euclid_7.miz
begin
reserve i,j,m,n for
Nat,
z,B0 for
set,
f,x0 for
real-valued
FinSequence;
::$Canceled
theorem ::
EUCLID_7:2
Th1: for R be
Relation, Y be
set st (
rng R)
c= Y holds (R
" Y)
= (
dom R)
proof
let R be
Relation, Y be
set;
assume (
rng R)
c= Y;
then ((
rng R)
/\ Y)
= (
rng R) by
XBOOLE_1: 28;
hence (R
" Y)
= (R
" (
rng R)) by
RELAT_1: 133
.= (
dom R) by
RELAT_1: 134;
end;
Lm1: for X be
set, Y be non
empty
set, f be
Function of X, Y st f is
bijective holds (
card X)
= (
card Y) by
EULER_1: 11;
theorem ::
EUCLID_7:3
(
<*z*>
*
<*1*>)
=
<*z*>
proof
A1: (
dom
<*z*>)
= (
Seg 1) by
FINSEQ_1: 38;
(
rng
<*1*>)
=
{1} by
FINSEQ_1: 38;
then
A2: (
dom (
<*z*>
*
<*1*>))
= (
dom
<*1*>) by
A1,
FINSEQ_1: 2,
RELAT_1: 27;
A3: (
dom
<*1*>)
= (
Seg 1) by
FINSEQ_1: 38;
then 1
in (
dom
<*1*>) by
FINSEQ_1: 2,
TARSKI:def 1;
then ((
<*z*>
*
<*1*>)
. 1)
= (
<*z*>
. (
<*1*>
. 1)) by
FUNCT_1: 13
.= (
<*z*>
. 1) by
FINSEQ_1: 40
.= z by
FINSEQ_1: 40;
hence thesis by
A3,
A2,
FINSEQ_1:def 8;
end;
theorem ::
EUCLID_7:4
for x be
Element of (
REAL
0 ) holds x
= (
<*>
REAL );
theorem ::
EUCLID_7:5
Th4: for a,b,c be
Element of (
REAL n) holds (((a
- b)
+ c)
+ b)
= (a
+ c)
proof
let a,b,c be
Element of (
REAL n);
reconsider a2 = a, b2 = b, c2 = c as
Element of (n
-tuples_on
REAL );
(((a2
- b2)
+ c2)
+ b2)
= (((a2
+ (
- b2))
+ b2)
+ c2) by
RFUNCT_1: 8
.= ((a2
+ (b2
+ (
- b2)))
+ c2) by
RFUNCT_1: 8
.= ((a2
+ (n
|-> (
In (
0 ,
REAL ))))
+ c2) by
RVSUM_1: 22
.= (a2
+ c2) by
RVSUM_1: 16;
hence thesis;
end;
registration
let f1,f2 be
FinSequence;
cluster
<:f1, f2:> ->
FinSequence-like;
coherence
proof
(
dom
<:f1, f2:>)
= ((
dom f1)
/\ (
dom f2)) by
FUNCT_3:def 7;
hence thesis by
VALUED_1: 19;
end;
end
definition
let D be
set, f1,f2 be
FinSequence of D;
:: original:
<:
redefine
func
<:f1,f2:> ->
FinSequence of
[:D, D:] ;
coherence
proof
A1:
[:(
rng f1), (
rng f2):]
c=
[:D, D:] by
ZFMISC_1: 96;
(
rng
<:f1, f2:>)
c=
[:(
rng f1), (
rng f2):] by
FUNCT_3: 51;
then (
rng
<:f1, f2:>)
c=
[:D, D:] by
A1;
hence thesis by
FINSEQ_1:def 4;
end;
end
Lm2: n
< m implies ex k be
Nat st m
= ((n
+ 1)
+ k)
proof
assume
A1: n
< m;
then
consider k1 be
Nat such that
A2: m
= (n
+ k1) by
NAT_1: 10;
k1
<>
0 by
A1,
A2;
then
consider n1 be
Nat such that
A3: k1
= (n1
+ 1) by
NAT_1: 6;
take n1;
thus m
= ((n
+ 1)
+ n1) by
A2,
A3;
end;
definition
let h be
real-valued
FinSequence;
:: original:
increasing
redefine
::
EUCLID_7:def1
attr h is
increasing means for i be
Nat st 1
<= i & i
< (
len h) holds (h
. i)
< (h
. (i
+ 1));
compatibility
proof
hereby
assume
A1: h is
increasing;
let i be
Nat such that
A2: 1
<= i and
A3: i
< (
len h);
A4: 1
<= (i
+ 1) by
NAT_1: 12;
(i
+ 1)
<= (
