (f
. (m1,n)) by
MESFUNC9:def 7;
hence ((
ProjMap2 (f,n))
. m)
>=
0 by
A1,
SUPINF_2: 51;
end;
hence (
ProjMap2 (f,n)) is
nonnegative by
SUPINF_2: 52;
end;
registration
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n be
Element of
NAT ;
cluster (
ProjMap1 (f,n)) ->
nonnegative;
correctness by
Lm10;
cluster (
ProjMap2 (f,n)) ->
nonnegative;
correctness by
Lm10;
end
theorem ::
DBLSEQ_3:63
Th63: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT holds (
ProjMap1 ((
Partial_Sums_in_cod2 f),m)) is
non-decreasing
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT ;
set PS = (
ProjMap1 ((
Partial_Sums_in_cod2 f),m));
for n,j be
Nat st j
<= n holds (PS
. j)
<= (PS
. n)
proof
let n,j be
Nat;
defpred
Q[
Nat] means (PS
. j)
<= (PS
. $1);
A6: for k be
Nat holds (PS
. k)
<= (PS
. (k
+ 1))
proof
let k be
Nat;
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
(PS
. (k
+ 1))
= ((
Partial_Sums_in_cod2 f)
. (m,(k
+ 1))) by
MESFUNC9:def 6
.= (((
Partial_Sums_in_cod2 f)
. (m,k1))
+ (f
. (m,(k
+ 1)))) by
DefCSM
.= ((PS
. k)
+ (f
. (m,(k
+ 1)))) by
MESFUNC9:def 6;
hence thesis by
SUPINF_2: 51,
XXREAL_3: 39;
end;
A8: for k be
Nat st k
>= j & (for l be
Nat st l
>= j & l
< k holds
Q[l]) holds
Q[k]
proof
let k be
Nat;
assume that
A9: k
>= j and
A10: for l be
Nat st l
>= j & l
< k holds
Q[l];
now
assume k
> j;
then
A11: k
>= (j
+ 1) by
NAT_1: 13;
per cases by
A11,
XXREAL_0: 1;
suppose k
= (j
+ 1);
hence thesis by
A6;
end;
suppose
A12: k
> (j
+ 1);
then
reconsider l = (k
- 1) as
Element of
NAT by
NAT_1: 20;
k
< (k
+ 1) by
NAT_1: 13;
then
A13: k
> l by
XREAL_1: 19;
k
= (l
+ 1);
then
A14: (PS
. l)
<= (PS
. k) by
A6;
(PS
. j)
<= (PS
. l) by
A10,
A13,
A12,
XREAL_1: 19;
hence thesis by
A14,
XXREAL_0: 2;
end;
end;
hence thesis by
A9,
XXREAL_0: 1;
end;
A15: for k be
Nat st k
>= j holds
Q[k] from
NAT_1:sch 9(
A8);
assume j
<= n;
hence thesis by
A15;
end;
hence thesis by
RINFSUP2: 7;
end;
theorem ::
DBLSEQ_3:64
Th64: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n be
Element of
NAT holds (
ProjMap2 ((
Partial_Sums_in_cod1 f),n)) is
non-decreasing
proof
let f be
nonnegative
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , n be
Element of
NAT ;
A1: (
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),n)) is
non-decreasing by
Th63;
(
Partial_Sums_in_cod2 (
~ f))
= (
~ (
Partial_Sums_in_cod1 f)) by
Th40;
hence (
ProjMap2 ((
Partial_Sums_in_cod1 f),n)) is
non-decreasing by
A1,
Th33;
end;
registration
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT ;
cluster (
ProjMap1 ((
Partial_Sums_in_cod2 f),m)) ->
non-decreasing;
correctness by
Th63;
cluster (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) ->
non-decreasing;
correctness by
Th64;
end
theorem ::
DBLSEQ_3:65
Th65: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL holds (f is
convergent_in_cod1 implies (
lim_in_cod1 f) is
nonnegative) & (f is
convergent_in_cod2 implies (
lim_in_cod2 f) is
nonnegative)
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
hereby
assume
A2: f is
convergent_in_cod1;
now
let n be
object;
assume n
in (
dom (
lim_in_cod1 f));
then
reconsider n1 = n as
Element of
NAT ;
A4: ((
lim_in_cod1 f)
. n)
= (
lim (
ProjMap2 (f,n1))) by
D1DEF5;
for k be
Nat holds
0
<= ((
ProjMap2 (f,n1))
. k) by
SUPINF_2: 51;
hence ((
lim_in_cod1 f)
. n)
>=
0 by
A2,
A4,
MESFUNC9: 10;
end;
hence (
lim_in_cod1 f) is
nonnegative by
SUPINF_2: 52;
end;
assume
A2: f is
convergent_in_cod2;
now
let n be
object;
assume n
in (
dom (
lim_in_cod2 f));
then
reconsider n1 = n as
Element of
NAT ;
A4: ((
lim_in_cod2 f)
. n)
= (
lim (
ProjMap1 (f,n1))) by
D1DEF6;
for k be
Nat holds
0
<= ((
ProjMap1 (f,n1))
. k) by
SUPINF_2: 51;
hence ((
lim_in_cod2 f)
. n)
>=
0 by
A2,
A4,
MESFUNC9: 10;
end;
hence thesis by
SUPINF_2: 52;
end;
theorem ::
DBLSEQ_3:66
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums_in_cod1 f) is
convergent_in_cod1 & (
Partial_Sums_in_cod2 f) is
convergent_in_cod2 by
RINFSUP2: 37;
theorem ::
DBLSEQ_3:67
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT st not (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_+infty holds for n be
Nat holds (f
. (n,m)) is
Real
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let m be
Element of
NAT ;
assume
A2: not (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_+infty;
given N be
Nat such that
A3: not (f
. (N,m)) is
Real;
not (f
. (N,m))
in
REAL by
A3;
then
A4: (f
. (N,m))
=
+infty or (f
. (N,m))
=
-infty by
XXREAL_0: 14;
reconsider N1 = N as
Element of
NAT by
ORDINAL1:def 12;
now
let g be
Real;
assume
0
< g;
take N;
hereby
let k be
Nat;
assume
A7: N
<= k;
per cases ;
suppose
A8: N
=
0 ;
((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. N)
= ((
Partial_Sums_in_cod1 f)
. (N1,m)) by
MESFUNC9:def 7
.= (f
. (N,m)) by
A8,
DefRSM;
then
A9: g
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. N) by
A4,
SUPINF_2: 51,
XXREAL_0: 3;
((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. N)
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. k) by
A7,
RINFSUP2: 7;
hence g
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. k) by
A9,
XXREAL_0: 2;
end;
suppose N
<>
0 ;
then
consider N2 be
Nat such that
A11: N
= (N2
+ 1) by
NAT_1: 6;
reconsider N3 = N2 as
Element of
NAT by
ORDINAL1:def 12;
A12: ((
Partial_Sums_in_cod1 f)
. (N3,m))
>=
0 by
SUPINF_2: 51;
((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. N1)
= ((
Partial_Sums_in_cod1 f)
. (N1,m)) by
MESFUNC9:def 7
.= (((
Partial_Sums_in_cod1 f)
. (N2,m))
+ (f
. (N1,m))) by
A11,
DefRSM;
then ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. N1)
=
+infty by
A4,
SUPINF_2: 51,
XXREAL_0: 4,
A12,
XXREAL_3: 39;
then ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. k)
=
+infty by
XXREAL_0: 4,
A7,
RINFSUP2: 7;
hence g
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. k) by
XXREAL_0: 3;
end;
end;
end;
hence contradiction by
A2,
MESFUNC5:def 9;
end;
theorem ::
DBLSEQ_3:68
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT st not (
ProjMap1 ((
Partial_Sums_in_cod2 f),m)) is
convergent_to_+infty holds for n be
Nat holds (f
. (m,n)) is
Real
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
let m be
Element of
NAT ;
assume
A2: not (
ProjMap1 ((
Partial_Sums_in_cod2 f),m)) is
convergent_to_+infty;
given N be
Nat such that
A3: not (f
. (m,N)) is
Real;
not (f
. (m,N))
in
REAL by
A3;
then
A4: (f
. (m,N))
=
+infty or (f
. (m,N))
=
-infty by
XXREAL_0: 14;
reconsider N1 = N as
Element of
NAT by
ORDINAL1:def 12;
now
let g be
Real;
assume
0
< g;
take N;
hereby
let k be
Nat;
assume
A7: N
<= k;
per cases ;
suppose
A8: N
=
0 ;
((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. N)
= ((
Partial_Sums_in_cod2 f)
. (m,N1)) by
MESFUNC9:def 6
.= (f
. (m,N)) by
A8,
DefCSM;
then
A9: g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. N) by
A4,
SUPINF_2: 51,
XXREAL_0: 3;
((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. N)
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. k) by
A7,
RINFSUP2: 7;
hence g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. k) by
A9,
XXREAL_0: 2;
end;
suppose N
<>
0 ;
then
consider N2 be
Nat such that
A11: N
= (N2
+ 1) by
NAT_1: 6;
reconsider N3 = N2 as
Element of
NAT by
ORDINAL1:def 12;
A12: ((
Partial_Sums_in_cod2 f)
. (m,N3))
>=
0 by
SUPINF_2: 51;
((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. N1)
= ((
Partial_Sums_in_cod2 f)
. (m,N1)) by
MESFUNC9:def 6
.= (((
Partial_Sums_in_cod2 f)
. (m,N2))
+ (f
. (m,N1))) by
A11,
DefCSM;
then ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. N1)
=
+infty by
A4,
SUPINF_2: 51,
XXREAL_0: 4,
A12,
XXREAL_3: 39;
then ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. k)
=
+infty by
XXREAL_0: 4,
A7,
RINFSUP2: 7;
hence g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m))
. k) by
XXREAL_0: 3;
end;
end;
end;
hence contradiction by
A2,
MESFUNC5:def 9;
end;
theorem ::
DBLSEQ_3:69
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat st (for i be
Nat st i
<= n holds (f
. (i,m)) is
Real) holds ((
Partial_Sums_in_cod1 f)
. (n,m))
<
+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
assume
A2: for i be
Nat st i
<= n holds (f
. (i,m)) is
Real;
defpred
P[
Nat] means $1
<= n implies ((
Partial_Sums_in_cod1 f)
. ($1,m))
<
+infty ;
((
Partial_Sums_in_cod1 f)
. (
0 ,m))
= (f
. (
0 ,m)) by
DefRSM;
then ((
Partial_Sums_in_cod1 f)
. (
0 ,m)) is
Real by
A2;
then
A4:
P[
0 ] by
XXREAL_0: 9,
XREAL_0:def 1;
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
now
assume
A7: (k
+ 1)
<= n;
then
A8: (f
. ((k
+ 1),m)) is
Real & (f
. ((k
+ 1),m))
>=
0 by
A2,
SUPINF_2: 51;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM;
hence ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
<
+infty by
A6,
A7,
A8,
NAT_1: 13,
XXREAL_3: 16,
XXREAL_0: 4;
end;
hence
P[(k
+ 1)];
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A4,
A5);
hence thesis;
end;
theorem ::
DBLSEQ_3:70
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat st (for i be
Nat st i
<= m holds (f
. (n,i)) is
Real) holds ((
Partial_Sums_in_cod2 f)
. (n,m))
<
+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
assume
A2: for i be
Nat st i
<= m holds (f
. (n,i)) is
Real;
defpred
P[
Nat] means $1
<= m implies ((
Partial_Sums_in_cod2 f)
. (n,$1))
<
+infty ;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then ((
Partial_Sums_in_cod2 f)
. (n,
0 )) is
Real by
A2;
then
A4:
P[
0 ] by
XXREAL_0: 9,
XREAL_0:def 1;
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
now
assume
A7: (k
+ 1)
<= m;
then
A8: (f
. (n,(k
+ 1))) is
Real & (f
. (n,(k
+ 1)))
>=
0 by
A2,
SUPINF_2: 51;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
<
+infty by
A6,
A7,
A8,
NAT_1: 13,
XXREAL_3: 16,
XXREAL_0: 4;
end;
hence
P[(k
+ 1)];
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A4,
A5);
hence thesis;
end;
theorem ::
DBLSEQ_3:71
for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums f) is
convergent_in_cod1_to_-infty implies ex m be
Element of
NAT st (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_-infty
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: (
Partial_Sums f) is
convergent_in_cod1_to_-infty;
A3: (
ProjMap2 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),
0 ))
= (
ProjMap2 ((
Partial_Sums f),
0 ))
.= (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 )) by
Th54;
assume for m be
Element of
NAT holds not (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_-infty;
hence contradiction by
A1,
A3;
end;
theorem ::
DBLSEQ_3:72
Th72: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat holds (for k be
Element of
NAT st k
<= m holds not (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty) iff (for k be
Element of
NAT st k
<= m holds (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
<
+infty )
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat;
hereby
assume
A1: for k be
Element of
NAT st k
<= m holds not (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty;
hereby
let k be
Element of
NAT ;
assume k
<= m;
then not (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty by
A1;
then (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_finite_number by
Th62;
then ex g be
Real st (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
= g & for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.(((
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
. m)
- (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))).|
< p by
MESFUNC9: 7;
hence (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
<
+infty by
XREAL_0:def 1,
XXREAL_0: 9;
end;
end;
assume
A2: for k be
Element of
NAT st k
<= m holds (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
<
+infty ;
now
let k be
Element of
NAT ;
assume k
<= m;
then (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
<
+infty by
A2;
hence not (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty by
MESFUNC9: 7;
end;
hence thesis;
end;
theorem ::
DBLSEQ_3:73
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat holds (for k be
Element of
NAT st k
<= m holds not (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty) iff (for k be
Element of
NAT st k
<= m holds (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
<
+infty )
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat;
hereby
assume
A1: for k be
Element of
NAT st k
<= m holds not (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty;
hereby
let k be
Element of
NAT ;
assume k
<= m;
then not (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty by
A1;
then (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_finite_number by
Th62;
then ex g be
Real st (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
= g & for p be
Real st
0
< p holds ex n be
Nat st for m be
Nat st n
<= m holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. m)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))).|
< p by
MESFUNC9: 7;
hence (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
<
+infty by
XREAL_0:def 1,
XXREAL_0: 9;
end;
end;
assume
A2: for k be
Element of
NAT st k
<= m holds (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
<
+infty ;
now
let k be
Element of
NAT ;
assume k
<= m;
then (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
<
+infty by
A2;
hence not (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty by
MESFUNC9: 7;
end;
hence thesis;
end;
theorem ::
DBLSEQ_3:74
Th74: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat holds ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
=
+infty iff ex k be
Element of
NAT st k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat;
hereby
assume
A2: ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
=
+infty ;
now
assume
A3: for k be
Element of
NAT holds k
> m or not (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty;
defpred
P[
Nat] means $1
<= m implies ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. $1)
<>
+infty ;
((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
.
0 )
= ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
.