len h) by
A3,
NAT_1: 13;
then
A5: (i
+ 1)
in (
dom h) by
A4,
FINSEQ_3: 25;
A6: i
< (i
+ 1) by
NAT_1: 16;
i
in (
dom h) by
A2,
A3,
FINSEQ_3: 25;
hence (h
. i)
< (h
. (i
+ 1)) by
A1,
A5,
A6,
VALUED_0:def 13;
end;
assume
A7: for i be
Nat st 1
<= i & i
< (
len h) holds (h
. i)
< (h
. (i
+ 1));
now
let n,m be
ExtReal;
assume
A8: n
in (
dom h);
assume
A9: m
in (
dom h);
then
reconsider m1 = m, n1 = n as
Element of
NAT by
A8;
defpred
X[
Nat] means (n1
+ $1)
< (
len h) implies (h
. n)
< (h
. ((n1
+ 1)
+ $1));
A10:
now
let a be
Nat such that
A11:
X[a];
thus
X[(a
+ 1)]
proof
A12: ((n1
+ a)
+
0 )
< ((n1
+ a)
+ 1) by
XREAL_1: 6;
assume
A13: (n1
+ (a
+ 1))
< (
len h);
1
<= (1
+ (n1
+ a)) by
NAT_1: 11;
then (h
. ((n1
+ 1)
+ a))
< (h
. (((n1
+ 1)
+ a)
+ 1)) by
A7,
A13;
hence thesis by
A11,
A13,
A12,
XXREAL_0: 2;
end;
end;
1
<= n by
A8,
FINSEQ_3: 25;
then
A14:
X[
0 ] by
A7;
A15: for k be
Nat holds
X[k] from
NAT_1:sch 2(
A14,
A10);
assume n
< m;
then
consider k be
Nat such that
A16: m1
= ((n1
+ 1)
+ k) by
Lm2;
m
<= (
len h) by
A9,
FINSEQ_3: 25;
then (m1
- 1)
< ((
len h)
-
0 ) by
XREAL_1: 15;
then (n1
+ k)
< (
len h) by
A16;
hence (h
. n)
< (h
. m) by
A16,
A15;
end;
hence thesis by
VALUED_0:def 13;
end;
end
theorem ::
EUCLID_7:6
Th5: for h be
real-valued
FinSequence st h is
increasing holds for i,j be
Nat st i
< j & 1
<= i & j
<= (
len h) holds (h
. i)
< (h
. j)
proof
let h be
real-valued
FinSequence;
assume
A1: h is
increasing;
let i,j be
Nat;
assume that
A2: i
< j and
A3: 1
<= i and
A4: j
<= (
len h);
defpred
P[
Nat] means ((i
+ 1)
+ $1)
<= (
len h) implies (h
. i)
< (h
. ((i
+ 1)
+ $1));
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
((i
+ 1)
+ (k
+ 1))
<= (
len h) implies (h
. i)
< (h
. ((i
+ 1)
+ (k
+ 1)))
proof
A7: (i
+ 1)
<= ((i
+ 1)
+ k) by
NAT_1: 11;
i
< (i
+ 1) by
XREAL_1: 29;
then i
< ((i
+ 1)
+ k) by
A7,
XXREAL_0: 2;
then
A8: 1
< ((i
+ 1)
+ k) by
A3,
XXREAL_0: 2;
k
< (k
+ 1) by
XREAL_1: 29;
then
A9: ((i
+ 1)
+ k)
< ((i
+ 1)
+ (k
+ 1)) by
XREAL_1: 6;
assume
A10: ((i
+ 1)
+ (k
+ 1))
<= (
len h);
then ((i
+ 1)
+ k)
< (
len h) by
A9,
XXREAL_0: 2;
then (h
. ((i
+ 1)
+ k))
< (h
. (((i
+ 1)
+ k)
+ 1)) by
A1,
A8;
hence (h
. i)
< (h
. ((i
+ 1)
+ (k
+ 1))) by
A6,
A10,
A9,
XXREAL_0: 2;
end;
hence
P[(k
+ 1)];
end;
i
< (
len h) by
A2,
A4,
XXREAL_0: 2;
then
A11:
P[
0 ] by
A1,
A3;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A11,
A5);
then
A12: ((i
+ 1)
+ (j
-' (i
+ 1)))
<= (
len h) implies (h
. i)
< (h
. ((i
+ 1)
+ (j
-' (i
+ 1))));
(i
+ 1)
<= j by
A2,
NAT_1: 13;
then (j
-' (i
+ 1))
= (j
- (i
+ 1)) by
XREAL_1: 233;
hence (h
. i)
< (h
. j) by
A4,
A12;
end;
theorem ::
EUCLID_7:7
Th6: for h be
real-valued
FinSequence st h is
increasing holds for i,j be
Nat st i
<= j & 1
<= i & j
<= (
len h) holds (h
. i)
<= (h
. j)
proof
let h be
real-valued
FinSequence;
assume
A1: h is
increasing;
let i,j be
Nat;
assume that
A2: i
<= j and
A3: 1
<= i and
A4: j
<= (
len h);
i
< j or i
= j by
A2,
XXREAL_0: 1;
hence thesis by
A1,
A3,
A4,
Th5;
end;
theorem ::
EUCLID_7:8
Th7: for h be
natural-valued
FinSequence st h is
increasing holds for i be
Nat st i
<= (
len h) & 1
<= (h
. 1) holds i
<= (h
. i)
proof
let h be
natural-valued
FinSequence;
assume
A1: h is
increasing;
defpred
P[
Nat] means $1
<= (
len h) implies $1
<= (h
. $1);
let i be
Nat;
assume that
A2: i
<= (
len h) and
A3: 1
<= (h
. 1);
A4: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A5:
P[k];
(k
+ 1)
<= (
len h) implies (k
+ 1)
<= (h
. (k
+ 1))
proof
A6: k
< (k
+ 1) by
XREAL_1: 29;
assume
A7: (k
+ 1)
<= (
len h);
then
A8: k
< (
len h) by
A6,
XXREAL_0: 2;
per cases ;
suppose k
=
0 ;
hence (k
+ 1)
<= (h
. (k
+ 1)) by
A3;
end;
suppose k
>
0 ;
then (
0
+ 1)
<= k by
NAT_1: 13;
then (h
. k)
< (h
. (k
+ 1)) by
A1,
A8;
then k
< (h
. (k
+ 1)) by
A5,
A7,
A6,
XXREAL_0: 2;
hence (k
+ 1)
<= (h
. (k
+ 1)) by
NAT_1: 13;
end;
end;
hence
P[(k
+ 1)];
end;
A9:
P[
0 ];
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A9,
A4);
hence i
<= (h
. i) by
A2;
end;
theorem ::
EUCLID_7:9
Th8: for V be
RealLinearSpace, X be
Subspace of V st V is
strict & X is
strict & the
carrier of X
= the
carrier of V holds X
= V
proof
let V be
RealLinearSpace, X be
Subspace of V;
assume that
A1: V is
strict and
A2: X is
strict and
A3: the
carrier of X
= the
carrier of V;
A4: the
Mult of X
= (the
Mult of V
|
[:
REAL , the
carrier of V:]) by
A3,
RLSUB_1:def 2
.= the
Mult of V;
A5: (
0. X)
= (
0. V) by
RLSUB_1:def 2;
the
addF of X
= (the
addF of V
|| the
carrier of V) by
A3,
RLSUB_1:def 2
.= (the
addF of V
|
[:the
carrier of V, the
carrier of V:]) by
REALSET1:def 2
.= the
addF of V;
hence X
= V by
A1,
A2,
A3,
A5,
A4;
end;
definition
let D be
set;
let F be
FinSequence of D;
let h be
Permutation of (
dom F);
::
EUCLID_7:def2
func F
(*) h ->
FinSequence of D equals (F
* h);
coherence
proof
reconsider G = F as
Function;
A1: (h
" (
dom G))
= (h
" (
rng h)) by
FUNCT_2:def 3
.= (
dom h) by
RELAT_1: 134;
(
dom (G
* h))
= (h
" (
dom G)) by
RELAT_1: 147
.= (
dom F) by
A1,
FUNCT_2: 52
.= (
Seg (
len F)) by
FINSEQ_1:def 3;
then
A2: (F
* h) is
FinSequence by
FINSEQ_1:def 2;
(
rng (F
* h))
c= D;
hence thesis by
A2,
FINSEQ_1:def 4;
end;
end
theorem ::
EUCLID_7:10
Th9: for D be non
empty
set, f be
FinSequence of D st 1
<= i & i
<= (
len f) & 1
<= j & j
<= (
len f) holds ((
Swap (f,i,j))
. i)
= (f
. j) & ((
Swap (f,i,j))
. j)
= (f
. i)
proof