0 ) by
MESFUNC9:def 1
.= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),
0 ))) by
D1DEF6;
then
A5:
P[
0 ] by
A3,
Th72;
A6: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A7:
P[k];
assume
A8: (k
+ 1)
<= m;
A10: ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. (k
+ 1))
= (((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. k)
+ ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. (k
+ 1))) by
MESFUNC9:def 1;
now
assume ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. (k
+ 1))
=
+infty ;
then (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),(k
+ 1))))
=
+infty by
D1DEF6;
hence contradiction by
A3,
A8,
Th72;
end;
hence ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. (k
+ 1))
<>
+infty by
A7,
A8,
NAT_1: 13,
A10,
XXREAL_3: 16;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A5,
A6);
hence contradiction by
A2;
end;
hence ex k be
Element of
NAT st k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty;
end;
given k be
Element of
NAT such that
B1: k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty;
(
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)))
=
+infty by
B1,
MESFUNC9: 7;
then
B2: ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. k)
=
+infty by
D1DEF6;
B3: (
Partial_Sums_in_cod2 f) is
convergent_in_cod2 by
RINFSUP2: 37;
then (
lim_in_cod2 (
Partial_Sums_in_cod2 f)) is
nonnegative by
Th65;
then
B5: ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. k)
>=
+infty by
B2,
Th4;
(
lim_in_cod2 (
Partial_Sums_in_cod2 f)) is
nonnegative by
B3,
Th65;
then ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. k)
<= ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m) by
B1,
RINFSUP2: 7,
MESFUNC9: 16;
hence ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
=
+infty by
B5,
XXREAL_0: 2,
XXREAL_0: 4;
end;
theorem ::
DBLSEQ_3:75
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat holds ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. m)
=
+infty iff ex k be
Element of
NAT st k
<= m & (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Nat;
hereby
assume
A1: ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. m)
=
+infty ;
(
lim_in_cod1 (
Partial_Sums_in_cod1 f))
= (
lim_in_cod2 (
~ (
Partial_Sums_in_cod1 f))) by
Th38
.= (
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f))) by
Th40;
then
consider k be
Element of
NAT such that
A2: k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),k)) is
convergent_to_+infty by
A1,
Th74;
(
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),k))
= (
ProjMap2 ((
~ (
Partial_Sums_in_cod2 (
~ f))),k)) by
Th32
.= (
ProjMap2 ((
Partial_Sums_in_cod1 (
~ (
~ f))),k)) by
Th40
.= (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) by
DBLSEQ_2: 7;
hence ex k be
Element of
NAT st k
<= m & (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty by
A2;
end;
given k be
Element of
NAT such that
A3: k
<= m & (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty;
A4: (
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
= (
ProjMap2 ((
Partial_Sums_in_cod1 (
~ (
~ f))),k)) by
DBLSEQ_2: 7
.= (
ProjMap2 ((
~ (
Partial_Sums_in_cod2 (
~ f))),k)) by
Th40
.= (
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),k)) by
Th32;
(
lim_in_cod1 (
Partial_Sums_in_cod1 f))
= (
lim_in_cod2 (
~ (
Partial_Sums_in_cod1 f))) by
Th38
.= (
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f))) by
Th40;
hence ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. m)
=
+infty by
A3,
A4,
Th74;
end;
theorem ::
DBLSEQ_3:76
Th76: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat holds ((
Partial_Sums_in_cod1 f)
. (n,m))
>= (f
. (n,m)) & ((
Partial_Sums_in_cod2 f)
. (n,m))
>= (f
. (n,m))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , n,m be
Nat;
defpred
P[
Nat] means $1
<= n implies ((
Partial_Sums_in_cod1 f)
. ($1,m))
>= (f
. ($1,m));
A2:
P[
0 ] by
DefRSM;
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat such that
P[k];
assume (k
+ 1)
<= n;
((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM;
hence ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
>= (f
. ((k
+ 1),m)) by
SUPINF_2: 51,
XXREAL_3: 39;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A2,
A5);
hence ((
Partial_Sums_in_cod1 f)
. (n,m))
>= (f
. (n,m));
defpred
Q[
Nat] means $1
<= m implies ((
Partial_Sums_in_cod2 f)
. (n,$1))
>= (f
. (n,$1));
A2:
Q[
0 ] by
DefCSM;
A5: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat such that
Q[k];
assume (k
+ 1)
<= m;
((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
hence ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
>= (f
. (n,(k
+ 1))) by
SUPINF_2: 51,
XXREAL_3: 39;
end;
for k be
Nat holds
Q[k] from
NAT_1:sch 2(
A2,
A5);
hence ((
Partial_Sums_in_cod2 f)
. (n,m))
>= (f
. (n,m));
end;
theorem ::
DBLSEQ_3:77
Th77: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT st (ex k be
Element of
NAT st k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty) holds (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m)) is
convergent_to_+infty & (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m)))
=
+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT ;
given k be
Element of
NAT such that
A2: k
<= m & (
ProjMap1 ((
Partial_Sums_in_cod2 f),k)) is
convergent_to_+infty;
A5: for g be
Real st
0
< g holds ex N be
Nat st for n be
Nat st N
<= n holds g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m))
. n)
proof
let g be
Real;
assume
0
< g;
then
consider N be
Nat such that
A4: for n be
Nat st N
<= n holds g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
. n) by
A2,
MESFUNC5:def 9;
now
let n be
Nat;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
assume N
<= n;
then g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),k))
. n) by
A4;
then
A7: g
<= ((
Partial_Sums_in_cod2 f)
. (k,n1)) by
MESFUNC9:def 6;
((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),n1))
. k)
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),n1))
. m) by
A2,
RINFSUP2: 7;
then ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (k,n1))
<= ((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),n1))
. m) by
MESFUNC9:def 7;
then
A10: ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (k,n1))
<= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,n1)) by
MESFUNC9:def 7;
((
Partial_Sums_in_cod2 f)
. (k,n1))
<= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (k,n1)) by
Th76;
then
A9: ((
Partial_Sums_in_cod2 f)
. (k,n1))
<= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,n)) by
A10,
XXREAL_0: 2;
((
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m))
. n1)
= ((
Partial_Sums f)
. (m,n)) by
MESFUNC9:def 6
.= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (m,n)) by
Lm8;
hence g
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m))
. n) by
A7,
A9,
XXREAL_0: 2;
end;
hence thesis;
end;
hence (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m)) is
convergent_to_+infty by
MESFUNC5:def 9;
thus (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m)))
=
+infty by
A5,
MESFUNC5:def 9,
MESFUNC9: 7;
end;
theorem ::
DBLSEQ_3:78
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT st (ex k be
Element of
NAT st k
<= m & (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty) holds (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),m)) is
convergent_to_+infty & (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),m)))
=
+infty
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , m be
Element of
NAT ;
given k be
Element of
NAT such that
A1: k
<= m & (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_+infty;
A3: (
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
= (
ProjMap2 ((
Partial_Sums_in_cod1 (
~ (
~ f))),k)) by
DBLSEQ_2: 7
.= (
ProjMap2 ((
~ (
Partial_Sums_in_cod2 (
~ f))),k)) by
Th40
.= (
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),k)) by
Th32;
(
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 (
~ f))),m))
= (
ProjMap2 ((
~ (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 (
~ f)))),m)) by
Th32
.= (
ProjMap2 ((
Partial_Sums_in_cod1 (
~ (
Partial_Sums_in_cod1 (
~ f)))),m)) by
Th40
.= (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
~ (
~ f)))),m)) by
Th40
.= (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),m)) by
DBLSEQ_2: 7;
hence thesis by
A3,
A1,
Th77;
end;
theorem ::
DBLSEQ_3:79
Th79: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums f) is
convergent_in_cod1_to_finite iff (
Partial_Sums_in_cod1 f) is
convergent_in_cod1_to_finite
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
hereby
assume
A1: (
Partial_Sums f) is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
defpred
P[
Nat] means for k be
Element of
NAT st k
= $1 holds (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_finite_number;
now
let k be
Element of
NAT ;
assume k
=
0 ;
then (
ProjMap2 ((
Partial_Sums f),k))
= (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) by
Th54;
hence (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_finite_number by
A1;
end;
then
A3:
P[
0 ];
A4: for m1 be
Nat st
P[m1] holds
P[(m1
+ 1)]
proof
let m1 be
Nat;
reconsider m = m1 as
Element of
NAT by
ORDINAL1:def 12;
assume
X1:
P[m1];
hereby
let k be
Element of
NAT ;
assume
B2: k
= (m1
+ 1);
then
reconsider k1 = (k
- 1) as
Element of
NAT by
NAT_1: 11,
NAT_1: 21;
F1: (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_finite_number by
X1;
not (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_+infty & not (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_-infty by
X1,
MESFUNC5: 50,
MESFUNC5: 51;
then
consider G0 be
Real such that
F3: (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)))
= G0 & (for e be
Real st
0
< e holds ex N be
Nat st for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)))).|
< e) & (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_finite_number by
F1,
MESFUNC5:def 12;
consider G1 be
Real such that
E3: (
lim (
ProjMap2 ((
Partial_Sums f),m)))
= G1 & (for e be
Real st
0
< e holds ex N be
Nat st for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))).|
< e) by
A1,
MESFUNC9: 7;
consider G2 be
Real such that
E4: (
lim (
ProjMap2 ((
Partial_Sums f),k)))
= G2 & (for e be
Real st
0
< e holds ex N be
Nat st for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))).|
< e) by
A1,
MESFUNC9: 7;
E7: (
- G1)
= (
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))) & (
- G2)
= (
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))) by
E3,
E4,
XXREAL_3:def 3;
then
E5: (G2
+ (
- G1))
= ((
lim (
ProjMap2 ((
Partial_Sums f),k)))
+ (
- (
lim (
ProjMap2 ((
Partial_Sums f),m))))) by
E4,
XXREAL_3:def 2;
now
let e be
Real;
assume
B6:
0
< e;
then
consider N1 be
Nat such that
B7: for n be
Nat st n
>= N1 holds
|.(((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))).|
< (e
/ 2) by
E3;
consider N2 be
Nat such that
B8: for n be
Nat st n
>= N2 holds
|.(((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))).|
< (e
/ 2) by
B6,
E4;
consider N0 be
Nat such that
B10: for n be
Nat st n
>= N0 holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)))).|
< e by
B6,
F3;
reconsider N = (
max ((
max (N1,N2)),N0)) as
Nat by
TARSKI: 1;
take N;
hereby
let n be
Nat;
assume
B9: n
>= N;
N
>= (
max (N1,N2)) & N
>= N0 & (
max (N1,N2))
>= N1 & (
max (N1,N2))
>= N2 by
XXREAL_0: 25;
then N
>= N1 & N
>= N2 & N
>= N0 by
XXREAL_0: 2;
then
K0: n
>= N1 & n
>= N2 & n
>= N0 by
B9,
XXREAL_0: 2;
then
|.(((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))).|
< (e
/ 2) &
|.(((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))).|
< (e
/ 2) &
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)))).|
< e by
B7,
B8,
B10;
then
B12: (
|.(((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))).|
+
|.(((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))).|)
< ((e
/ 2)
+ (e
/ 2)) by
XXREAL_3: 64;
K2:
now
assume ((
ProjMap2 ((
Partial_Sums f),m))
. n)
=
+infty ;
then (((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m))))
=
+infty by
E3,
XXREAL_3: 13;
hence contradiction by
K0,
B7,
XXREAL_0: 3,
EXTREAL1: 30;
end;
KK2:
now
assume ((
ProjMap2 ((
Partial_Sums f),m))
. n)
=
-infty ;
then (((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m))))
=
-infty by
E3,
XXREAL_3: 14;
hence contradiction by
K0,
B7,
XXREAL_0: 3,
EXTREAL1: 30;
end;
K5:
now
assume ((
ProjMap2 ((
Partial_Sums f),k))
. n)
=
+infty ;
then (((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k))))
=
+infty by
E4,
XXREAL_3: 13;
hence contradiction by
K0,
B8,
XXREAL_0: 3,
EXTREAL1: 30;
end;
KK5:
now
assume ((
ProjMap2 ((
Partial_Sums f),k))
. n)
=
-infty ;
then (((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k))))
=
-infty by
E4,
XXREAL_3: 14;
hence contradiction by
K0,
B8,
XXREAL_0: 3,
EXTREAL1: 30;
end;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
XX2: ((
ProjMap2 ((
Partial_Sums f),k))
. n)
= ((
Partial_Sums f)
. (n1,k)) by
MESFUNC9:def 7
.= (((
Partial_Sums f)
. (n1,m))
+ ((
Partial_Sums_in_cod1 f)
. (n1,k))) by
B2,
DefCSM
.= (((
ProjMap2 ((
Partial_Sums f),m))
. n)
+ ((
Partial_Sums_in_cod1 f)
. (n1,k))) by
MESFUNC9:def 7
.= (((
ProjMap2 ((
Partial_Sums f),m))
. n)
+ ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)) by
MESFUNC9:def 7;
((
ProjMap2 ((
Partial_Sums f),k))
. n)
in
REAL by
K5,
KK5,
XXREAL_0: 14;
then
reconsider r4 = ((
ProjMap2 ((
Partial_Sums f),k))
. n) as
Real;
((
ProjMap2 ((
Partial_Sums f),m))
. n)
in
REAL by
K2,
KK2,
XXREAL_0: 14;
then
reconsider r5 = ((
ProjMap2 ((
Partial_Sums f),m))
. n) as
Real;
r4
= (r5
+ ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)) by
XX2;
then ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
<>
+infty & ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
<>
-infty ;
then ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
in
REAL by
XXREAL_0: 14;
then
reconsider r1 = ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n) as
Real;
T1: (((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
- ((
lim (
ProjMap2 ((
Partial_Sums f),k)))
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))))
= (((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
- (G2
- G1)) by
E5
.= (r1
+ (
- (G2
- G1))) by
XXREAL_3:def 2;
T2: (r5
+ r1)
= (((
ProjMap2 ((
Partial_Sums f),m))
. n)
+ ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)) by
XXREAL_3:def 2;
E8: (((
ProjMap2 ((
Partial_Sums f),k))
. n)
+ (
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))))
= (r4
+ (
- G2)) & (((
ProjMap2 ((
Partial_Sums f),m))
. n)
+ (
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))))
= (r5
+ (
- G1)) by
E7,
XXREAL_3:def 2;
then (
- (((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))))
= (
- (r5
- G1)) by
XXREAL_3:def 3;
then ((((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k))))
+ (
- (((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m))))))
= ((r4
- G2)
+ (
- (r5
- G1))) by
E8,
XXREAL_3:def 2;
then
T3: ((((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k))))
- (((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))))
= ((r4
- G2)
- (r5
- G1));
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
- ((
lim (
ProjMap2 ((
Partial_Sums f),k)))
- (
lim (
ProjMap2 ((
Partial_Sums f),m))))).|
<= (
|.(((
ProjMap2 ((
Partial_Sums f),k))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),k)))).|
+
|.(((
ProjMap2 ((
Partial_Sums f),m))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums f),m)))).|) by
T1,
T2,
XX2,
T3,
EXTREAL1: 32;
hence
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. n)
- (G2
- G1)).|
< e by
B12,
E5,
XXREAL_0: 2;
end;
end;
hence (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)) is
convergent_to_finite_number by
MESFUNC5:def 8;
end;
end;
for m1 be
Nat holds
P[m1] from
NAT_1:sch 2(
A3,
A4);
hence (
ProjMap2 ((
Partial_Sums_in_cod1 f),m)) is
convergent_to_finite_number;
end;
hence (
Partial_Sums_in_cod1 f) is
convergent_in_cod1_to_finite;
end;
assume
C0: (
Partial_Sums_in_cod1 f) is
convergent_in_cod1_to_finite;
now
let m be
Element of
NAT ;
defpred
P[
Nat] means for k be
Element of
NAT st k
= $1 holds (
ProjMap2 ((
Partial_Sums f),k)) is
convergent_to_finite_number;
(
ProjMap2 ((
Partial_Sums f),
0 ))
= (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 )) by
Th54;
then
C1:
P[
0 ] by
C0;
C2: for m be
Nat st
P[m] holds
P[(m
+ 1)]
proof
let m be
Nat;
assume
C3:
P[m];
reconsider m1 = m as
Element of
NAT by
ORDINAL1:def 12;
hereby
let k be
Element of
NAT ;
assume
C4: k
= (m
+ 1);
then
reconsider k1 = (k
- 1) as
Element of
NAT by
NAT_1: 11,
NAT_1: 21;
reconsider f1 = (
ProjMap2 ((
Partial_Sums f),m1)), f2 = (
ProjMap2 ((
Partial_Sums_in_cod1 f),(m1
+ 1))) as
without-infty
ExtREAL_sequence;
for n be
Element of
NAT holds ((
ProjMap2 ((
Partial_Sums f),k))
. n)
= (((
ProjMap2 ((
Partial_Sums f),m1))
+ (
ProjMap2 ((
Partial_Sums_in_cod1 f),(m1
+ 1))))
. n)
proof
let n be
Element of
NAT ;
((
ProjMap2 ((
Partial_Sums f),k))
. n)
= ((
Partial_Sums f)
. (n,(m1
+ 1))) by
C4,
MESFUNC9:def 7
.= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,(m1
+ 1))) by
Lm8
.= (((
Partial_Sums_in_cod1 f)
. (n,(m1
+ 1)))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,m1))) by
Th47
.= (((
Partial_Sums_in_cod1 f)
. (n,(m1
+ 1)))
+ ((
Partial_Sums f)
. (n,m1))) by
Lm8
.= (((
ProjMap2 ((
Partial_Sums_in_cod1 f),(m1
+ 1)))
. n)
+ ((
Partial_Sums f)
. (n,m1))) by
MESFUNC9:def 7
.= (((
ProjMap2 ((
Partial_Sums_in_cod1 f),(m1
+ 1)))
. n)
+ ((
ProjMap2 ((
Partial_Sums f),m1))
. n)) by
MESFUNC9:def 7;
hence thesis by
Th7;
end;
then
C5: (
ProjMap2 ((
Partial_Sums f),k))
= (f1
+ f2) by
FUNCT_2:def 8;
(
ProjMap2 ((
Partial_Sums f),m1)) is
convergent_to_finite_number & (
ProjMap2 ((
Partial_Sums_in_cod1 f),(m1
+ 1))) is
convergent_to_finite_number by
C3,
C0;
hence (
ProjMap2 ((
Partial_Sums f),k)) is
convergent_to_finite_number by
C5,
Th23;
end;
end;
for m be
Nat holds
P[m] from
NAT_1:sch 2(
C1,
C2);
hence (
ProjMap2 ((
Partial_Sums f),m)) is
convergent_to_finite_number;
end;
hence thesis;
end;
theorem ::
DBLSEQ_3:80
Th80: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL st (
Partial_Sums f) is
convergent_in_cod1_to_finite holds for m be
Element of
NAT holds ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. m)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),m)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: (
Partial_Sums f) is
convergent_in_cod1_to_finite;
then
A2: (
Partial_Sums_in_cod1 f) is
convergent_in_cod1_to_finite by
Th79;
let m be
Element of
NAT ;
defpred
P[
Nat] means for k be
Element of
NAT st k
<= $1 holds ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. k)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),k)));
now
let k be
Element of
NAT ;
assume k
<=
0 ;
then
A5: k
=
0 ;
A6: ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
.
0 )
= ((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
.