let D be non
empty
set, f be
FinSequence of D;
assume that
A1: 1
<= i and
A2: i
<= (
len f) and
A3: 1
<= j and
A4: j
<= (
len f);
A5: (
len (
Replace (f,i,(f
/. j))))
= (
len f) by
FINSEQ_7: 5;
A6: (
len (
Swap (f,i,j)))
= (
len f) by
FINSEQ_7: 18;
then ((
Swap (f,i,j))
/. j)
= ((
Swap (f,i,j))
. j) by
A3,
A4,
FINSEQ_4: 15;
then
A7: ((
Swap (f,i,j))
. j)
= ((
Replace ((
Replace (f,i,(f
/. j))),j,(f
/. i)))
/. j) by
A1,
A2,
A3,
A4,
FINSEQ_7:def 2
.= (f
/. i) by
A3,
A4,
A5,
FINSEQ_7: 8
.= (f
. i) by
A1,
A2,
FINSEQ_4: 15;
A8: (
Swap (f,i,j))
= (
Swap (f,j,i)) by
FINSEQ_7: 21;
A9: (
len (
Replace (f,j,(f
/. i))))
= (
len f) by
FINSEQ_7: 5;
((
Swap (f,i,j))
/. i)
= ((
Swap (f,i,j))
. i) by
A1,
A2,
A6,
FINSEQ_4: 15;
then ((
Swap (f,i,j))
. i)
= ((
Replace ((
Replace (f,j,(f
/. i))),i,(f
/. j)))
/. i) by
A1,
A2,
A3,
A4,
A8,
FINSEQ_7:def 2
.= (f
/. j) by
A1,
A2,
A9,
FINSEQ_7: 8
.= (f
. j) by
A3,
A4,
FINSEQ_4: 15;
hence thesis by
A7;
end;
theorem ::
EUCLID_7:11
Th10:
{} is
Permutation of
{}
proof
A1: (
rng
{} )
=
{} ;
(
dom
{} )
=
{} ;
then
reconsider f =
{} as
Function of
{} ,
{} by
A1,
FUNCT_2: 1;
f is
one-to-one
onto
Function of
{} ,
{} by
A1,
FUNCT_2:def 3;
hence thesis;
end;
theorem ::
EUCLID_7:12
<*1*> is
Permutation of
{1}
proof
set g =
<*1*>;
A1: (
rng g)
=
{1} by
FINSEQ_1: 38;
(
dom g)
=
{1} by
FINSEQ_1: 2,
FINSEQ_1: 38;
then
reconsider f = g as
Function of
{1},
{1} by
A1,
FUNCT_2: 1;
f is
one-to-one
onto
Function of
{1},
{1} by
A1,
FUNCT_2:def 3;
hence thesis;
end;
theorem ::
EUCLID_7:13
Th12: for h be
FinSequence of
REAL holds h is
one-to-one iff (
sort_a h) is
one-to-one
proof
let h be
FinSequence of
REAL ;
A1: (h,(
sort_a h))
are_fiberwise_equipotent by
RFINSEQ2:def 6;
then ex H be
Function st (
dom H)
= (
dom (
sort_a h)) & (
rng H)
= (
dom h) & H is
one-to-one & (
sort_a h)
= (h
* H) by
CLASSES1: 77;
hence h is
one-to-one implies (
sort_a h) is
one-to-one;
ex G be
Function st (
dom G)
= (
dom h) & (
rng G)
= (
dom (
sort_a h)) & G is
one-to-one & h
= ((
sort_a h)
* G) by
A1,
CLASSES1: 77;
hence (
sort_a h) is
one-to-one implies h is
one-to-one;
end;
theorem ::
EUCLID_7:14
Th13: for h be
FinSequence of
NAT st h is
one-to-one holds ex h3 be
Permutation of (
dom h), h2 be
FinSequence of
NAT st h2
= (h
* h3) & h2 is
increasing & (
dom h)
= (
dom h2) & (
rng h)
= (
rng h2)
proof
let h be
FinSequence of
NAT ;
assume
A1: h is
one-to-one;
per cases ;
suppose
A2: (
dom h)
<>
{} ;
(
rng h)
c=
REAL by
NUMBERS: 19;
then
reconsider hr = h as
FinSequence of
REAL by
FINSEQ_1:def 4;
A3: (hr,(
sort_a hr))
are_fiberwise_equipotent by
RFINSEQ2:def 6;
then
consider H be
Function such that
A4: (
dom H)
= (
dom (
sort_a hr)) and