0 ) by
MESFUNC9:def 1
.= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))) by
D1DEF5;
consider G be
Real such that
A7: (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 )))
= G & for p be
Real st
0
< p holds ex N be
Nat st for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))
. n)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 )))).|
< p by
A2,
MESFUNC9: 7;
reconsider G1 = G as
R_eal by
XXREAL_0:def 1;
(
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),
0 ))
= (
ProjMap2 ((
Partial_Sums f),
0 )) by
Lm8;
then
A8: (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),
0 )) is
convergent_to_finite_number by
A1;
for p be
Real st
0
< p holds ex N be
Nat st for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),
0 ))
. n)
- G1).|
< p
proof
let p be
Real;
assume
0
< p;
then
consider N be
Nat such that
A11: for n be
Nat st N
<= n holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))
. n)
- G1).|
< p by
A7;
now
let n be
Nat;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
assume
A12: N
<= n;
((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),
0 ))
. n1)
= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (n,
0 )) by
MESFUNC9:def 7
.= ((
Partial_Sums f)
. (n,
0 )) by
Lm8
.= ((
ProjMap2 ((
Partial_Sums f),
0 ))
. n1) by
MESFUNC9:def 7
.= ((
ProjMap2 ((
Partial_Sums_in_cod1 f),
0 ))
. n1) by
Th54;
hence
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),
0 ))
. n)
- G1).|
< p by
A11,
A12;
end;
hence thesis;
end;
hence ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. k)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),k))) by
A5,
A6,
A7,
A8,
MESFUNC5:def 12;
end;
then
A13:
P[
0 ];
A14: for n be
Nat st
P[n] holds
P[(n
+ 1)]
proof
let n be
Nat;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
assume
A15:
P[n];
now
let k be
Element of
NAT ;
assume
A16: k
<= (n
+ 1);
per cases ;
suppose k
< (n
+ 1);
then k
<= n by
NAT_1: 13;
hence ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. k)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),k))) by
A15;
end;
suppose k
>= (n
+ 1);
then
A17: k
= (n
+ 1) by
A16,
XXREAL_0: 1;
then
A18: ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. k)
= (((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. n)
+ ((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
. (n
+ 1))) by
MESFUNC9:def 1
.= ((
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),n1)))
+ ((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
. (n
+ 1))) by
A15
.= ((
lim (
ProjMap2 ((
Partial_Sums f),n1)))
+ ((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
. (n
+ 1))) by
Lm8
.= ((
lim (
ProjMap2 ((
Partial_Sums f),n1)))
+ (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))) by
A17,
D1DEF5;
consider Gn be
Real such that
A19: (
lim (
ProjMap2 ((
Partial_Sums f),n1)))
= Gn & for p be
Real st
0
< p holds ex I be
Nat st for i be
Nat st I
<= i holds
|.(((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- (
lim (
ProjMap2 ((
Partial_Sums f),n1)))).|
< p by
A1,
MESFUNC9: 7;
consider Gn1 be
Real such that
A20: (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))
= Gn1 & for p be
Real st
0
< p holds ex I be
Nat st for i be
Nat st I
<= i holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))).|
< p by
A2,
MESFUNC9: 7;
reconsider Gna = Gn, Gn1a = Gn1 as
R_eal by
XXREAL_0:def 1;
set G = (Gna
+ Gn1a);
A21: (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),k)))
= ((
lim_in_cod1 (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)))
. k) by
D1DEF5
.= ((
lim_in_cod1 (
Partial_Sums f))
. k) by
Lm8
.= (
lim (
ProjMap2 ((
Partial_Sums f),k))) by
D1DEF5;
A22: (
ProjMap2 ((
Partial_Sums f),k)) is
convergent_to_finite_number by
A1;
for p be
Real st
0
< p holds ex I be
Nat st for i be
Nat st I
<= i holds
|.(((
ProjMap2 ((
Partial_Sums f),k))
. i)
- G).|
< p
proof
let p be
Real;
assume
A24:
0
< p;
then
consider I1 be
Nat such that
A25: for i be
Nat st I1
<= i holds
|.(((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- (
lim (
ProjMap2 ((
Partial_Sums f),n1)))).|
< (p
/ 2) by
A19;
consider I2 be
Nat such that
A26: for i be
Nat st I2
<= i holds
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),k)))).|
< (p
/ 2) by
A20,
A24;
reconsider I = (
max (I1,I2)) as
Nat by
XXREAL_0: 16;
A27: I
>= I1 & I
>= I2 by
XXREAL_0: 25;
now
let i be
Nat;
reconsider i1 = i as
Element of
NAT by
ORDINAL1:def 12;
assume I
<= i;
then I1
<= i & I2
<= i by
A27,
XXREAL_0: 2;
then
|.(((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gn).|
< (p
/ 2) &
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1).|
< (p
/ 2) by
A19,
A20,
A25,
A26;
then
A28: (
|.(((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gn).|
+
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1).|)
< ((p
/ 2)
+ (p
/ 2)) by
XXREAL_3: 64;
A29: ((
ProjMap2 ((
Partial_Sums f),k))
. i1)
= ((
Partial_Sums f)
. (i,k)) by
MESFUNC9:def 7
.= ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,k)) by
Lm8
.= (((
Partial_Sums_in_cod1 f)
. (i,k))
+ ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))
. (i,n))) by
A17,
Th47
.= (((
Partial_Sums f)
. (i,n))
+ ((
Partial_Sums_in_cod1 f)
. (i,k))) by
Lm8
.= (((
ProjMap2 ((
Partial_Sums f),n1))
. i1)
+ ((
Partial_Sums_in_cod1 f)
. (i,k))) by
MESFUNC9:def 7
.= (((
ProjMap2 ((
Partial_Sums f),n1))
. i1)
+ ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i1)) by
MESFUNC9:def 7;
((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i1)
<>
-infty by
SUPINF_2: 51;
then
A30: (((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i1)
- Gn1a)
<>
-infty by
XXREAL_3: 19;
then
A31: ((((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i1)
- Gn1a)
+ Gn1a)
<>
-infty by
XXREAL_3: 17;
((
ProjMap2 ((
Partial_Sums f),n1))
. i1)
>=
0 by
SUPINF_2: 51;
then
A32: (((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
<>
-infty by
XXREAL_3: 19;
((
ProjMap2 ((
Partial_Sums f),n1))
. i)
= ((((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
+ Gna) & ((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
= ((((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a)
+ Gn1a) by
XXREAL_3: 22;
then ((
ProjMap2 ((
Partial_Sums f),k))
. i)
= ((((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
+ (((((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a)
+ Gn1a)
+ Gna)) by
A29,
A31,
A32,
XXREAL_3: 29
.= ((((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
+ ((((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a)
+ (Gn1a
+ Gna))) by
XXREAL_3: 29
.= (((((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
+ (((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a))
+ (Gna
+ Gn1a)) by
A30,
A32,
XXREAL_3: 29;
then (((
ProjMap2 ((
Partial_Sums f),k))
. i)
- G)
= ((((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna)
+ (((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a)) by
XXREAL_3: 22;
then
|.(((
ProjMap2 ((
Partial_Sums f),k))
. i)
- G).|
<= (
|.(((
ProjMap2 ((
Partial_Sums f),n1))
. i)
- Gna).|
+
|.(((
ProjMap2 ((
Partial_Sums_in_cod1 f),k))
. i)
- Gn1a).|) by
EXTREAL1: 24;
hence
|.(((
ProjMap2 ((
Partial_Sums f),k))
. i)
- G).|
< p by
A28,
XXREAL_0: 2;
end;
hence thesis;
end;
hence ((
Partial_Sums (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
. k)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)),k))) by
A18,
A19,
A20,
A21,
A22,
MESFUNC5:def 12;
end;
end;
hence
P[(n
+ 1)];
end;
for n be
Nat holds
P[n] from
NAT_1:sch 2(
A13,
A14);
hence thesis;
end;
theorem ::
DBLSEQ_3:81
for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
Partial_Sums f) is
convergent_in_cod2_to_finite iff (
Partial_Sums_in_cod2 f) is
convergent_in_cod2_to_finite
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
hereby
assume (
Partial_Sums f) is
convergent_in_cod2_to_finite;
then (
~ (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f))) is
convergent_in_cod1_to_finite by
Th36;
then (
Partial_Sums_in_cod1 (
~ (
Partial_Sums_in_cod1 f))) is
convergent_in_cod1_to_finite by
Th40;
then (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
~ f))) is
convergent_in_cod1_to_finite by
Th40;
then (
Partial_Sums (
~ f)) is
convergent_in_cod1_to_finite by
Lm8;
then (
Partial_Sums_in_cod1 (
~ f)) is
convergent_in_cod1_to_finite by
Th79;
then (
~ (
Partial_Sums_in_cod2 f)) is
convergent_in_cod1_to_finite by
Th40;
hence (
Partial_Sums_in_cod2 f) is
convergent_in_cod2_to_finite by
Th36;
end;
assume (
Partial_Sums_in_cod2 f) is
convergent_in_cod2_to_finite;
then (
~ (
Partial_Sums_in_cod2 f)) is
convergent_in_cod1_to_finite by
Th36;
then (
Partial_Sums_in_cod1 (
~ f)) is
convergent_in_cod1_to_finite by
Th40;
then (
Partial_Sums (
~ f)) is
convergent_in_cod1_to_finite by
Th79;
then (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
~ f))) is
convergent_in_cod1_to_finite by
Lm8;
then (
Partial_Sums_in_cod1 (
~ (
Partial_Sums_in_cod1 f))) is
convergent_in_cod1_to_finite by
Th40;
then (
~ (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f))) is
convergent_in_cod1_to_finite by
Th40;
hence (
Partial_Sums f) is
convergent_in_cod2_to_finite by
Th36;
end;
theorem ::
DBLSEQ_3:82
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL st (
Partial_Sums f) is
convergent_in_cod2_to_finite holds for m be
Element of
NAT holds ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume (
Partial_Sums f) is
convergent_in_cod2_to_finite;
then (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f)) is
convergent_in_cod2_to_finite by
Lm8;
then (
~ (
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 f))) is
convergent_in_cod1_to_finite by
Th36;
then (
Partial_Sums_in_cod2 (
~ (
Partial_Sums_in_cod2 f))) is
convergent_in_cod1_to_finite by
Th40;
then
A1: (
Partial_Sums (
~ f)) is
convergent_in_cod1_to_finite by
Th40;
hereby
let m be
Element of
NAT ;
(
lim_in_cod2 (
Partial_Sums_in_cod2 f))
= (
lim_in_cod1 (
~ (
Partial_Sums_in_cod2 f))) by
Th38
.= (
lim_in_cod1 (
Partial_Sums_in_cod1 (
~ f))) by
Th40;
then ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
Partial_Sums_in_cod2 (
~ f))),m))) by
A1,
Th80
.= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 (
~ (
Partial_Sums_in_cod1 f))),m))) by
Th40
.= (
lim (
ProjMap2 ((
~ (
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f))),m))) by
Th40;
hence ((
Partial_Sums (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. m)
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 (
Partial_Sums_in_cod1 f)),m))) by
Th32;
end;
end;
theorem ::
DBLSEQ_3:83
Th83: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence st (for m be
Element of
NAT holds (seq
. m)
= (
lim_inf (
ProjMap2 (f,m)))) holds (
Sum seq)
<= (
lim_inf (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence;
assume
A1: for m be
Element of
NAT holds (seq
. m)
= (
lim_inf (
ProjMap2 (f,m)));
A2: for m be
Element of
NAT holds for N,n be
Element of
NAT st n
>= N holds ((
inferior_realsequence (
ProjMap2 (f,m)))
. N)
<= (f
. (n,m))
proof
let m be
Element of
NAT ;
let N,n be
Element of
NAT ;
assume n
>= N;
then
A4: ((
inferior_realsequence (
ProjMap2 (f,m)))
. N)
<= ((
inferior_realsequence (
ProjMap2 (f,m)))
. n) by
RINFSUP2: 7;
((
inferior_realsequence (
ProjMap2 (f,m)))
. n)
<= ((
ProjMap2 (f,m))
. n) by
RINFSUP2: 8;
then ((
inferior_realsequence (
ProjMap2 (f,m)))
. n)
<= (f
. (n,m)) by
MESFUNC9:def 7;
hence ((
inferior_realsequence (
ProjMap2 (f,m)))
. N)
<= (f
. (n,m)) by
A4,
XXREAL_0: 2;
end;
deffunc
F(
Element of
NAT ) = (
inferior_realsequence (
ProjMap2 (f,$1)));
deffunc
G(
Element of
NAT ,
Element of
NAT ) = ((
inferior_realsequence (
ProjMap2 (f,$2)))
. $1);
consider g be
Function of
[:
NAT ,
NAT :],
ExtREAL such that
A5: for n be
Element of
NAT holds for m be
Element of
NAT holds (g
. (n,m))
=
G(n,m) from
BINOP_1:sch 4;
now
let z be
object;
per cases ;
suppose z
in (
dom g);
then
consider n,m be
object such that
D1: n
in
NAT & m
in
NAT & z
=
[n, m] by
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
D1;
(g
. (n,m))
= ((
inferior_realsequence (
ProjMap2 (f,m)))
. n) by
A5;
then
consider Y be non
empty
Subset of
ExtREAL such that
D2: Y
= { ((
ProjMap2 (f,m))
. k) where k be
Nat : n
<= k } & (g
. z)
= (
inf Y) by
D1,
RINFSUP2:def 6;
for x be
ExtReal st x
in Y holds
0
<= x
proof
let x be
ExtReal;
assume x
in Y;
then ex k be
Nat st x
= ((
ProjMap2 (f,m))
. k) & n
<= k by
D2;
hence
0
<= x by
SUPINF_2: 51;
end;
then
0 is
LowerBound of Y by
XXREAL_2:def 2;
hence
0
<= (g
. z) by
D2,
XXREAL_2:def 4;
end;
suppose not z
in (
dom g);
hence
0
<= (g
. z) by
FUNCT_1:def 2;
end;
end;
then g is
nonnegative by
SUPINF_2: 51;
then
reconsider g as
nonnegative
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A6: for m be
Element of
NAT holds for N,n be
Element of
NAT st n
>= N holds ((
Partial_Sums_in_cod2 g)
. (N,m))
<= ((
Partial_Sums_in_cod2 f)
. (n,m))
proof
let m be
Element of
NAT ;
let N,n be
Element of
NAT ;
assume
A7: n
>= N;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 g)
. (N,$1))
<= ((
Partial_Sums_in_cod2 f)
. (n,$1));
A8: ((
Partial_Sums_in_cod2 g)
. (N,
0 ))
= (g
. (N,
0 )) by
DefCSM
.= ((
inferior_realsequence (
ProjMap2 (f,
0 )))
. N) by
A5;
((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
then
A9:
P[
0 ] by
A2,
A7,
A8;
A10: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
assume
A11:
P[k];
(g
. (N,(k1
+ 1)))
= ((
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))
. N) by
A5;
then
A12: (g
. (N,(k1
+ 1)))
<= (f
. (n,(k1
+ 1))) by
A2,
A7;
((
Partial_Sums_in_cod2 g)
. (N,(k
+ 1)))
= (((
Partial_Sums_in_cod2 g)
. (N,k))
+ (g
. (N,(k1
+ 1)))) & ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k1
+ 1)))) by
DefCSM;
hence
P[(k
+ 1)] by
A11,
A12,
XXREAL_3: 36;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A9,
A10);
hence thesis;
end;
A13: for m be
Element of
NAT holds for N,n be
Element of
NAT st n
>= N holds ((
Partial_Sums_in_cod2 g)
. (N,m))
<= ((
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. n)
proof
let m be
Element of
NAT ;
let N,n be
Element of
NAT ;
assume
A14: n
>= N;
consider Y be non
empty
Subset of
ExtREAL such that
A15: Y
= { ((
ProjMap2 ((
Partial_Sums_in_cod2 f),m))
. k) where k be
Nat : n
<= k } & ((
inferior_realsequence (
ProjMap2 ((
Partial_Sums_in_cod2 f),m)))
. n)
= (
inf Y) by
RINFSUP2:def 6;
for x be
ExtReal st x
in Y holds ((
Partial_Sums_in_cod2 g)
. (N,m))
<= x
proof
let x be
ExtReal;
assume x
in Y;
then
consider k be
Nat such that
A17: x
= ((
ProjMap2 ((
Partial_Sums_in_cod2 f),m))
. k) & n
<= k by
A15;
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
A18: N
<= k1 by
A14,
A17,
XXREAL_0: 2;
x
= ((
Partial_Sums_in_cod2 f)
. (k1,m)) by
A17,
MESFUNC9:def 7;
hence ((
Partial_Sums_in_cod2 g)
. (N,m))
<= x by
A6,
A18;
end;
then ((
Partial_Sums_in_cod2 g)
. (N,m)) is
LowerBound of Y by
XXREAL_2:def 2;
then
A19: ((
Partial_Sums_in_cod2 g)
. (N,m))
<= ((
inferior_realsequence (
ProjMap2 ((
Partial_Sums_in_cod2 f),m)))
. n) by
A15,
XXREAL_2:def 4;
consider Z be non
empty
Subset of
ExtREAL such that
A20: Z
= { ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. k) where k be
Nat : n
<= k } & ((
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. n)
= (
inf Z) by
RINFSUP2:def 6;
for z be
ExtReal st z
in Z holds ex y be
ExtReal st y
in Y & y
<= z
proof
let z be
ExtReal;
assume z
in Z;
then
consider j be
Nat such that
A21: z
= ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. j) & n
<= j by
A20;
reconsider j1 = j as
Element of
NAT by
ORDINAL1:def 12;
z
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),j1))) by
A21,
D1DEF6;
then
A23: z
= (
sup (
ProjMap1 ((
Partial_Sums_in_cod2 f),j1))) by
RINFSUP2: 37;
set y = ((
ProjMap2 ((
Partial_Sums_in_cod2 f),m))
. j1);
take y;
y
= ((
Partial_Sums_in_cod2 f)
. (j1,m)) by
MESFUNC9:def 7
.= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),j1))
. m) by
MESFUNC9:def 6;
hence y
in Y & y
<= z by
A15,
A21,
A23,
RINFSUP2: 23;
end;
then (
inf Y)
<= (
inf Z) by
XXREAL_2: 64;
hence thesis by
A15,
A20,
A19,
XXREAL_0: 2;
end;
defpred
Q[
Nat] means for m be
Element of
NAT st m
= $1 holds ((
Partial_Sums seq)
. m)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),m)));
now
let m be
Element of
NAT ;
assume
A24: m
=
0 ;
then ((
Partial_Sums seq)
. m)
= (seq
.
0 ) by
MESFUNC9:def 1
.= (
lim_inf (
ProjMap2 (f,
0 ))) by
A1
.= (
sup (
inferior_realsequence (
ProjMap2 (f,
0 )))) by
RINFSUP2:def 9;
then
A26: ((
Partial_Sums seq)
. m)
= (
lim (
inferior_realsequence (
ProjMap2 (f,
0 )))) by
RINFSUP2: 37;
now
let n be
Element of
NAT ;
((
ProjMap2 ((
Partial_Sums_in_cod2 g),
0 ))
. n)
= ((
Partial_Sums_in_cod2 g)
. (n,
0 )) by
MESFUNC9:def 7
.= (g
. (n,
0 )) by
DefCSM
.= ((
inferior_realsequence (
ProjMap2 (f,
0 )))
. n) by
A5;
hence ((
inferior_realsequence (
ProjMap2 (f,
0 )))
. n)
= ((
ProjMap2 ((
Partial_Sums_in_cod2 g),
0 ))
. n);
end;
hence ((
Partial_Sums seq)
. m)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),m))) by
A24,
A26,
FUNCT_2: 63;
end;
then
A28:
Q[
0 ];
P1: for m be
Element of
NAT holds (
ProjMap2 ((
Partial_Sums_in_cod2 g),m)) is
convergent
proof
let m be
Element of
NAT ;
for j,i be
Nat st i
<= j holds ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. i)
<= ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. j)
proof
let j,i be
Nat;
reconsider i1 = i, j1 = j as
Element of
NAT by
ORDINAL1:def 12;
assume
B2: i
<= j;
B3: ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. i1)
= ((
Partial_Sums_in_cod2 g)
. (i,m)) & ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. j1)
= ((
Partial_Sums_in_cod2 g)
. (j,m)) by
MESFUNC9:def 7;
defpred
R[
Nat] means ((
Partial_Sums_in_cod2 g)
. (i,$1))
<= ((
Partial_Sums_in_cod2 g)
. (j,$1));
B4: ((
Partial_Sums_in_cod2 g)
. (i,
0 ))
= (g
. (i,
0 )) by
DefCSM
.= ((
inferior_realsequence (
ProjMap2 (f,
0 )))
. i1) by
A5;
((
Partial_Sums_in_cod2 g)
. (j,
0 ))
= (g
. (j,
0 )) by
DefCSM
.= ((
inferior_realsequence (
ProjMap2 (f,
0 )))
. j1) by
A5;
then
B5:
R[
0 ] by
B2,
B4,
RINFSUP2: 7;
B6: for l be
Nat st
R[l] holds
R[(l
+ 1)]
proof
let l be
Nat;
reconsider l1 = l as
Element of
NAT by
ORDINAL1:def 12;
assume
B7:
R[l];
(g
. (i,(l
+ 1)))
= ((
inferior_realsequence (
ProjMap2 (f,(l1
+ 1))))
. i1) & (g
. (j,(l
+ 1)))
= ((
inferior_realsequence (
ProjMap2 (f,(l1
+ 1))))
. j1) by
A5;
then
B8: (g
. (i,(l
+ 1)))
<= (g
. (j,(l
+ 1))) by
B2,
RINFSUP2: 7;
((
Partial_Sums_in_cod2 g)
. (i,(l
+ 1)))
= (((
Partial_Sums_in_cod2 g)
. (i,l))
+ (g
. (i,(l
+ 1)))) & ((
Partial_Sums_in_cod2 g)
. (j,(l
+ 1)))
= (((
Partial_Sums_in_cod2 g)
. (j,l))
+ (g
. (j,(l
+ 1)))) by
DefCSM;
hence
R[(l
+ 1)] by
B7,
B8,
XXREAL_3: 36;
end;
for l be
Nat holds
R[l] from
NAT_1:sch 2(
B5,
B6);
hence thesis by
B3;
end;
then (
ProjMap2 ((
Partial_Sums_in_cod2 g),m)) is
non-decreasing by
RINFSUP2: 7;
hence (
ProjMap2 ((
Partial_Sums_in_cod2 g),m)) is
convergent by
RINFSUP2: 37;
end;
A29: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
assume
A30:
Q[k];
now
let m be
Element of
NAT ;
assume
B00: m
= (k
+ 1);
then
B0: ((
Partial_Sums seq)
. m)
= (((
Partial_Sums seq)
. k)
+ (seq
. (k
+ 1))) by
MESFUNC9:def 1
.= ((
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),k1)))
+ (seq
. (k
+ 1))) by
A30
.= ((
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),k1)))
+ (
lim_inf (
ProjMap2 (f,(k
+ 1))))) by
A1;
B1: (
lim_inf (
ProjMap2 (f,(k
+ 1))))
= (
sup (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))) by
RINFSUP2:def 9
.= (
lim (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))) by
RINFSUP2: 37;
B9: (
ProjMap2 ((
Partial_Sums_in_cod2 g),k1)) is
convergent by
P1;
B10: (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1)))) is
convergent by
RINFSUP2: 37;
for n be
object st n
in (
dom (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))) holds
0.
<= ((
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))
. n)
proof
let n be
object;
assume n
in (
dom (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1)))));
then (g
. (n,(k1
+ 1)))
= ((
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))
. n) by
A5;
hence thesis by
SUPINF_2: 51;
end;
then
C2: (
inferior_realsequence (
ProjMap2 (f,(k1
+ 1)))) is
nonnegative by
SUPINF_2: 52;
for i be
Nat holds ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. i)
= (((
ProjMap2 ((
Partial_Sums_in_cod2 g),k1))
. i)
+ ((
inferior_realsequence (
ProjMap2 (f,(k1
+ 1))))
. i))
proof
let i be
Nat;
reconsider i1 = i as
Element of
NAT by
ORDINAL1:def 12;
((
ProjMap2 ((
Partial_Sums_in_cod2 g),m))
. i1)
= ((
Partial_Sums_in_cod2 g)
. (i,m)) by
MESFUNC9:def 7
.= (((
Partial_Sums_in_cod2 g)
. (i,k))
+ (g
. (i,(k
+ 1)))) by
B00,
DefCSM
.= (((
ProjMap2 ((
Partial_Sums_in_cod2 g),k1))
. i1)
+ (g
. (i,(k
+ 1)))) by
MESFUNC9:def 7;
hence thesis by
A5;
end;
hence ((
Partial_Sums seq)
. m)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),m))) by
B0,
B1,
B9,
B10,
C2,
MESFUNC9: 11;
end;
hence
Q[(k
+ 1)];
end;
A30: for k be
Nat holds
Q[k] from
NAT_1:sch 2(
A28,
A29);
A31: for m be
Nat holds ((
Partial_Sums seq)
. m)
<= (
lim_inf (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
proof
let m be
Nat;
reconsider m1 = m as
Element of
NAT by
ORDINAL1:def 12;
A32: for n be
Nat holds ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m1))
. n)
<= ((
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. n)
proof
let n be
Nat;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
((
ProjMap2 ((
Partial_Sums_in_cod2 g),m1))
. n1)
= ((
Partial_Sums_in_cod2 g)
. (n,m)) by
MESFUNC9:def 7;
hence ((
ProjMap2 ((
Partial_Sums_in_cod2 g),m1))
. n)
<= ((
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
. n) by
A13;
end;
A33: (
ProjMap2 ((
Partial_Sums_in_cod2 g),m1)) is
convergent by
P1;
(
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f))) is
convergent by
RINFSUP2: 37;
then (
lim (
ProjMap2 ((
Partial_Sums_in_cod2 g),m1)))
<= (
lim (
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))) by
A32,
A33,
RINFSUP2: 38;
then ((
Partial_Sums seq)
. m)
<= (
lim (
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))) by
A30;
then ((
Partial_Sums seq)
. m)
<= (
sup (
inferior_realsequence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))) by
RINFSUP2: 37;
hence thesis by
RINFSUP2:def 9;
end;
for m be
object st m
in (
dom seq) holds
0
<= (seq
. m)
proof
let m be
object;
assume m
in (
dom seq);
then
reconsider m1 = m as
Element of
NAT ;
E1: (seq
. m)
= (
lim_inf (
ProjMap2 (f,m1))) by
A1
.= (
sup (
inferior_realsequence (
ProjMap2 (f,m1)))) by
RINFSUP2:def 9;
for n be
object st n
in (
dom (
inferior_realsequence (
ProjMap2 (f,m1)))) holds
0.
<= ((
inferior_realsequence (
ProjMap2 (f,m1)))
. n)
proof
let n be
object;
assume n
in (
dom (
inferior_realsequence (
ProjMap2 (f,m1))));
then (g
. (n,m1))
= ((
inferior_realsequence (
ProjMap2 (f,m1)))
. n) by
A5;
hence thesis by
SUPINF_2: 51;
end;
then ((
inferior_realsequence (
ProjMap2 (f,m1)))
.
0 )
>=
0 by
SUPINF_2: 51,
SUPINF_2: 52;
hence thesis by
E1,
RINFSUP2: 23;
end;
then seq is
nonnegative by
SUPINF_2: 52;
then (
Partial_Sums seq) is
non-decreasing by
MESFUNC9: 16;
then (
lim (
Partial_Sums seq))
<= (
lim_inf (
lim_in_cod2 (
Partial_Sums_in_cod2 f))) by
A31,
MESFUNC9: 9,
RINFSUP2: 37;
hence thesis by
MESFUNC9:def 3;
end;
theorem ::
DBLSEQ_3:84
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence st (for m be
Element of
NAT holds (seq
. m)
= (
lim_inf (
ProjMap1 (f,m)))) holds (
Sum seq)
<= (
lim_inf (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence;
assume
A1: for m be
Element of
NAT holds (seq
. m)
= (
lim_inf (
ProjMap1 (f,m)));
now
let m be
Element of
NAT ;
(
ProjMap1 (f,m))
= (
ProjMap2 ((
~ f),m)) by
Th32;
hence (seq
. m)
= (
lim_inf (
ProjMap2 ((
~ f),m))) by
A1;
end;
then
A2: (
Sum seq)
<= (
lim_inf (
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f)))) by
Th83;
(
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f)))
= (
lim_in_cod1 (
~ (
Partial_Sums_in_cod2 (
~ f)))) by
Th38
.= (
lim_in_cod1 (
Partial_Sums_in_cod1 (
~ (
~ f)))) by
Th40;
hence (
Sum seq)
<= (
lim_inf (
lim_in_cod1 (
Partial_Sums_in_cod1 f))) by
A2,
DBLSEQ_2: 7;
end;
theorem ::
DBLSEQ_3:85
Th85: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence, n,m be
Nat holds ((for i,j be
Nat holds (f
. (i,j))
<= (seq
. i)) implies ((
Partial_Sums_in_cod1 f)
. (n,m))
<= ((
Partial_Sums seq)
. n)) & ((for i,j be
Nat holds (f
. (i,j))
<= (seq
. j)) implies ((
Partial_Sums_in_cod2 f)
. (n,m))
<= ((
Partial_Sums seq)
. m))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence, n,m be
Nat;
hereby
assume
A1: for i,j be
Nat holds (f
. (i,j))
<= (seq
. i);
defpred
P[
Nat] means ((
Partial_Sums_in_cod1 f)
. ($1,m))
<= ((
Partial_Sums seq)
. $1);
A2: ((
Partial_Sums_in_cod1 f)
. (
0 ,m))
= (f
. (
0 ,m)) by
DefRSM;
((
Partial_Sums seq)
.
0 )
= (seq
.
0 ) by
MESFUNC9:def 1;
then
A3:
P[
0 ] by
A1,
A2;
A4: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A5:
P[k];
A6: (f
. ((k
+ 1),m))
<= (seq
. (k
+ 1)) by
A1;
A7: ((
Partial_Sums_in_cod1 f)
. ((k
+ 1),m))
= (((
Partial_Sums_in_cod1 f)
. (k,m))
+ (f
. ((k
+ 1),m))) by
DefRSM;
((
Partial_Sums seq)
. (k
+ 1))
= (((
Partial_Sums seq)
. k)
+ (seq
. (k
+ 1))) by
MESFUNC9:def 1;
hence thesis by
A5,
A6,
A7,
XXREAL_3: 36;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A3,
A4);
hence ((
Partial_Sums_in_cod1 f)
. (n,m))
<= ((
Partial_Sums seq)
. n);
end;
assume
A1: for i,j be
Nat holds (f
. (i,j))
<= (seq
. j);
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (n,$1))
<= ((
Partial_Sums seq)
. $1);
A2: ((
Partial_Sums_in_cod2 f)
. (n,
0 ))
= (f
. (n,
0 )) by
DefCSM;
((
Partial_Sums seq)
.
0 )
= (seq
.
0 ) by
MESFUNC9:def 1;
then
A3:
P[
0 ] by
A1,
A2;
A4: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A5:
P[k];
A6: (f
. (n,(k
+ 1)))
<= (seq
. (k
+ 1)) by
A1;
A7: ((
Partial_Sums_in_cod2 f)
. (n,(k
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (n,k))
+ (f
. (n,(k
+ 1)))) by
DefCSM;
((
Partial_Sums seq)
. (k
+ 1))
= (((
Partial_Sums seq)
. k)
+ (seq
. (k
+ 1))) by
MESFUNC9:def 1;
hence thesis by
A5,
A6,
A7,
XXREAL_3: 36;
end;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A3,
A4);
hence thesis;
end;
theorem ::
DBLSEQ_3:86
Th86: for seq be
ExtREAL_sequence, r be
R_eal st (for n be
Nat holds (seq
. n)
<= r) holds (
lim_sup seq)
<= r
proof
let seq be
ExtREAL_sequence, r be
R_eal;
assume
A1: for n be
Nat holds (seq
. n)
<= r;
deffunc
F(
Element of
NAT ) = r;
consider f be
Function of
NAT ,
ExtREAL such that
A2: for n be
Element of
NAT holds (f
. n)
=
F(n) from
FUNCT_2:sch 4;
A4: for n be
Nat holds (f
. n)
= r
proof
let n be
Nat;
n is
Element of
NAT by
ORDINAL1:def 12;
hence (f
. n)
= r by
A2;
end;
then
A5: f is
convergent & (
lim f)
= r by
MESFUNC5: 60;
for n be
Nat holds (seq
. n)
<= (f
. n)
proof
let n be
Nat;
(f
. n)
= r by
A4;
hence (seq
. n)
<= (f
. n) by
A1;
end;
then (
lim_sup seq)
<= (
lim_sup f) by
MESFUN10: 3;
hence (
lim_sup seq)
<= r by
A5,
RINFSUP2: 41;
end;
theorem ::
DBLSEQ_3:87
Th87: for seq be
ExtREAL_sequence, r be
R_eal st (for n be
Nat holds r
<= (seq
. n)) holds r
<= (
lim_inf seq)
proof
let seq be
ExtREAL_sequence, r be
R_eal;
assume
A1: for n be
Nat holds r
<= (seq
. n);
deffunc
F(
Element of
NAT ) = r;
consider f be
Function of
NAT ,
ExtREAL such that
A2: for n be
Element of
NAT holds (f
. n)
=
F(n) from
FUNCT_2:sch 4;
A4: for n be
Nat holds (f
. n)
= r
proof
let n be
Nat;
n is
Element of
NAT by
ORDINAL1:def 12;
hence (f
. n)
= r by
A2;
end;
then
A5: f is
convergent & (
lim f)
= r by
MESFUNC5: 60;
for n be
Nat holds (f
. n)
<= (seq
. n)
proof
let n be
Nat;
(f
. n)
= r by
A4;
hence (f
. n)
<= (seq
. n) by
A1;
end;
then (
lim_inf f)
<= (
lim_inf seq) by
MESFUN10: 3;
hence r
<= (
lim_inf seq) by
A5,
RINFSUP2: 41;
end;
theorem ::
DBLSEQ_3:88
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL holds (for i1,i2,j be
Nat st i1
<= i2 holds ((
Partial_Sums_in_cod1 f)
. (i1,j))
<= ((
Partial_Sums_in_cod1 f)
. (i2,j))) & (for i,j1,j2 be
Nat st j1
<= j2 holds ((
Partial_Sums_in_cod2 f)
. (i,j1))
<= ((
Partial_Sums_in_cod2 f)
. (i,j2)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A2:
now
let i1,i2,j be
natural
number;
assume i1
<= i2;
then
consider k be
Nat such that
A3: i2
= (i1
+ k) by
NAT_1: 10;
defpred
P[
Nat] means $1
<= k implies ((
Partial_Sums_in_cod1 f)
. (i1,j))
<= ((
Partial_Sums_in_cod1 f)
. ((i1
+ $1),j));
A4:
P[
0 ];
A5: for l be
Nat st
P[l] holds
P[(l
+ 1)]
proof
let l be
Nat;
assume
A6:
P[l];
now
assume
A7: (l
+ 1)
<= k;
((
Partial_Sums_in_cod1 f)
. (((i1
+ l)
+ 1),j))
= (((
Partial_Sums_in_cod1 f)
. ((i1
+ l),j))
+ (f
. (((i1
+ l)
+ 1),j))) by
DefRSM;
then ((
Partial_Sums_in_cod1 f)
. ((i1
+ l),j))
<= ((
Partial_Sums_in_cod1 f)
. (((i1
+ l)
+ 1),j)) by
SUPINF_2: 51,
XXREAL_3: 39;
hence ((
Partial_Sums_in_cod1 f)
. (i1,j))
<= ((
Partial_Sums_in_cod1 f)
. ((i1
+ (l
+ 1)),j)) by
A6,
A7,
NAT_1: 13,
XXREAL_0: 2;
end;
hence
P[(l
+ 1)];
end;
for l be
Nat holds
P[l] from
NAT_1:sch 2(
A4,
A5);
hence ((
Partial_Sums_in_cod1 f)
. (i1,j))
<= ((
Partial_Sums_in_cod1 f)
. (i2,j)) by
A3;
end;
now
let i,j1,j2 be
natural
number;
assume j1
<= j2;
then
consider k be
Nat such that
B3: j2
= (j1
+ k) by
NAT_1: 10;
defpred
P[