A5: (
rng H)
= (
dom hr) and
A6: H is
one-to-one and
A7: (
sort_a hr)
= (hr
* H) by
CLASSES1: 77;
(
rng (
sort_a hr))
= (
rng hr) by
A3,
CLASSES1: 75;
then
reconsider h4 = (
sort_a hr) as
FinSequence of
NAT by
FINSEQ_1:def 4;
A8: (
rng h)
= (
rng h4) by
A5,
A7,
RELAT_1: 28;
A9: (
dom h4)
= (
Seg (
len h4)) by
FINSEQ_1:def 3;
for i be
Nat st 1
<= i & i
< (
len h4) holds (h4
. i)
< (h4
. (i
+ 1))
proof
let i be
Nat;
assume that
A10: 1
<= i and
A11: i
< (
len h4);
A12: (i
+ 1)
<= (
len h4) by
A11,
NAT_1: 13;
A13: i
in (
dom h4) by
A9,
A10,
A11,
FINSEQ_1: 1;
1
< (i
+ 1) by
A10,
XREAL_1: 29;
then
A14: (i
+ 1)
in (
dom h4) by
A9,
A12,
FINSEQ_1: 1;
A15: h4 is
one-to-one by
A1,
Th12;
A16:
now
assume (h4
. i)
= (h4
. (i
+ 1));
then i
= (i
+ 1) by
A14,
A13,
A15,
FUNCT_1:def 4;
hence contradiction;
end;
(h4
. i)
<= (h4
. (i
+ 1)) by
A14,
A13,
INTEGRA2:def 1;
hence (h4
. i)
< (h4
. (i
+ 1)) by
A16,
XXREAL_0: 1;
end;
then
A17: h4 is
increasing;
(
dom (
sort_a hr))
= (
dom hr) by
RFINSEQ2: 31;
then
reconsider H2 = H as
Function of (
dom h), (
dom h) by
A4,
A5,
FUNCT_2: 1;
H2 is
onto by
A5,
FUNCT_2:def 3;
then
reconsider h5 = H as
Permutation of (
dom h) by
A6;
A18: h4
= (h
* h5) by
A7;
(
dom h)
= (
dom h5) by
A2,
FUNCT_2:def 1
.= (
dom h4) by
A4;
hence thesis by
A18,
A17,
A8;
end;
suppose
A19: (
dom h)
=
{} ;
then (
rng h)
=
{} by
RELAT_1: 42;
then
reconsider h5 = h as
Function of
{} ,
{} by
A19,
FUNCT_2: 1;
reconsider h22 = (h
* h5) as
FinSequence of
NAT ;
h22 is
increasing;
hence thesis by
A19,
Th10;
end;
end;
begin
definition
let B0 be
set;
::
EUCLID_7:def3
attr B0 is
R-orthogonal means
:
Def3: for x,y be
real-valued
FinSequence st x
in B0 & y
in B0 & x
<> y holds
|(x, y)|
=
0 ;
end
registration
cluster
empty ->
R-orthogonal for
set;
coherence ;
end
theorem ::
EUCLID_7:15
B0 is
R-orthogonal iff for x,y be
real-valued
FinSequence st x
in B0 & y
in B0 & x
<> y holds (x,y)
are_orthogonal
proof
thus B0 is
R-orthogonal implies for x,y be
real-valued
FinSequence st x
in B0 & y
in B0 & x
<> y holds (x,y)
are_orthogonal ;
assume
A1: for x,y be
real-valued
FinSequence st x
in B0 & y
in B0 & x
<> y holds (x,y)
are_orthogonal ;
let x,y be
real-valued
FinSequence;
assume that
A2: x
in B0 and
A3: y
in B0 and
A4: x
<> y;
(x,y)
are_orthogonal by
A1,
A2,
A3,
A4;
hence
|(x, y)|
=
0 ;
end;
definition
let B0 be
set;
::
EUCLID_7:def4
attr B0 is
R-normal means
:
Def4: for x be
real-valued
FinSequence st x
in B0 holds
|.x.|
= 1;
end
registration
cluster
empty ->
R-normal for
set;
coherence ;
end
registration
cluster
R-normal for
set;
existence
proof
take
{} ;
thus thesis;
end;
end
registration
let B0,B1 be
R-normal
set;
cluster (B0
\/ B1) ->
R-normal;
coherence
proof
let x be
real-valued
FinSequence;
assume x
in (B0
\/ B1);
then x
in B0 or x
in B1 by
XBOOLE_0:def 3;
hence thesis by
Def4;
end;
end
theorem ::
EUCLID_7:16
Th15:
|.f.|
= 1 implies
{f} is
R-normal by
TARSKI:def 1;
theorem ::
EUCLID_7:17
Th16: B0 is
R-normal &
|.x0.|
= 1 implies (B0
\/
{x0}) is
R-normal
proof
assume that
A1: B0 is
R-normal and
A2:
|.x0.|
= 1;
{x0} is
R-normal by
A2,
Th15;
hence (B0
\/
{x0}) is
R-normal by
A1;
end;
definition
let B0 be
set;
::
EUCLID_7:def5
attr B0 is
R-orthonormal means B0 is
R-orthogonal
R-normal;
end
registration
cluster
R-orthonormal ->
R-orthogonal
R-normal for
set;
coherence ;
cluster
R-orthogonal
R-normal ->
R-orthonormal for
set;
coherence ;
end
registration
cluster
{
<*1*>} ->
R-orthonormal;
coherence
proof
set B0 =
{
<*1*>};
thus for x,y be
real-valued
FinSequence st x
in B0 & y
in B0 & x
<> y holds
|(x, y)|
=
0
proof
let x,y be
real-valued
FinSequence;
assume that
A1: x
in B0 and
A2: y
in B0;
x
=
<*1*> by
A1,
TARSKI:def 1;
hence thesis by
A2,
TARSKI:def 1;
end;
let x be
real-valued
FinSequence;
assume x
in B0;
then x
=
<*1*> by
TARSKI:def 1;
then (
sqr x)
=
<*(1
^2 )*> by
RVSUM_1: 55;
hence thesis by
RVSUM_1: 73,
SQUARE_1: 18;
end;
end
registration
cluster
R-orthonormal non
empty for
set;
existence
proof
take
{
<*1*>};
thus thesis;
end;
end
registration
let n;
cluster
R-orthonormal for
Subset of (
REAL n);
existence
proof
(
{} (
REAL n)) is
R-orthonormal;
hence thesis;
end;
end
definition
let n be
Nat;
let B0 be
Subset of (
REAL n);
::
EUCLID_7:def6
attr B0 is
complete means
:
Def6: for B be
R-orthonormal
Subset of (
REAL n) st B0
c= B holds B
= B0;
end
definition
let n be
Nat, B0 be
Subset of (
REAL n);
::
EUCLID_7:def7
attr B0 is
orthogonal_basis means B0 is
R-orthonormal
complete;
end
registration
let n be
Nat;
cluster
orthogonal_basis ->
R-orthonormal
complete for
Subset of (
REAL n);
coherence ;
cluster
R-orthonormal
complete ->
orthogonal_basis for
Subset of (
REAL n);
coherence ;
end
theorem ::
EUCLID_7:18
Th17: for B0 be
Subset of (
REAL
0 ) st B0 is
orthogonal_basis holds B0
=
{}
proof
let B0 be
Subset of (
REAL
0 );
assume that
A1: B0 is
orthogonal_basis and
A2: B0
<>
{} ;
set x = the
Element of B0;
x
in B0 by
A2;
then
reconsider x0 = x as
Element of (
REAL
0 );
|((
0* (
len x0)), (
0* (
len x0)))|
=
0 by
EUCLID_2: 9;
then
|.(
0* (
len x0)).|
=
0 by
EUCLID_2: 8;
hence contradiction by
A1,
A2,
Def4;
end;
theorem ::
EUCLID_7:19
for B0 be
Subset of (
REAL n), y be
Element of (
REAL n) st B0 is
orthogonal_basis & (for x be
Element of (
REAL n) st x
in B0 holds
|(x, y)|
=
0 ) holds y
= (
0* n)
proof
let B0 be
Subset of (
REAL n), y be