Nat] means $1
<= k implies ((
Partial_Sums_in_cod2 f)
. (i,j1))
<= ((
Partial_Sums_in_cod2 f)
. (i,(j1
+ $1)));
B4:
P[
0 ];
B5: for l be
Nat st
P[l] holds
P[(l
+ 1)]
proof
let l be
Nat;
assume
B6:
P[l];
now
assume
B7: (l
+ 1)
<= k;
((
Partial_Sums_in_cod2 f)
. (i,((j1
+ l)
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (i,(j1
+ l)))
+ (f
. (i,((j1
+ l)
+ 1)))) by
DefCSM;
then ((
Partial_Sums_in_cod2 f)
. (i,(j1
+ l)))
<= ((
Partial_Sums_in_cod2 f)
. (i,((j1
+ l)
+ 1))) by
SUPINF_2: 51,
XXREAL_3: 39;
hence ((
Partial_Sums_in_cod2 f)
. (i,j1))
<= ((
Partial_Sums_in_cod2 f)
. (i,(j1
+ (l
+ 1)))) by
B6,
B7,
NAT_1: 13,
XXREAL_0: 2;
end;
hence
P[(l
+ 1)];
end;
for l be
Nat holds
P[l] from
NAT_1:sch 2(
B4,
B5);
hence ((
Partial_Sums_in_cod2 f)
. (i,j1))
<= ((
Partial_Sums_in_cod2 f)
. (i,j2)) by
B3;
end;
hence thesis by
A2;
end;
theorem ::
DBLSEQ_3:89
Th89: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , i,j,k be
Nat st (for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
non-decreasing) & i
<= j holds ((
Partial_Sums_in_cod2 f)
. (i,k))
<= ((
Partial_Sums_in_cod2 f)
. (j,k))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , i,j,k be
Nat;
assume that
A1: for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
non-decreasing and
A2: i
<= j;
reconsider i1 = i, j1 = j as
Element of
NAT by
ORDINAL1:def 12;
defpred
P[
Nat] means ((
Partial_Sums_in_cod2 f)
. (i,$1))
<= ((
Partial_Sums_in_cod2 f)
. (j,$1));
((
ProjMap2 (f,
0 ))
. i1)
= (f
. (i,
0 )) & ((
ProjMap2 (f,
0 ))
. j1)
= (f
. (j,
0 )) by
MESFUNC9:def 7;
then ((
ProjMap2 (f,
0 ))
. i1)
= ((
Partial_Sums_in_cod2 f)
. (i,
0 )) & ((
ProjMap2 (f,
0 ))
. j1)
= ((
Partial_Sums_in_cod2 f)
. (j,
0 )) by
DefCSM;
then
A4:
P[
0 ] by
A1,
A2,
RINFSUP2: 7;
A5: for n be
Nat st
P[n] holds
P[(n
+ 1)]
proof
let n be
Nat;
assume
A6:
P[n];
A7: ((
Partial_Sums_in_cod2 f)
. (i,(n
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (i,n))
+ (f
. (i,(n
+ 1)))) & ((
Partial_Sums_in_cod2 f)
. (j,(n
+ 1)))
= (((
Partial_Sums_in_cod2 f)
. (j,n))
+ (f
. (j,(n
+ 1)))) by
DefCSM;
A8: ((
ProjMap2 (f,(n
+ 1)))
. i)
<= ((
ProjMap2 (f,(n
+ 1)))
. j) by
A1,
A2,
RINFSUP2: 7;
((
ProjMap2 (f,(n
+ 1)))
. i1)
= (f
. (i,(n
+ 1))) & ((
ProjMap2 (f,(n
+ 1)))
. j1)
= (f
. (j,(n
+ 1))) by
MESFUNC9:def 7;
hence
P[(n
+ 1)] by
A6,
A7,
A8,
XXREAL_3: 36;
end;
for n be
Nat holds
P[n] from
NAT_1:sch 2(
A4,
A5);
hence thesis;
end;
theorem ::
DBLSEQ_3:90
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , i,j,k be
Nat st (for n be
Element of
NAT holds (
ProjMap1 (f,n)) is
non-decreasing) & i
<= j holds ((
Partial_Sums_in_cod1 f)
. (k,i))
<= ((
Partial_Sums_in_cod1 f)
. (k,j))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , i,j,k be
Nat;
assume that
A1: for n be
Element of
NAT holds (
ProjMap1 (f,n)) is
non-decreasing and
A2: i
<= j;
for n be
Element of
NAT holds (
ProjMap2 ((
~ f),n)) is
non-decreasing
proof
let n be
Element of
NAT ;
(
ProjMap1 (f,n))
= (
ProjMap2 ((
~ f),n)) by
Th32;
hence thesis by
A1;
end;
then ((
Partial_Sums_in_cod2 (
~ f))
. (i,k))
<= ((
Partial_Sums_in_cod2 (
~ f))
. (j,k)) by
A2,
Th89;
then ((
Partial_Sums_in_cod1 f)
. (k,i))
<= ((
Partial_Sums_in_cod2 (
~ f))
. (j,k)) by
Th39;
hence thesis by
Th39;
end;
theorem ::
DBLSEQ_3:91
Th91: for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence st (for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
non-decreasing & (seq
. m)
= (
lim (
ProjMap2 (f,m)))) holds (
lim_in_cod2 (
Partial_Sums_in_cod2 f)) is
non-decreasing & (
Sum seq)
= (
lim (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence;
assume
A1: for m be
Element of
NAT holds (
ProjMap2 (f,m)) is
non-decreasing & (seq
. m)
= (
lim (
ProjMap2 (f,m)));
now
let m be
Element of
NAT ;
(
ProjMap2 (f,m)) is
non-decreasing by
A1;
then (
ProjMap2 (f,m)) is
convergent by
RINFSUP2: 37;
then (
lim_inf (
ProjMap2 (f,m)))
= (
lim (
ProjMap2 (f,m))) by
RINFSUP2: 41;
hence (seq
. m)
= (
lim_inf (
ProjMap2 (f,m))) by
A1;
end;
then
A2: (
Sum seq)
<= (
lim_inf (
lim_in_cod2 (
Partial_Sums_in_cod2 f))) by
Th83;
A3: for n,m be
Nat holds (f
. (n,m))
<= (seq
. m)
proof
let n,m be
Nat;
reconsider m1 = m as
Element of
NAT by
ORDINAL1:def 12;
(
ProjMap2 (f,m1)) is
non-decreasing by
A1;
then (
lim (
ProjMap2 (f,m1)))
= (
sup (
ProjMap2 (f,m1))) by
RINFSUP2: 37;
then
A4: (seq
. m)
= (
sup (
ProjMap2 (f,m1))) by
A1;
A5: n is
Element of
NAT by
ORDINAL1:def 12;
(
dom (
ProjMap2 (f,m1)))
=
NAT by
FUNCT_2:def 1;
then ((
ProjMap2 (f,m1))
. n)
<= (
sup (
rng (
ProjMap2 (f,m1)))) by
A5,
FUNCT_1: 3,
XXREAL_2: 4;
then ((
ProjMap2 (f,m1))
. n)
<= (
sup (
ProjMap2 (f,m1))) by
RINFSUP2:def 1;
hence thesis by
A4,
A5,
MESFUNC9:def 7;
end;
for n,m be
Nat st m
<= n holds ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. m)
<= ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. n)
proof
let n,m be
Nat;
assume
C1: m
<= n;
reconsider m1 = m, n1 = n as
Element of
NAT by
ORDINAL1:def 12;
C2: ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. m)
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),m1))) & ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. n)
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))) by
D1DEF6;
C3: (
ProjMap1 ((
Partial_Sums_in_cod2 f),m1)) is
convergent & (
ProjMap1 ((
Partial_Sums_in_cod2 f),n1)) is
convergent by
RINFSUP2: 37;
for k be
Nat holds ((
ProjMap1 ((
Partial_Sums_in_cod2 f),m1))
. k)
<= ((
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))
. k)
proof
let k be
Nat;
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
((
ProjMap1 ((
Partial_Sums_in_cod2 f),m1))
. k1)
= ((
Partial_Sums_in_cod2 f)
. (m1,k1)) & ((
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))
. k1)
= ((
Partial_Sums_in_cod2 f)
. (n1,k1)) by
MESFUNC9:def 6;
hence thesis by
A1,
C1,
Th89;
end;
hence thesis by
C2,
C3,
RINFSUP2: 38;
end;
hence (
lim_in_cod2 (
Partial_Sums_in_cod2 f)) is
non-decreasing by
RINFSUP2: 7;
then
B1: (
lim_in_cod2 (
Partial_Sums_in_cod2 f)) is
convergent by
RINFSUP2: 37;
then
B3: (
Sum seq)
<= (
lim (
lim_in_cod2 (
Partial_Sums_in_cod2 f))) by
A2,
RINFSUP2: 41;
for n be
Nat holds ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. n)
<= (
Sum seq)
proof
let n be
Nat;
reconsider n1 = n as
Element of
NAT by
ORDINAL1:def 12;
A6: ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. n)
= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))) by
D1DEF6;
A7: (
ProjMap1 ((
Partial_Sums_in_cod2 f),n1)) is
convergent by
RINFSUP2: 37;
for m be
Element of
NAT holds
0
<= (seq
. m)
proof
let m be
Element of
NAT ;
for n be
Nat holds
0.
<= ((
ProjMap2 (f,m))
. n)
proof
let n be
Nat;
n is
Element of
NAT by
ORDINAL1:def 12;
hence
0.
<= ((
ProjMap2 (f,m))
. n) by
SUPINF_2: 39;
end;
then
B2:
0.
<= (
lim_inf (
ProjMap2 (f,m))) by
Th87;
(
ProjMap2 (f,m)) is
non-decreasing by
A1;
then (
ProjMap2 (f,m)) is
convergent by
RINFSUP2: 37;
then (
lim_inf (
ProjMap2 (f,m)))
= (
lim (
ProjMap2 (f,m))) by
RINFSUP2: 41;
hence
0
<= (seq
. m) by
B2,
A1;
end;
then seq is
nonnegative by
SUPINF_2: 39;
then (
Partial_Sums seq) is
non-decreasing by
MESFUNC9: 16;
then
A8: (
Partial_Sums seq) is
convergent by
RINFSUP2: 37;
for m be
Nat holds ((
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))
. m)
<= ((
Partial_Sums seq)
. m)
proof
let m be
Nat;
m is
Element of
NAT by
ORDINAL1:def 12;
then ((
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))
. m)
= ((
Partial_Sums_in_cod2 f)
. (n,m)) by
MESFUNC9:def 6;
hence ((
ProjMap1 ((
Partial_Sums_in_cod2 f),n1))
. m)
<= ((
Partial_Sums seq)
. m) by
A3,
Th85;
end;
then (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 f),n1)))
<= (
lim (
Partial_Sums seq)) by
A7,
A8,
RINFSUP2: 38;
hence ((
lim_in_cod2 (
Partial_Sums_in_cod2 f))
. n)
<= (
Sum seq) by
A6,
MESFUNC9:def 3;
end;
then (
lim_sup (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
<= (
Sum seq) by
Th86;
then (
lim (
lim_in_cod2 (
Partial_Sums_in_cod2 f)))
<= (
Sum seq) by
B1,
RINFSUP2: 41;
hence (
Sum seq)
= (
lim (
lim_in_cod2 (
Partial_Sums_in_cod2 f))) by
B3,
XXREAL_0: 1;
end;
theorem ::
DBLSEQ_3:92
for f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence st (for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
non-decreasing & (seq
. m)
= (
lim (
ProjMap1 (f,m)))) holds (
lim_in_cod1 (
Partial_Sums_in_cod1 f)) is
non-decreasing & (
Sum seq)
= (
lim (
lim_in_cod1 (
Partial_Sums_in_cod1 f)))
proof
let f be
nonnegative
Function of
[:
NAT ,
NAT :],
ExtREAL , seq be
ExtREAL_sequence;
assume
A1: for m be
Element of
NAT holds (
ProjMap1 (f,m)) is
non-decreasing & (seq
. m)
= (
lim (
ProjMap1 (f,m)));
for m be
Element of
NAT holds (
ProjMap2 ((
~ f),m)) is
non-decreasing & (seq
. m)
= (
lim (
ProjMap2 ((
~ f),m)))
proof
let m be
Element of
NAT ;
(
ProjMap1 (f,m)) is
non-decreasing by
A1;
hence (
ProjMap2 ((
~ f),m)) is
non-decreasing by
Th32;
(seq
. m)
= (
lim (
ProjMap1 (f,m))) by
A1;
hence (seq
. m)
= (
lim (
ProjMap2 ((
~ f),m))) by
Th32;
end;
then
A2: (
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f))) is
non-decreasing & (
Sum seq)
= (
lim (
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f)))) by
Th91;
for n be
Element of
NAT holds ((
lim_in_cod2 (
Partial_Sums_in_cod2 (
~ f)))
. n)
= ((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
. n)
proof
let n be
Element of
NAT ;
((
lim_in_cod1 (
Partial_Sums_in_cod1 f))
. n)
= (
lim (
ProjMap2 ((
Partial_Sums_in_cod1 f),n))) by
D1DEF5
.= (
lim (
ProjMap1 ((
~ (
Partial_Sums_in_cod1 f)),n))) by
Th33
.= (
lim (
ProjMap1 ((
Partial_Sums_in_cod2 (
~ f)),n))) by
Th40;
hence thesis by
D1DEF6;
end;
hence thesis by
A2,
FUNCT_2:def 8;
end;
begin
theorem ::
DBLSEQ_3:93
Th93: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL st f is
P-convergent_to_+infty holds not f is
P-convergent_to_-infty & not f is
P-convergent_to_finite_number
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: f is
P-convergent_to_+infty;
hereby
assume f is
P-convergent_to_-infty;
then
consider N1 be
Nat such that
A3: for n,m be
Nat st n
>= N1 & m
>= N1 holds (f
. (n,m))
<= (
- 1);
consider N2 be
Nat such that
A4: for n,m be
Nat st n
>= N2 & m
>= N2 holds 1
<= (f
. (n,m)) by
A1;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A5: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then (f
. (N,N))
<= (
- 1) by
A3;
hence contradiction by
A4,
A5;
end;
assume f is
P-convergent_to_finite_number;
then
consider p be
Real such that
A6: for e be
Real st
0
< e holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- p) qua
ExtReal.|
< e;
reconsider p1 = p as
ExtReal;
per cases ;
suppose
A9: p
>
0 ;
then
consider N1 be
Nat such that
A7: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p1).|
< p by
A6;
A8:
now
let n,m be
Nat;
assume n
>= N1 & m
>= N1;
then
|.((f
. (n,m))
- p) qua
ExtReal.|
< p by
A7;
then ((f
. (n,m))
- p1)
< p by
EXTREAL1: 21;
then (f
. (n,m))
< (p1
+ p1) by
XXREAL_3: 54;
then (f
. (n,m))
< (2
* p1) by
XXREAL_3: 94;
hence (f
. (n,m))
< (2
* p) by
XXREAL_3:def 5;
end;
consider N2 be
Nat such that
A10: for n,m be
Nat st n
>= N2 & m
>= N2 holds (2
* p)
<= (f
. (n,m)) by
A1,
A9;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A11: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then (f
. (N,N))
< (2
* p) by
A8;
hence contradiction by
A11,
A10;
end;
suppose
A12: p
=
0 ;
consider N1 be
Nat such that
A13: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p).|
< 1 by
A6;
consider N2 be
Nat such that
A14: for n,m be
Nat st n
>= N2 & m
>= N2 holds 1
<= (f
. (n,m)) by
A1;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider jj = 1 as
ExtReal;
set N = (
max (N1,N2));
A15: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then
|.((f
. (N,N))
- p1).|
< jj by
A13;
then ((f
. (N,N))
- p1)
< jj by
EXTREAL1: 21;
then (f
. (N,N))
< (jj
+ p1) by
XXREAL_3: 54;
then (f
. (N,N))
< (1
+
0 ) by
A12,
XXREAL_3:def 2;
hence contradiction by
A14,
A15;
end;
suppose p
<
0 ;
then
consider N1 be
Nat such that
A17: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p).|
< (
- p) by
A6;
A18:
now
let n,m be
Nat;
assume n
>= N1 & m
>= N1;
then
|.((f
. (n,m))
- p).|
< (
- p) by
A17;
then ((f
. (n,m))
- p1)
< (
- p) by
EXTREAL1: 21;
then (f
. (n,m))
< (p1
+ (
- p)) by
XXREAL_3: 54;
then (f
. (n,m))
< (p
+ (
- p)) by
XXREAL_3:def 2;
hence (f
. (n,m))
<
0 ;
end;
consider N2 be
Nat such that
A19: for n,m be
Nat st n
>= N2 & m
>= N2 holds 1
<= (f
. (n,m)) by
A1;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A20: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then (f
. (N,N))
<
0 by
A18;
hence contradiction by
A19,
A20;
end;
end;
theorem ::
DBLSEQ_3:94
Th94: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL st f is
P-convergent_to_-infty holds not f is
P-convergent_to_+infty & not f is
P-convergent_to_finite_number
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: f is
P-convergent_to_-infty;
hereby
assume f is
P-convergent_to_+infty;
then