Element of (
REAL n);
assume that
A1: B0 is
orthogonal_basis and
A2: for x be
Element of (
REAL n) st x
in B0 holds
|(x, y)|
=
0 ;
now
reconsider y1 = ((1
/
|.y.|)
* y) as
Element of (
REAL n);
reconsider B1 = (B0
\/
{y1}) as
Subset of (
REAL n);
y1
in
{y1} by
TARSKI:def 1;
then
A3: y1
in B1 by
XBOOLE_0:def 3;
A4: (
len y)
= n by
CARD_1:def 7;
for x2,y2 be
real-valued
FinSequence st x2
in B1 & y2
in B1 & x2
<> y2 holds
|(x2, y2)|
=
0
proof
let x2,y2 be
real-valued
FinSequence;
assume that
A5: x2
in B1 and
A6: y2
in B1 and
A7: x2
<> y2;
reconsider X2 = x2, Y2 = y2 as
Element of (
REAL n) by
A5,
A6;
A8: (
len Y2)
= n by
CARD_1:def 7;
per cases by
A5,
XBOOLE_0:def 3;
suppose
A9: x2
in B0;
per cases by
A6,
XBOOLE_0:def 3;
suppose y2
in B0;
hence
|(x2, y2)|
=
0 by
A1,
A7,
A9,
Def3;
end;
suppose
A10: y2
in
{y1};
A11: (
len X2)
= n by
CARD_1:def 7;
|(x2, y)|
=
0 by
A2,
A9;
then
A12: ((1
/
|.y.|)
*
|(x2, y)|)
=
0 ;
y2
= y1 by
A10,
TARSKI:def 1;
hence
|(x2, y2)|
=
0 by
A4,
A11,
A12,
RVSUM_1: 121;
end;
end;
suppose
A13: x2
in
{y1};
then x2
= y1 by
TARSKI:def 1;
then not y2
in
{y1} by
A7,
TARSKI:def 1;
then y2
in B0 by
A6,
XBOOLE_0:def 3;
then
|(y2, y)|
=
0 by
A2;
then ((1
/
|.y.|)
*
|(y2, y)|)
=
0 ;
then
|(Y2, ((1
/
|.y.|)
* y))|
=
0 by
A4,
A8,
RVSUM_1: 121;
hence
|(x2, y2)|
=
0 by
A13,
TARSKI:def 1;
end;
end;
then
A14: B1 is
R-orthogonal;
assume
A15: y
<> (
0* n);
A16:
|.y1.|
= (
|.(1
/
|.y.|).|
*
|.y.|) by
EUCLID: 11
.= ((1
/
|.y.|)
*
|.y.|) by
ABSVALUE:def 1
.= 1 by
A15,
EUCLID: 8,
XCMPLX_1: 106;
for x be
real-valued
FinSequence st x
in B1 holds
|.x.|
= 1
proof
let x be
real-valued
FinSequence;
assume x
in B1;
then x
in B0 or x
in
{y1} by
XBOOLE_0:def 3;
hence
|.x.|
= 1 by
A1,
A16,
Def4,
TARSKI:def 1;
end;
then
A17: B1 is
R-normal;
A18: (
len y1)
= n by
CARD_1:def 7;
A19:
now
assume y1
in B0;
then
|(y1, y)|
=
0 by
A2;
then ((1
/
|.y.|)
*
|(y1, y)|)
=
0 ;
then
|(y1, ((1
/
|.y.|)
* y))|
=
0 by
A4,
A18,
RVSUM_1: 121;
hence contradiction by
A16,
EUCLID_2: 8;
end;
B0
c= B1 by
XBOOLE_1: 7;
hence contradiction by
A1,
A19,
A3,
A14,
A17,
Def6;
end;
hence y
= (
0* n);
end;
begin
definition
let n;
let X be
Subset of (
REAL n);
::
EUCLID_7:def8
attr X is
linear_manifold means for x,y be
Element of (
REAL n), a,b be
Element of
REAL st x
in X & y
in X holds ((a
* x)
+ (b
* y))
in X;
end
registration
let n;
cluster (
[#] (
REAL n)) ->
linear_manifold;
coherence
proof
let x,y be
Element of (
REAL n), a,b be
Element of
REAL ;
assume that x
in (
[#] (
REAL n)) and y
in (
[#] (
REAL n));
((a
* x)
+ (b
* y))
in (
REAL n);
hence thesis by
SUBSET_1:def 3;
end;
end
theorem ::
EUCLID_7:20
Th19:
{(
0* n)} is
linear_manifold
proof
let x,y be