consider N1 be
Nat such that
A3: for n,m be
Nat st n
>= N1 & m
>= N1 holds (f
. (n,m))
>= 1;
consider N2 be
Nat such that
A4: for n,m be
Nat st n
>= N2 & m
>= N2 holds (
- 1)
>= (f
. (n,m)) by
A1;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A5: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then (f
. (N,N))
>= 1 by
A3;
hence contradiction by
A4,
A5;
end;
assume f is
P-convergent_to_finite_number;
then
consider p be
Real such that
A6: for e be
Real st
0
< e holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- p) qua
ExtReal.|
< e;
reconsider p1 = p as
ExtReal;
per cases ;
suppose
A9: p
>
0 ;
then
consider N1 be
Nat such that
A7: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p1).|
< p by
A6;
A8:
now
let n,m be
Nat;
assume n
>= N1 & m
>= N1;
then
|.((f
. (n,m))
- p) qua
ExtReal.|
< p by
A7;
then (
- p1)
< ((f
. (n,m))
- p) by
EXTREAL1: 21;
then ((
- p1)
+ p)
< (f
. (n,m)) by
XXREAL_3: 53;
hence
0
< (f
. (n,m)) by
XXREAL_3: 7;
end;
consider N2 be
Nat such that
A10: for n,m be
Nat st n
>= N2 & m
>= N2 holds (
- (2
* p))
>= (f
. (n,m)) by
A1,
A9;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A11: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then
0
< (f
. (N,N)) by
A8;
hence contradiction by
A9,
A11,
A10;
end;
suppose
A12: p
=
0 ;
consider N1 be
Nat such that
A13: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p).|
< 1 by
A6;
consider N2 be
Nat such that
A14: for n,m be
Nat st n
>= N2 & m
>= N2 holds (
- 1)
>= (f
. (n,m)) by
A1;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider jj = 1 as
ExtReal;
set N = (
max (N1,N2));
A15: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then
|.((f
. (N,N))
- p1).|
< jj by
A13;
then (
- jj)
< ((f
. (N,N))
- p1) by
EXTREAL1: 21;
then ((
- jj)
+ p)
< (f
. (N,N)) by
XXREAL_3: 53;
then (
- jj)
< (f
. (N,N)) by
A12,
XXREAL_3: 4;
then (
- 1)
< (f
. (N,N)) by
XXREAL_3:def 3;
hence contradiction by
A14,
A15;
end;
suppose
A16: p
<
0 ;
then
consider N1 be
Nat such that
A17: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- p).|
< (
- p) by
A6;
A18:
now
let n,m be
Nat;
assume n
>= N1 & m
>= N1;
then
|.((f
. (n,m))
- p) qua
ExtReal.|
< (
- p) by
A17;
then (
- (
- p1))
< ((f
. (n,m))
- p1) by
EXTREAL1: 21;
then (p1
+ p1)
< (f
. (n,m)) by
XXREAL_3: 53;
then (2
* p1)
< (f
. (n,m)) by
XXREAL_3: 94;
hence (2
* p)
< (f
. (n,m)) by
XXREAL_3:def 5;
end;
consider N2 be
Nat such that
A19: for n,m be
Nat st n
>= N2 & m
>= N2 holds (f
. (n,m))
<= (2
* p) by
A1,
A16;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
A20: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then (2
* p)
< (f
. (N,N)) by
A18;
hence contradiction by
A19,
A20;
end;
end;
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
::
DBLSEQ_3:def17
attr f is
P-convergent means f is
P-convergent_to_finite_number or f is
P-convergent_to_+infty or f is
P-convergent_to_-infty;
end
definition
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: f is
P-convergent;
::
DBLSEQ_3:def18
func
P-lim f ->
ExtReal means
:
Def5: (ex p be
Real st it
= p & (for e be
Real st
0
< e holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- it ).|
< e) & f is
P-convergent_to_finite_number) or (it
=
+infty & f is
P-convergent_to_+infty) or (it
=
-infty & f is
P-convergent_to_-infty);
existence
proof
per cases by
A1;
suppose
A2: f is
P-convergent_to_finite_number;
then
consider p be
Real such that
A3: for e be
Real st
0
< e holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- p).|
< e;
reconsider p as
R_eal by
XXREAL_0:def 1;
take p;
thus thesis by
A2,
A3;
end;
suppose f is
P-convergent_to_+infty or f is
P-convergent_to_-infty;
hence thesis;
end;
end;
uniqueness
proof
defpred
P[
ExtReal] means (ex g be
Real st $1
= g & (for p be
Real st
0
< p holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- $1).|
< p) & f is
P-convergent_to_finite_number) or ($1
=
+infty & f is
P-convergent_to_+infty) or ($1
=
-infty & f is
P-convergent_to_-infty);
given g1,g2 be
ExtReal such that
A4:
P[g1] and
A5:
P[g2] and
A6: g1
<> g2;
per cases by
A1;
suppose
A7: f is
P-convergent_to_finite_number;
then
consider g be
Real such that
A8: g1
= g and
A9: for p be
Real st
0
< p holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- g1).|
< p and f is
P-convergent_to_finite_number by
A4,
Th93,
Th94;
consider h be
Real such that
A10: g2
= h and
A11: for p be
Real st
0
< p holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- g2).|
< p and f is
P-convergent_to_finite_number by
A5,
A7,
Th93,
Th94;
reconsider g, h as
Complex;
A12: (g
- h)
<>
0 by
A6,
A8,
A10;
then
consider N1 be
Nat such that
A13: for n,m be
Nat st n
>= N1 & m
>= N1 holds
|.((f
. (n,m))
- g1).|
< (
|.(g
- h).|
/ 2) by
A9;
consider N2 be
Nat such that
A14: for n,m be
Nat st n
>= N2 & m
>= N2 holds
|.((f
. (n,m))
- g2).|
< (
|.(g
- h).|
/ 2) by
A11,
A12;
reconsider N1, N2 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (N1,N2));
B1: N
>= N1 & N
>= N2 by
XXREAL_0: 25;
then
A15:
|.((f
. (N,N))
- g1).|
< (
|.(g
- h).|
/ 2) by
A13;
A16:
|.((f
. (N,N))
- g2).|
< (
|.(g
- h).|
/ 2) by
A14,
B1;
reconsider g, h as
Complex;
A17: ((f
. (N,N))
- g2)
< (
|.(g
- h).|
/ 2) by
A16,
EXTREAL1: 21;
A18: (
- (
|.(g
- h).|
/ 2) qua
ExtReal)
< ((f
. (N,N))
- g2) by
A16,
EXTREAL1: 21;
then
reconsider w = ((f
. (N,N))
- g2) as
Element of
REAL by
A17,
XXREAL_0: 48;
A19: ((f
. (N,N))
- g2)
in
REAL by
A18,
A17,
XXREAL_0: 48;
then
A20: (f
. (N,N))
<>
+infty by
A10;
A21: ((
- (f
. (N,N)))
+ g1)
= (
- ((f
. (N,N))
- g1)) by
XXREAL_3: 26;
then
A22:
|.((
- (f
. (N,N)))
+ g1).|
< (
|.(g
- h).|
/ 2) by
A15,
EXTREAL1: 29;
then
A23: ((
- (f
. (N,N)))
+ g1)
< (
|.(g
- h).|
/ 2) by
EXTREAL1: 21;
(
- (
|.(g
- h).|
/ 2) qua
ExtReal)
< ((
- (f
. (N,N)))
+ g1) by
A22,
EXTREAL1: 21;
then
A24: ((
- (f
. (N,N)))
+ g1)
in
REAL by
A23,
XXREAL_0: 48;
A25: (f
. (N,N))
<>
-infty by
A10,
A19;
|.(g1
- g2).|
=
|.((g1
+
0. )
- g2).| by
XXREAL_3: 4
.=
|.((g1
+ ((f
. (N,N))
+ (
- (f
. (N,N)))))
- g2).| by
XXREAL_3: 7
.=
|.((((
- (f
. (N,N)))
+ g1)
+ (f
. (N,N)))
- g2).| by
A8,
A20,
A25,
XXREAL_3: 29
.=
|.(((
- (f
. (N,N)))
+ g1)
+ ((f
. (N,N))
- g2)).| by
A10,
A24,
XXREAL_3: 30;
then
|.(g1
- g2).|
<= (
|.((
- (f
. (N,N)))
+ g1).|
+
|.((f
. (N,N))
- g2).|) by
EXTREAL1: 24;
then
A26:
|.(g1
- g2).|
<= (
|.((f
. (N,N))
- g1).|
+
|.((f
. (N,N))
- g2).|) by
A21,
EXTREAL1: 29;
|.((f
. (N,N))
- g2).|
<
+infty &
|.((f
. (N,N))
- g2).|
>=
0 by
A19,
EXTREAL1: 41,
EXTREAL1: 14;
then
|.((f
. (N,N))
- g2).|
in
REAL by
XXREAL_0: 14;
then
A27: (
|.((f
. (N,N))
- g1).|
+
|.((f
. (N,N))
- g2).|)
< ((
|.(g
- h).|
/ 2) qua
ExtReal
+
|.((f
. (N,N))
- g2).|) by
A15,
XXREAL_3: 43;
(
|.(g
- h).|
/ 2)
in
REAL by
XREAL_0:def 1;
then ((
|.(g
- h).|
/ 2) qua
ExtReal
+
|.((f
. (N,N))
- g2).|)
< ((
|.(g
- h).|
/ 2) qua
ExtReal
+ (
|.(g
- h).|
/ 2)) by
A16,
XXREAL_3: 43;
then
A28: (
|.((f
. (N,N))
- g1).|
+
|.((f
. (N,N))
- g2).|)
< ((
|.(g
- h).|
/ 2) qua
ExtReal
+ (
|.(g
- h).|
/ 2)) by
A27,
XXREAL_0: 2;
(g
- h)
= (g1
- g2) by
A8,
A10,
SUPINF_2: 3;
hence contradiction by
A28,
A26,
EXTREAL1: 12;
end;
suppose f is
P-convergent_to_+infty or f is
P-convergent_to_-infty;
hence contradiction by
A4,
A5,
A6,
Th93,
Th94;
end;
end;
end
theorem ::
DBLSEQ_3:95
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , r be
Real st (for n,m be
Nat holds (f
. (n,m))
= r) holds f is
P-convergent_to_finite_number & (
P-lim f)
= r
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , r be
Real;
assume
A1: for n,m be
Nat holds (f
. (n,m))
= r;
A2:
now
reconsider N = 1 as
Nat;
let p be
Real;
assume
A3:
0
< p;
take N;
let n,m be
Nat such that n
>= N & m
>= N;
(f
. (n,m))
= r by
A1;
hence
|.((f
. (n,m))
- r).|
< p by
A3,
XXREAL_3: 7,
EXTREAL1: 16;
end;
hence
A4: f is
P-convergent_to_finite_number;
then f is
P-convergent;
hence thesis by
A2,
A4,
Def5;
end;
theorem ::
DBLSEQ_3:96
Th96: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL st (for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f
. (n1,m1))
<= (f
. (n2,m2))) holds f is
P-convergent & (
P-lim f)
= (
sup (
rng f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f
. (n1,m1))
<= (f
. (n2,m2));
A2:
now
let n,m be
Nat;
reconsider n1 = n, m1 = m as
Element of
NAT by
ORDINAL1:def 12;
[n1, m1]
in
[:
NAT ,
NAT :] & (
dom f)
=
[:
NAT ,
NAT :] by
ZFMISC_1:def 2,
FUNCT_2:def 1;
then
A3: (f
. (n1,m1))
in (
rng f) by
FUNCT_1:def 3;
(
sup (
rng f)) is
UpperBound of (
rng f) by
XXREAL_2:def 3;
hence (f
. (n,m))
<= (
sup (
rng f)) by
A3,
XXREAL_2:def 1;
end;
per cases ;
suppose
A4: not ex n0,m0 be
Nat st
-infty
< (f
. (n0,m0));
now
let x be
ExtReal;
assume x
in (
rng f);
then
consider z be
object such that
B1: z
in (
dom f) & x
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
B2: n
in
NAT & m
in
NAT & z
=
[n, m] by
B1,
ZFMISC_1:def 2;
reconsider n, m as
Nat by
B2;
not (
-infty
< (f
. (n,m))) by
A4;
hence x
<=
-infty by
B1,
B2;
end;
then
A5:
-infty is
UpperBound of (
rng f) by
XXREAL_2:def 1;
for y be
UpperBound of (
rng f) holds
-infty
<= y by
XXREAL_0: 5;
then
A6:
-infty
= (
sup (
rng f)) by
A5,
XXREAL_2:def 3;
now
reconsider N0 =
0 as
Nat;
let K be
Real such that K
<
0 ;
take N0;
let n,m be
Nat such that N0
<= n & N0
<= m;
(f
. (n,m))
=
-infty by
A4,
XXREAL_0: 6;
hence (f
. (n,m))
<= K by
XXREAL_0: 5;
end;
then
A7: f is
P-convergent_to_-infty;
then f is
P-convergent;
hence thesis by
A7,
A6,
Def5;
end;
suppose ex n0,m0 be
Nat st
-infty
< (f
. (n0,m0));
then
consider n0,m0 be
Nat such that
A8:
-infty
< (f
. (n0,m0));
reconsider n0, m0 as
Element of
NAT by
ORDINAL1:def 12;
per cases ;
suppose ex K be
Real st for n,m be
Nat holds (f
. (n,m))
< K;
then
consider K be
Real such that
A9: for n,m be
Nat holds (f
. (n,m))
< K;
now
let x be
ExtReal;
assume x
in (
rng f);
then
consider z be
object such that
C1: z
in (
dom f) & x
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
C2: n
in
NAT & m
in
NAT & z
=
[n, m] by
C1,
ZFMISC_1:def 2;
reconsider n, m as
Nat by
C2;
(f
. (n,m))
< K by
A9;
hence x
<= K by
C1,
C2;
end;
then K is
UpperBound of (
rng f) by
XXREAL_2:def 1;
then (
sup (
rng f))
<= K by
XXREAL_2:def 3;
then
A11: (
sup (
rng f))
<>
+infty by
XXREAL_0: 9,
XREAL_0:def 1;
A12: (
sup (
rng f))
<>
-infty by
A2,
A8;
then
reconsider h = (
sup (
rng f)) as
Element of
REAL by
A11,
XXREAL_0: 14;
A13: for p be
Real st
0
< p holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds
|.((f
. (n,m))
- (
sup (
rng f))).|
< p
proof
let p be
Real;
assume
A14:
0
< p;
reconsider e = p as
R_eal by
XXREAL_0:def 1;
(
sup (
rng f))
in
REAL by
A12,
A11,
XXREAL_0: 14;
then
consider y be
ExtReal such that
A15: y
in (
rng f) and
A16: ((
sup (
rng f))
- e)
< y by
A14,
MEASURE6: 6;
consider x be
object such that
A17: x
in (
dom f) and
A18: y
= (f
. x) by
A15,
FUNCT_1:def 3;
consider i,j be
object such that
B1: i
in
NAT & j
in
NAT & x
=
[i, j] by
A17,
ZFMISC_1:def 2;
reconsider i, j as
Nat by
B1;
reconsider Ni = i, Nj = j as
Element of
NAT by
B1;
set N0 = (
max (Ni,n0)), M0 = (
max (Nj,m0)), N = (
max (N0,M0));
take N;
hereby
let n,m be
Nat;
Ni
<= N0 & n0
<= N0 & Nj
<= M0 & m0
<= M0 & N0
<= N & M0
<= N by
XXREAL_0: 25;
then
B2: Ni
<= N & Nj
<= N by
XXREAL_0: 2;
assume N
<= n & N
<= m;
then Ni
<= n & Nj
<= m by
B2,
XXREAL_0: 2;
then (f
. (Ni,Nj))
<= (f
. (n,m)) by
A1;
then ((
sup (
rng f))
- e)
< (f
. (n,m)) by
A16,
A18,
B1,
XXREAL_0: 2;
then (
sup (
rng f))
< ((f
. (n,m))
+ e) by
XXREAL_3: 54;
then ((
sup (
rng f))
- (f
. (n,m)))
< e by
XXREAL_3: 55;
then (
- e)
< (
- ((
sup (
rng f))
- (f
. (n,m)))) by
XXREAL_3: 38;
then
A20: (
- e)
< ((f
. (n,m))
- (
sup (
rng f))) by
XXREAL_3: 26;
A21: (f
. (n,m))
<= (
sup (
rng f)) by
A2;
A22:
now
assume
A23: (
sup (
rng f))
= ((
sup (
rng f))
+ e);
((e
+ (
sup (
rng f)))
+ (
- (
sup (
rng f))))
= (e
+ ((
sup (
rng f))
+ (
- (
sup (
rng f))))) by
A12,
A11,
XXREAL_3: 29
.= (e
+
0 ) by
XXREAL_3: 7
.= e by
XXREAL_3: 4;
hence contradiction by
A14,
A23,
XXREAL_3: 7;
end;
((
sup (
rng f))
+
0 qua
ExtReal)
<= ((
sup (
rng f))
+ e) by
A14,
XXREAL_3: 36;
then (
sup (
rng f))
<= ((
sup (
rng f))
+ e) by
XXREAL_3: 4;
then (
sup (
rng f))
< ((
sup (
rng f))
+ e) by
A22,
XXREAL_0: 1;
then (f
. (n,m))
< ((
sup (
rng f))
+ e) by
A21,
XXREAL_0: 2;
then ((f
. (n,m))
- (
sup (
rng f)))
< e by
XXREAL_3: 55;
hence
|.((f
. (n,m))
- (
sup (
rng f))).|
< p by
A20,
EXTREAL1: 22;
end;
end;
A24: h
= (
sup (
rng f));
then
A25: f is
P-convergent_to_finite_number by
A13;
hence f is
P-convergent;
hence thesis by
A13,
A24,
A25,
Def5;
end;
suppose
A26: not (ex K be
Real st
0
< K & for n,m be
Nat holds (f
. (n,m))
< K);
now
let K be
Real;
assume
0
< K;
then
consider N0,N1 be
Nat such that
A27: K
<= (f
. (N0,N1)) by
A26;
reconsider n0 = N0, n1 = N1 as
Element of
NAT by
ORDINAL1:def 12;
set N = (
max (n0,n1));
B3: N
>= N0 & N
>= N1 by
XXREAL_0: 25;
reconsider N as
Nat;
now
let n,m be
Nat;
assume N
<= n & N
<= m;
then N0
<= n & N1
<= m by
B3,
XXREAL_0: 2;
then (f
. (N0,N1))
<= (f
. (n,m)) by
A1;
hence K
<= (f
. (n,m)) by
A27,
XXREAL_0: 2;
end;
hence ex N be
Nat st for n,m be
Nat st N
<= n & N
<= m holds K
<= (f
. (n,m));
end;
then
A28: f is
P-convergent_to_+infty;
hence
A29: f is
P-convergent;
now
assume
A30: (
sup (
rng f))
<>
+infty ;
(f
. (n0,m0))
<= (
sup (
rng f)) by
A2;
then
reconsider h = (
sup (
rng f)) as
Element of
REAL by
A8,
A30,
XXREAL_0: 14;
set K = (
max (
0 ,h));
0
<= K by
XXREAL_0: 25;
then
consider N0,M0 be
Nat such that
A31: (K
+ 1)
<= (f
. (N0,M0)) by
A26;
(h
+
0 )
< (K
+ 1) by
XREAL_1: 8,
XXREAL_0: 25;
then (
sup (
rng f))
< (f
. (N0,M0)) by
A31,
XXREAL_0: 2;
hence contradiction by
A2;
end;
hence thesis by
A28,
A29,
Def5;
end;
end;
end;
theorem ::
DBLSEQ_3:97
for f1,f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL st (for n,m be
Nat holds (f1
. (n,m))
<= (f2
. (n,m))) holds (
sup (
rng f1))
<= (
sup (
rng f2))
proof
let f1,f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume
A1: for n,m be
Nat holds (f1
. (n,m))
<= (f2
. (n,m));
A2:
now
let n,m be
Element of
NAT ;
(
dom f2)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
then
[n, m]
in (
dom f2) by
ZFMISC_1: 87;
then
A3: (f2
. (n,m))
in (
rng f2) by
FUNCT_1:def 3;
A4: (f1
. (n,m))
<= (f2
. (n,m)) by
A1;
(
sup (
rng f2)) is
UpperBound of (
rng f2) by
XXREAL_2:def 3;
then (f2
. (n,m))
<= (
sup (
rng f2)) by
A3,
XXREAL_2:def 1;
hence (f1
. (n,m))
<= (
sup (
rng f2)) by
A4,
XXREAL_0: 2;
end;
now
let x be
ExtReal;
assume x
in (
rng f1);
then
consider z be
object such that
A5: z
in (
dom f1) & x
= (f1
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A6: n
in
NAT & m
in
NAT & z
=
[n, m] by
A5,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A6;
x
= (f1
. (n,m)) by
A5,
A6;
hence x
<= (
sup (
rng f2)) by
A2;
end;
then (
sup (
rng f2)) is
UpperBound of (
rng f1) by
XXREAL_2:def 1;
hence thesis by
XXREAL_2:def 3;
end;
theorem ::
DBLSEQ_3:98
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL holds for n,m be
Nat holds (f
. (n,m))
<= (
sup (
rng f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL ;
hereby
let n,m be
Nat;
A0: n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
(
dom f)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
then
[n, m]
in (
dom f) by
A0,
ZFMISC_1: 87;
then
A1: (f
. (n,m))
in (
rng f) by
FUNCT_1:def 3;
(
sup (
rng f)) is
UpperBound of (
rng f) by
XXREAL_2:def 3;
hence (f
. (n,m))
<= (
sup (
rng f)) by
A1,
XXREAL_2:def 1;
end;
end;
theorem ::
DBLSEQ_3:99
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , K be
R_eal st (for n,m be
Nat holds (f
. (n,m))
<= K) holds (
sup (
rng f))
<= K
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , K be
R_eal;
assume
A1: for n,m be
Nat holds (f
. (n,m))
<= K;
now
let x be
ExtReal;
assume x
in (
rng f);
then
consider z be
object such that
A2: z
in (
dom f) & x
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A3: n
in
NAT & m
in
NAT & z
=
[n, m] by
A2,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A3;
x
= (f
. (n,m)) by
A2,
A3;
hence x
<= K by
A1;
end;
then K is
UpperBound of (
rng f) by
XXREAL_2:def 1;
hence thesis by
XXREAL_2:def 3;
end;
theorem ::
DBLSEQ_3:100
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , K be
R_eal st K
<>
+infty & (for n,m be
Nat holds (f
. (n,m))
<= K) holds (
sup (
rng f))
<
+infty
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , K be
R_eal;
assume
A1: K
<>
+infty & (for n,m be
Nat holds (f
. (n,m))
<= K);
now
let x be
ExtReal;
assume x
in (
rng f);
then
consider z be
object such that
A2: z
in (
dom f) & x
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A3: n
in
NAT & m
in
NAT & z
=
[n, m] by
A2,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A3;
x
= (f
. (n,m)) by
A2,
A3;
hence x
<= K by
A1;
end;
then K is
UpperBound of (
rng f) by
XXREAL_2:def 1;
then (
sup (
rng f))
<= K by
XXREAL_2:def 3;
hence thesis by
A1,
XXREAL_0: 2,
XXREAL_0: 4;
end;
theorem ::
DBLSEQ_3:101
Th101: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
sup (
rng f))
<>
+infty iff ex K be
Real st
0
< K & for n,m be
Nat holds (f
. (n,m))
<= K
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A1:
-infty
< (f
. (1,1)) by
MESFUNC5:def 5;
A2: (
dom f)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
then
[1, 1]
in (
dom f) by
ZFMISC_1: 87;
then
A3: (f
. (1,1))
<= (
sup (
rng f)) by
FUNCT_1: 3,
XXREAL_2: 4;
A4:
now
assume (
sup (
rng f))
<>
+infty ;
then not (
sup (
rng f))
in
{
-infty ,
+infty } by
A1,
A3,
TARSKI:def 2;
then (
sup (
rng f))
in
REAL by
XBOOLE_0:def 3,
XXREAL_0:def 4;
then
reconsider S = (
sup (
rng f)) as
Real;
take K = (
max (S,1));
thus
0
< K by
XXREAL_0: 25;
let n,m be
Nat;
n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
[n, m]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
then
A5: (f
. (n,m))
<= (
sup (
rng f)) by
A2,
FUNCT_1: 3,
XXREAL_2: 4;
S
<= K by
XXREAL_0: 25;
hence (f
. (n,m))
<= K by
A5,
XXREAL_0: 2;
end;
now
given K be
Real such that
0
< K and
A6: for n,m be
Nat holds (f
. (n,m))
<= K;
now
let w be
ExtReal;
assume w
in (
rng f);
then
consider z be
object such that
A7: z
in (
dom f) & w
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A8: n
in
NAT & m
in
NAT & z
=
[n, m] by
A7,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A8;
w
= (f
. (n,m)) by
A7,
A8;
hence w
<= K by
A6;
end;
then K is
UpperBound of (
rng f) by
XXREAL_2:def 1;
then (
sup (
rng f))
<= K by
XXREAL_2:def 3;
hence (
sup (
rng f))
<>
+infty by
XXREAL_0: 9,
XREAL_0:def 1;
end;
hence thesis by
A4;
end;
theorem ::
DBLSEQ_3:102
Th102: for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
ExtReal st (for n,m be
Nat holds (f
. (n,m))
= c) holds f is
P-convergent & (
P-lim f)
= c & (
P-lim f)
= (
sup (
rng f))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
ExtReal;
reconsider cc = c as
R_eal by
XXREAL_0:def 1;
A1: (
dom f)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
c
in
ExtREAL by
XXREAL_0:def 1;
then
A2: c
in
REAL or c
in
{
-infty ,
+infty } by
XBOOLE_0:def 3,
XXREAL_0:def 4;
assume
A3: for n,m be
Nat holds (f
. (n,m))
= c;
then
A4: (f
. (1,1))
= c;
now
let v be
ExtReal;
assume v
in (
rng f);
then
consider z be
object such that
A7: z
in (
dom f) & v
= (f
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A8: n
in
NAT & m
in
NAT & z
=
[n, m] by
A7,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A8;
v
= (f
. (n,m)) by
A7,
A8;
hence v
<= c by
A3;
end;
then
A5: c is
UpperBound of (
rng f) by
XXREAL_2:def 1;
per cases by
A2,
TARSKI:def 2;
suppose c
in
REAL ;
then
reconsider rc = c as
Real;
A6:
now
reconsider N =
0 as
Nat;
let p be
Real;
assume
A7:
0
< p;
take N;
let n1,m1 be
Nat such that N
<= n1 & N
<= m1;
((f
. (n1,m1))
- rc)
= ((f
. (n1,m1))
- (f
. (n1,m1))) by
A3;
hence
|.((f
. (n1,m1))
- rc).|
< p by
A7,
EXTREAL1: 16,
XXREAL_3: 7;
end;
then
A8: f is
P-convergent_to_finite_number;
hence f is
P-convergent;
hence
A9: (
P-lim f)
= c by
A6,
A8,
Def5;
[1, 1]
in (
dom f) by
A1,
ZFMISC_1: 87;
hence thesis by
A9,
A5,
A4,
FUNCT_1: 3,
XXREAL_2: 55;
end;
suppose
A10: c
=
-infty ;
for p be
Real st p
<
0 holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds (f
. (n,m))
<= p
proof
let p be
Real such that p
<
0 ;
take
0 ;
now
let n,m be
Nat such that
0
<= n &
0
<= m;
(f
. (n,m))
=
-infty by
A3,
A10;
hence (f
. (n,m))
<= p by
XREAL_0:def 1,
XXREAL_0: 12;
end;
hence thesis;
end;
then
A12: f is
P-convergent_to_-infty;
hence f is
P-convergent;
hence
A13: (
P-lim f)
= c by
A10,
A12,
Def5;
[1, 1]
in (
dom f) by
A1,
ZFMISC_1: 87;
hence thesis by
A5,
A4,
A13,
FUNCT_1: 3,
XXREAL_2: 55;
end;
suppose
A14: c
=
+infty ;
for p be
Real st
0
< p holds ex N be
Nat st for n,m be
Nat st n
>= N & m
>= N holds p
<= (f
. (n,m))
proof
let p be
Real such that
0
< p;
take
0 ;
now
let n,m be
Nat such that n
>=
0 & m
>=
0 ;
(f
. (n,m))
=
+infty by
A3,
A14;
hence p
<= (f
. (n,m)) by
XREAL_0:def 1,
XXREAL_0: 9;
end;
hence thesis;
end;
then
A16: f is
P-convergent_to_+infty;
hence f is
P-convergent;
hence
A17: (
P-lim f)
= c by
A14,
A16,
Def5;
[1, 1]
in (
dom f) by
A1,
ZFMISC_1: 87;
hence thesis by
A5,
A4,
A17,
FUNCT_1: 3,
XXREAL_2: 55;
end;
end;
Lm8: for f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL holds (
sup (
rng f))
in
REAL or (
sup (
rng f))
=
+infty
proof
let f be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
A1: not
-infty
in (
rng f) by
MESFUNC5:def 3;
(
dom f)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
then
[1, 1]
in (
dom f) by
ZFMISC_1: 87;
then (f
. (1,1))
in (
rng f) by
FUNCT_1: 3;
then not
-infty is
UpperBound of (
rng f) by
A1,
XXREAL_0: 6,
XXREAL_2:def 1;
then (
sup (
rng f))
<>
-infty by
XXREAL_2:def 3;
hence thesis by
XXREAL_0: 14;
end;
theorem ::
DBLSEQ_3:103
for f be
Function of
[:
NAT ,
NAT :],
ExtREAL , f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL st (for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f1
. (n1,m1))
<= (f1
. (n2,m2))) & (for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f2
. (n1,m1))
<= (f2
. (n2,m2))) & (for n,m be
Nat holds ((f1
. (n,m))
+ (f2
. (n,m)))
= (f
. (n,m))) holds f is
P-convergent & (
P-lim f)
= (
sup (
rng f)) & (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2)) & (
sup (
rng f))
= ((
sup (
rng f1))
+ (
sup (
rng f2)))
proof
let f be
Function of
[:
NAT ,
NAT :],
ExtREAL , f1,f2 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL ;
assume that
A1: for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f1
. (n1,m1))
<= (f1
. (n2,m2)) and
A2: for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f2
. (n1,m1))
<= (f2
. (n2,m2)) and
A5: for n,m be
Nat holds ((f1
. (n,m))
+ (f2
. (n,m)))
= (f
. (n,m));
A6: (
dom f1)
=
[:
NAT ,
NAT :] & (
dom f2)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
B0: f1 is
P-convergent & (
P-lim f1)
= (
sup (
rng f1)) & f2 is
P-convergent & (
P-lim f2)
= (
sup (
rng f2)) by
A1,
A2,
Th96;
now
let n1,m1,n2,m2 be
Nat;
assume n1
<= n2 & m1
<= m2;
then (f1
. (n1,m1))
<= (f1
. (n2,m2)) & (f2
. (n1,m1))
<= (f2
. (n2,m2)) by
A1,
A2;
then ((f1
. (n1,m1))
+ (f2
. (n1,m1)))
<= ((f1
. (n2,m2))
+ (f2
. (n2,m2))) by
XXREAL_3: 36;
then (f
. (n1,m1))
<= ((f1
. (n2,m2))
+ (f2
. (n2,m2))) by
A5;
hence (f
. (n1,m1))
<= (f
. (n2,m2)) by
A5;
end;
hence
A7: f is
P-convergent & (
P-lim f)
= (
sup (
rng f)) by
Th96;
P1:
now
per cases by
Lm8;
suppose
A9: (
sup (
rng f1))
in
REAL ;
set SE1 = (
sup (
rng f1));
per cases by
Lm8;
suppose
A10: (
sup (
rng f2))
in
REAL ;
set SE2 = (
sup (
rng f2));
B1:
now
let p be
Real;
assume
A11:
0
< p;
then
consider x1 be
ExtReal such that
A12: x1
in (
rng f1) & ((
sup (
rng f1))
- (p
/ 2))
< x1 by
A9,
MEASURE6: 6;
consider z1 be
object such that
A13: z1
in (
dom f1) & x1
= (f1
. z1) by
A12,
FUNCT_1:def 3;
consider n1,m1 be
object such that
A14: n1
in
NAT & m1
in
NAT & z1
=
[n1, m1] by
A13,
ZFMISC_1:def 2;
reconsider n1, m1 as
Element of
NAT by
A14;
consider x2 be
ExtReal such that
A15: x2
in (
rng f2) & ((
sup (
rng f2))
- (p
/ 2))
< x2 by
A10,
A11,
MEASURE6: 6;
consider z2 be
object such that
A16: z2
in (
dom f2) & x2
= (f2
. z2) by
A15,
FUNCT_1:def 3;
consider n2,m2 be
object such that
A17: n2
in
NAT & m2
in
NAT & z2
=
[n2, m2] by
A16,
ZFMISC_1:def 2;
reconsider n2, m2 as
Element of
NAT by
A17;
reconsider N = (
max ((
max (n1,m1)),(
max (n2,m2)))) as
Nat;
take N;
hereby
let n,m be
Nat;
assume
A18: n
>= N & m
>= N;
N
>= (
max (n1,m1)) & N
>= (
max (n2,m2)) & (
max (n1,m1))
>= n1 & (
max (n1,m1))
>= m1 & (
max (n2,m2))
>= n2 & (
max (n2,m2))
>= m2 by
XXREAL_0: 25;
then N
>= n1 & N
>= m1 & N
>= n2 & N
>= m2 by
XXREAL_0: 2;
then n
>= n1 & n
>= n2 & m
>= m1 & m
>= m2 by
A18,
XXREAL_0: 2;
then
A22: (f1
. (n1,m1))
<= (f1
. (n,m)) & (f2
. (n2,m2))
<= (f2
. (n,m)) by
A1,
A2;
then (SE1
- (f1
. (n,m)))
<= (SE1
- x1) & (SE2
- (f2
. (n,m)))
<= (SE2
- x2) by
A13,
A14,
A16,
A17,
XXREAL_3: 37;
then
A19: ((SE1
- (f1
. (n,m)))
+ (SE2
- (f2
. (n,m))))
<= ((SE1
- x1)
+ (SE2
- x2)) by
XXREAL_3: 36;
A20: (p
/ 2)
in
REAL by
XREAL_0:def 1;
SE1
< ((p
/ 2)
+ x1) & SE2
< ((p
/ 2)
+ x2) by
A12,
A15,
XXREAL_3: 54;
then
A21: (SE1
- x1)
< (p
/ 2) & (SE2
- x2)
< (p
/ 2) by
XXREAL_3: 55;
then
A24: ((p
/ 2)
+ (SE2
- x2))
< ((p
/ 2)
+ (p
/ 2)) by
A20,
XXREAL_3: 43;
n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
[n, m]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
then
B1: (f1
. (n,m))
in (
rng f1) & (f2
. (n,m))
in (
rng f2) & (f1
. (n,m))
<= SE1 & (f2
. (n,m))
<= SE2 by
A6,
FUNCT_1: 3,
XXREAL_2: 4;
then
B2: (f1
. (n,m))
<
+infty & (f2
. (n,m))
<
+infty by
A9,
A10,
XXREAL_0: 2,
XXREAL_0: 9;
B3:
-infty
<> (f1
. (n,m)) &
-infty
<> (f2
. (n,m)) by
B1,
MESFUNC5:def 3;
-infty
<> x1 &
-infty
<> x2 by
A12,
A15,
MESFUNC5:def 3;
then x1
in
REAL & x2
in
REAL by
B2,
A22,
A13,
A14,
A16,
A17,
XXREAL_0: 14;
then (SE2
- x2)
in
REAL by
A10,
XREAL_0:def 1;
then ((SE1
- x1)
+ (SE2
- x2))
< ((p
/ 2)
+ (SE2
- x2)) by
A21,
XXREAL_3: 43;
then ((SE1
- x1)
+ (SE2
- x2))
< ((p
/ 2)
+ (p
/ 2)) by
A24,
XXREAL_0: 2;
then
A26: ((SE1
- (f1
. (n,m)))
+ (SE2
- (f2
. (n,m))))
< p by
A19,
XXREAL_0: 2;
B5: ((SE1
- (f1
. (n,m)))
+ (SE2
- (f2
. (n,m))))
= (((SE1
- (f1
. (n,m)))
+ SE2)
- (f2
. (n,m))) by
A10,
B2,
B3,
XXREAL_3: 30
.= (((SE2
+ SE1)
- (f1
. (n,m)))
- (f2
. (n,m))) by
A9,
A10,
XXREAL_3: 30
.= ((SE1
+ SE2)
- ((f1
. (n,m))
+ (f2
. (n,m)))) by
A9,
A10,
B3,
XXREAL_3: 31
.= ((SE1
+ SE2)
- (f
. (n,m))) by
A5;
(SE1
- (f1
. (n,m)))
>=
0 & (SE2
- (f2
. (n,m)))
>=
0 by
B1,
XXREAL_3: 40;
then
|.((SE1
+ SE2)
- (f
. (n,m))).|
< p by
B5,
A26,
EXTREAL1:def 1;
then
|.(
- ((f
. (n,m))
- (SE1
+ SE2))).|
< p by
XXREAL_3: 26;
hence
|.((f
. (n,m))
- (SE1
+ SE2)).|
< p by
EXTREAL1: 29;
end;
end;
then f is
P-convergent_to_finite_number by
A9,
A10;
hence (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2)) by
A9,
A10,
B0,
B1,
A7,
Def5;
end;
suppose
C1: (
sup (
rng f2))
=
+infty ;
then
C2: ((
P-lim f1)
+ (
P-lim f2))
=
+infty by
B0,
A9,
XXREAL_3:def 2;
now
let g be
Real;
assume
0
< g;
then
consider e1 be
ExtReal such that
C5: e1
in (
rng f1) & (SE1
- (g
/ 2))
< e1 by
A9,
MEASURE6: 6;
consider z1 be
object such that
C6: z1
in (
dom f1) & e1
= (f1
. z1) by
C5,
FUNCT_1:def 3;
consider n1,m1 be
object such that
C7: n1
in
NAT & m1
in
NAT & z1
=
[n1, m1] by
C6,
ZFMISC_1:def 2;
reconsider n1, m1 as
Element of
NAT by
C7;
(g
- (SE1
- (g
/ 2)))
in
REAL & (SE1
- (g
/ 2))
in
REAL by
A9,
XREAL_0:def 1;
then
consider e2 be
Element of
ExtREAL such that
C8: e2
in (
rng f2) & (g
- (SE1
- (g
/ 2)))
< e2 by
C1,
XXREAL_0: 9,
XXREAL_2: 94;
consider z2 be
object such that
C9: z2
in (
dom f2) & e2
= (f2
. z2) by
C8,
FUNCT_1:def 3;
consider n2,m2 be
object such that
C10: n2
in
NAT & m2
in
NAT & z2
=
[n2, m2] by
C9,
ZFMISC_1:def 2;
reconsider n2, m2 as
Element of
NAT by
C10;
reconsider N = (
max ((
max (n2,m2)),(
max (n1,m1)))) as
Nat;
take N;
N
>= (
max (n1,m1)) & N
>= (
max (n2,m2)) & (
max (n1,m1))
>= n1 & (
max (n1,m1))
>= m1 & (
max (n2,m2))
>= n2 & (
max (n2,m2))
>= m2 by
XXREAL_0: 25;
then
C13: N
>= n1 & N
>= m1 & N
>= n2 & N
>= m2 by
XXREAL_0: 2;
hereby
let n,m be
Nat;
assume n
>= N & m
>= N;
then n
>= n1 & m
>= m1 & n
>= n2 & m
>= m2 by
C13,
XXREAL_0: 2;
then (f1
. (n,m))
>= (f1
. (n1,m1)) & (f2
. (n,m))
>= (f2
. (n2,m2)) by
A1,
A2;
then
C14: (SE1
- (g
/ 2))
< (f1
. (n,m)) & (g
- (SE1
- (g
/ 2)))
< (f2
. (n,m)) by
C5,
C6,
C7,
C8,
C9,
C10,
XXREAL_0: 2;
((g
- (SE1
- (g
/ 2)))
+ (SE1
- (g
/ 2)))
= g by
A9,
XXREAL_3: 22;
then g
< ((f1
. (n,m))
+ (f2
. (n,m))) by
C14,
XXREAL_3: 64;
hence g
<= (f
. (n,m)) by
A5;
end;
end;
then f is
P-convergent_to_+infty;
hence (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2)) by
C2,
A7,
Def5;
end;
end;
suppose
D1: (
sup (
rng f1))
=
+infty ;
per cases by
Lm8;
suppose
D3: (
sup (
rng f2))
in
REAL ;
set SE2 = (
sup (
rng f2));
D2: ((
P-lim f1)
+ (
P-lim f2))
=
+infty by
B0,
D1,
D3,
XXREAL_3:def 2;
now
let g be
Real;
assume
0
< g;
then
consider e2 be
ExtReal such that
D5: e2
in (
rng f2) & (SE2
- (g
/ 2))
< e2 by
D3,
MEASURE6: 6;
consider z2 be
object such that
D6: z2
in (
dom f2) & e2
= (f2
. z2) by
D5,
FUNCT_1:def 3;
consider n1,m1 be
object such that
D7: n1
in
NAT & m1
in
NAT & z2
=
[n1, m1] by
D6,
ZFMISC_1:def 2;
reconsider n1, m1 as
Element of
NAT by
D7;
(g
- (SE2
- (g
/ 2)))
in
REAL & (SE2
- (g
/ 2))
in
REAL by
D3,
XREAL_0:def 1;
then
consider e1 be
Element of
ExtREAL such that
D8: e1
in (
rng f1) & (g
- (SE2
- (g
/ 2)))
< e1 by
D1,
XXREAL_0: 9,
XXREAL_2: 94;
consider z1 be
object such that
D9: z1
in (
dom f1) & e1
= (f1
. z1) by
D8,
FUNCT_1:def 3;
consider n2,m2 be
object such that
D10: n2
in
NAT & m2
in
NAT & z1
=
[n2, m2] by
D9,
ZFMISC_1:def 2;
reconsider n2, m2 as
Element of
NAT by
D10;
reconsider N = (
max ((
max (n2,m2)),(
max (n1,m1)))) as
Nat;
take N;
N
>= (
max (n1,m1)) & N
>= (
max (n2,m2)) & (
max (n1,m1))
>= n1 & (
max (n1,m1))
>= m1 & (
max (n2,m2))
>= n2 & (
max (n2,m2))
>= m2 by
XXREAL_0: 25;
then
D13: N
>= n1 & N
>= m1 & N
>= n2 & N
>= m2 by
XXREAL_0: 2;
hereby
let n,m be
Nat;
assume n
>= N & m
>= N;
then n
>= n1 & m
>= m1 & n
>= n2 & m
>= m2 by
D13,
XXREAL_0: 2;
then (f1
. (n,m))
>= (f1
. (n2,m2)) & (f2
. (n,m))
>= (f2
. (n1,m1)) by
A1,
A2;
then
D14: (SE2
- (g
/ 2))
< (f2
. (n,m)) & (g
- (SE2
- (g
/ 2)))
< (f1
. (n,m)) by
D5,
D6,
D7,
D8,
D9,
D10,
XXREAL_0: 2;
((g
- (SE2
- (g
/ 2)))
+ (SE2
- (g
/ 2)))
= g by
D3,
XXREAL_3: 22;
then g
< ((f1
. (n,m))
+ (f2
. (n,m))) by
D14,
XXREAL_3: 64;
hence g
<= (f
. (n,m)) by
A5;
end;
end;
then f is
P-convergent_to_+infty;
hence (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2)) by
D2,
A7,
Def5;
end;
suppose
E1: (
sup (
rng f2))
=
+infty ;
now
let p be
Real;
assume
E2:
0
< p;
then
consider n1,m1 be
Nat such that
E3: (f1
. (n1,m1))
> (p
/ 2) by
D1,
Th101;
consider n2,m2 be
Nat such that
E4: (f2
. (n2,m2))
> (p
/ 2) by
E1,
E2,
Th101;
reconsider n1, n2, m1, m2 as
Element of
NAT by
ORDINAL1:def 12;
reconsider N = (
max ((
max (n2,m2)),(
max (n1,m1)))) as
Nat;
take N;
N
>= (
max (n1,m1)) & N
>= (
max (n2,m2)) & (
max (n1,m1))
>= n1 & (
max (n1,m1))
>= m1 & (
max (n2,m2))
>= n2 & (
max (n2,m2))
>= m2 by
XXREAL_0: 25;
then
E5: N
>= n1 & N
>= m1 & N
>= n2 & N
>= m2 by
XXREAL_0: 2;
hereby
let n,m be
Nat;
assume n
>= N & m
>= N;
then n
>= n1 & m
>= m1 & n
>= n2 & m
>= m2 by
E5,
XXREAL_0: 2;
then (f1
. (n,m))
>= (f1
. (n1,m1)) & (f2
. (n,m))
>= (f2
. (n2,m2)) by
A1,
A2;
then (f1
. (n,m))
>= (p
/ 2) & (f2
. (n,m))
>= (p
/ 2) by
E3,
E4,
XXREAL_0: 2;
then ((p
/ 2)
+ (p
/ 2))
<= ((f1
. (n,m))
+ (f2
. (n,m))) by
XXREAL_3: 36;
hence p
<= (f
. (n,m)) by
A5;
end;
end;
then f is
P-convergent_to_+infty;
then (
P-lim f)
=
+infty & (
P-lim f1)
=
+infty & (
P-lim f2)
=
+infty by
A7,
Def5,
D1,
E1,
A1,
A2,
Th96;
hence (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2)) by
XXREAL_3:def 2;
end;
end;
end;
hence (
P-lim f)
= ((
P-lim f1)
+ (
P-lim f2));
P2: (
P-lim f1)
= (
sup (
rng f1)) & (
P-lim f2)
= (
sup (
rng f2)) by
A1,
A2,
Th96;
now
let n1,m1,n2,m2 be
Nat;
assume n1
<= n2 & m1
<= m2;
then (f1
. (n1,m1))
<= (f1
. (n2,m2)) & (f2
. (n1,m1))
<= (f2
. (n2,m2)) by
A1,
A2;
then ((f1
. (n1,m1))
+ (f2
. (n1,m1)))
<= ((f1
. (n2,m2))
+ (f2
. (n2,m2))) by
XXREAL_3: 36;
then (f
. (n1,m1))
<= ((f1
. (n2,m2))
+ (f2
. (n2,m2))) by
A5;
hence (f
. (n1,m1))
<= (f
. (n2,m2)) by
A5;
end;
hence thesis by
P1,
P2,
Th96;
end;
theorem ::
DBLSEQ_3:104
Th104: for f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
Real st
0
<= c & (for n,m be
Nat holds (f2
. (n,m))
= (c
* (f1
. (n,m)))) holds (
sup (
rng f2))
= (c
* (
sup (
rng f1))) & f2 is
without-infty
proof
let f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
Real;
assume that
A1:
0
<= c and
A3: for n,m be
Nat holds (f2
. (n,m))
= (c
* (f1
. (n,m)));
A6: (
dom f1)
=
[:
NAT ,
NAT :] & (
dom f2)
=
[:
NAT ,
NAT :] by
FUNCT_2:def 1;
C6:
now
assume
-infty
in (
rng f2);
then
consider z be
object such that
C1: z
in (
dom f2) &
-infty
= (f2
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
C2: n
in
NAT & m
in
NAT & z
=
[n, m] by
C1,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
C2;
(f2
. (n,m))
=
-infty by
C1,
C2;
then
C3: (c
* (f1
. (n,m)))
=
-infty by
A3;
then
C4: (f1
. (n,m))
=
-infty or (f1
. (n,m))
=
+infty by
XXREAL_3: 70;
z
in
[:
NAT ,
NAT :] by
C1;
then
[n, m]
in (
dom f1) by
C2,
FUNCT_2:def 1;
then
-infty
in (
rng f1) by
A1,
C3,
C4,
FUNCT_1: 3;
hence contradiction by
MESFUNC5:def 3;
end;
then
C6a: f2 is
without-infty by
MESFUNC5:def 3;
now
per cases by
Lm8;
suppose
A4: (
sup (
rng f1))
in
REAL ;
A5: for y be
UpperBound of (
rng f2) holds (c
* (
sup (
rng f1)))
<= y
proof
let y be
UpperBound of (
rng f2);
reconsider y as
R_eal by
XXREAL_0:def 1;
per cases ;
suppose
A8: c
=
0 ;
[1, 1]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
then (f2
. (1,1))
<= y by
A6,
FUNCT_1: 3,
XXREAL_2:def 1;
then (c
* (f1
. (1,1)))
<= y by
A3;
hence thesis by
A8;
end;
suppose
A10: c
<>
0 ;
now
let x be
ExtReal;
assume x
in (
rng f1);
then
consider z be
object such that
A11: z
in (
dom f1) & x
= (f1
. z) by
FUNCT_1:def 3;
consider n,m be
object such that
A12: n
in
NAT & m
in
NAT & z
=
[n, m] by
A11,
ZFMISC_1:def 2;
reconsider n, m as
Element of
NAT by
A12;
A14: (f2
. (n,m))
in (
rng f2) by
A11,
A12,
A6,
FUNCT_1: 3;
(f2
. (n,m))
= (c
* (f1
. (n,m))) by
A3;
then ((c
* (f1
. (n,m)))
/ c)
<= (y
/ c) by
A1,
A10,
A14,
XXREAL_2:def 1,
XXREAL_3: 79;
hence x
<= (y
/ c) by
A10,
A11,
A12,
XXREAL_3: 88;
end;
then
B14: (y
/ c) is
UpperBound of (
rng f1) by
XXREAL_2:def 1;
A15:
now
assume
A16: y
=
-infty ;
A17:
[1, 1]
in
[:
NAT ,
NAT :] by
ZFMISC_1: 87;
then (f2
. (1,1))
in (
rng f2) by
A6,
FUNCT_1: 3;
then (f2
. (1,1))
=
-infty by
A16,
XXREAL_0: 6,
XXREAL_2:def 1;
then
A19: (c
* (f1
. (1,1)))
=
-infty by
A3;
(f1
. (1,1))
<= (
sup (
rng f1)) by
A6,
A17,
FUNCT_1: 3,
XXREAL_2: 4;
then
A20: (f1
. (1,1))
<
+infty by
A4,
XXREAL_0: 2,
XXREAL_0: 9;
A21: not
-infty
in (
rng f1) by
MESFUNC5:def 3;
(f1
. (1,1))
in (
rng f1) by
A6,
A17,
FUNCT_1: 3;
hence contradiction by
A19,
A20,
A21,
XXREAL_3: 70;
end;
per cases by
A15,
XXREAL_0: 14;
suppose y
=
+infty ;
hence thesis by
XXREAL_0: 4;
end;
suppose y
in
REAL ;
then
reconsider ry = y as
Real;
reconsider SE1 = (
sup (
rng f1)) as
Real by
A4;
(y
/ c)
= (ry
/ c);
then (SE1
* c)
<= ry by
A1,
A10,
B14,
XXREAL_2:def 3,
XREAL_1: 83;
hence thesis by
XXREAL_3:def 5;
end;
end;
end;
now
let x be
ExtReal;
assume x
in (
rng f2);
then
consider z2 be
object such that
A22: z2
in (
dom f2) & x
= (f2
. z2) by
FUNCT_1:def 3;
consider n2,m2 be
object such that
A23: n2
in
NAT & m2
in
NAT & z2
=
[n2, m2] by
A22,
ZFMISC_1:def 2;
reconsider n2, m2 as
Element of
NAT by
A23;
A24: (
sup (
rng f1)) is
UpperBound of (
rng f1) by
XXREAL_2:def 3;
[n2, m2]
in (
dom f1) by
A6,
ZFMISC_1: 87;
then
A25: (f1
. (n2,m2))
<= (
sup (
rng f1)) by
A24,
XXREAL_2:def 1,
FUNCT_1: 3;
x
= (f2
. (n2,m2)) by
A22,
A23;
then x
= (c
* (f1
. (n2,m2))) by
A3;
hence x
<= (c
* (
sup (
rng f1))) by
A1,
A25,
XXREAL_3: 71;
end;
then (c
* (
sup (
rng f1))) is
UpperBound of (
rng f2) by
XXREAL_2:def 1;
hence (
sup (
rng f2))
= (c
* (
sup (
rng f1))) by
A5,
XXREAL_2:def 3;
end;
suppose
A30: (
sup (
rng f1))
=
+infty ;
per cases ;
suppose
A27: c
=
0 ;
A28:
now
let n,m be
Nat;
(f2
. (n,m))
= (c
* (f1
. (n,m))) by
A3;
hence (f2
. (n,m))
=
0 by
A27;
end;
then (
P-lim f2)
= (
sup (
rng f2)) by
Th102;
hence (
sup (
rng f2))
= (c
* (
sup (
rng f1))) by
A27,
A28,
Th102;
end;
suppose
A29: c
<>
0 ;
A34:
now
let k be
Real;
reconsider k1 = k as
Real;
assume
C5:
0
< k;
then
consider n,m be
Nat such that
A31: (k
/ c)
< (f1
. (n,m)) by
A1,
A29,
A30,
Th101;
C10: (f2
. (n,m))
= (c
* (f1
. (n,m))) by
A3;
n
in
NAT & m
in
NAT by
ORDINAL1:def 12;
then
[n, m]
in (
dom f2) by
A6,
ZFMISC_1: 87;
then
C3: (f2
. (n,m))
<>
-infty by
C6,
FUNCT_1: 3;
now
per cases by
C3,
XXREAL_0: 14;
suppose
C7: (f2
. (n,m))
in
REAL ;
(f1
. (n,m))
>
0 &
0
>=
-infty by
A1,
C5,
A31;
then
C9: (f1
. (n,m))
in
REAL or (f1
. (n,m))
=
+infty by
XXREAL_0: 14;
now
assume (f1
. (n,m))
=
+infty ;
then (c
* (f1
. (n,m)))
=
+infty by
A1,
A29,
XXREAL_3:def 5;
hence contradiction by
A3,
C7;
end;
then
reconsider ES1 = (f1
. (n,m)) as
Real by
C9;
k
< (c
* ES1) by
A1,
A29,
A31,
XREAL_1: 77;
hence k
< (f2
. (n,m)) by
C10,
XXREAL_3:def 5;
end;
suppose (f2
. (n,m))
=
+infty ;
hence k
< (f2
. (n,m)) by
XREAL_0:def 1,
XXREAL_0: 9;
end;
end;
hence ex n,m be
Nat st not (f2
. (n,m))
<= k;
end;
(c
* (
sup (
rng f1)))
=
+infty by
A1,
A29,
A30,
XXREAL_3:def 5;
hence (
sup (
rng f2))
= (c
* (
sup (
rng f1))) by
C6a,
A34,
Th101;
end;
end;
end;
hence (
sup (
rng f2))
= (c
* (
sup (
rng f1)));
thus f2 is
without-infty by
C6,
MESFUNC5:def 3;
end;
theorem ::
DBLSEQ_3:105
for f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
Real st
0
<= c & (for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f1
. (n1,m1))
<= (f1
. (n2,m2))) & (for n,m be
Nat holds (f2
. (n,m))
= (c
* (f1
. (n,m)))) holds (for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f2
. (n1,m1))
<= (f2
. (n2,m2))) & f2 is
without-infty & f2 is
P-convergent & (
P-lim f2)
= (
sup (
rng f2)) & (
P-lim f2)
= (c
* (
P-lim f1))
proof
let f1 be
without-infty
Function of
[:
NAT ,
NAT :],
ExtREAL , f2 be
Function of
[:
NAT ,
NAT :],
ExtREAL , c be
Real;
assume that
A1:
0
<= c and
A2: for n1,m1,n2,m2 be
Nat st n1
<= n2 & m1
<= m2 holds (f1
. (n1,m1))
<= (f1
. (n2,m2)) and
A3: for n,m be
Nat holds (f2
. (n,m))
= (c
* (f1
. (n,m)));
A5: (
sup (
rng f1))
= (
P-lim f1) by
A2,
Th96;
now
let n1,m1,n2,m2 be
Nat;
assume n1
<= n2 & m1
<= m2;
then (c
* (f1
. (n1,m1)))
<= (c
* (f1
. (n2,m2))) by
A1,
A2,
XXREAL_3: 71;
then (f2
. (n1,m1))
<= (c
* (f1
. (n2,m2))) by
A3;
hence (f2
. (n1,m1))
<= (f2
. (n2,m2)) by
A3;
end;
thus f2 is
without-infty by
A1,
A3,
Th104;
thus f2 is
P-convergent & (
P-lim f2)
= (
sup (
rng f2)) by
A6,
Th96;
(
sup (
rng f2))
= (
P-lim f2) by
A6,
Th96;
hence thesis by
A1,
A3,
A5,
Th104;
end;