Element of (
REAL n), a,b be
Element of
REAL ;
assume that
A1: x
in
{(
0* n)} and
A2: y
in
{(
0* n)};
reconsider nn = n as
Element of
NAT by
ORDINAL1:def 12;
A3: y
= (
0* n) by
A2,
TARSKI:def 1;
x
= (
0* n) by
A1,
TARSKI:def 1;
then ((a
* x)
+ (b
* y))
= ((
0* nn)
+ (b
* (
0* nn))) by
A3,
EUCLID_4: 2
.= ((
0* nn)
+ (
0* nn)) by
EUCLID_4: 2
.= (
0* nn) by
EUCLID_4: 1;
hence ((a
* x)
+ (b
* y))
in
{(
0* n)} by
TARSKI:def 1;
end;
registration
let n;
cluster
{(
0* n)} ->
linear_manifold;
coherence by
Th19;
end
definition
let n;
let X be
Subset of (
REAL n);
::
EUCLID_7:def9
func
L_Span X ->
Subset of (
REAL n) equals (
meet { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y });
correctness
proof
X
c= (
REAL n);
then X
c= (
[#] (
REAL n)) by
SUBSET_1:def 3;
then (
[#] (
REAL n))
in { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y };
hence thesis by
SETFAM_1: 7;
end;
end
registration
let n;
let X be
Subset of (
REAL n);
cluster (
L_Span X) ->
linear_manifold;
coherence
proof
X
c= (
REAL n);
then X
c= (
[#] (
REAL n)) by
SUBSET_1:def 3;
then (
[#] (
REAL n))
in { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y };
then
reconsider Z = { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y } as non
empty
set;
let x,y be
Element of (
REAL n), a,b be
Element of
REAL ;
assume that
A1: x
in (
L_Span X) and
A2: y
in (
L_Span X);
A3: for Y4 be
set st Y4
in { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y } holds ((a
* x)
+ (b
* y))
in Y4
proof
let Y4 be
set;
assume
A4: Y4
in { Y where Y be
Subset of (
REAL n) : Y is
linear_manifold & X
c= Y };
then
A5: ex Y0 be
Subset of (
REAL n) st Y4
= Y0 & Y0 is
linear_manifold & X
c= Y0;
A6: y
in Y4 by
A2,
A4,
SETFAM_1:def 1;
x
in Y4 by
A1,
A4,
SETFAM_1:def 1;
hence ((a
* x)
+ (b
* y))
in Y4 by
A6,
A5;
end;
Z
<>
{} ;
hence ((a
* x)
+ (b
* y))
in (
L_Span X) by
A3,
SETFAM_1:def 1;
end;
end
definition
let n be
Nat, f be
FinSequence of (
REAL n);
::
EUCLID_7:def10
func
accum f ->
FinSequence of (
REAL n) means
:
Def10: (
len f)
= (
len it ) & (f
. 1)
= (it
. 1) & for i be
Nat st 1
<= i & i
< (
len f) holds (it
. (i
+ 1))
= ((it
/. i)
+ (f
/. (i
+ 1)));
existence
proof
per cases ;
suppose
A1: (
len f)
>
0 ;
reconsider q =
<*(f
/. 1)*> as
FinSequence of (
REAL n);
A2: (
0
+ 1)
<= (
len f) by
A1,
NAT_1: 13;
then (f
/. 1)
= (f
. 1) by
FINSEQ_4: 15;
then
A3: (q
. 1)
= (f
. 1) by
FINSEQ_1: 40;
defpred
P[
Nat] means ($1
+ 1)
<= (
len f) implies ex g be
FinSequence of (
REAL n) st ($1
+ 1)
= (
len g) & (f
. 1)
= (g
. 1) & (for i be
Nat st 1
<= i & i
< ($1
+ 1) holds (g
. (i
+ 1))
= ((g
/. i)
+ (f
/. (i
+ 1))));
A4: for i be
Nat st 1
<= i & i
< (
0
+ 1) holds (q
. (i
+ 1))
= ((q
/. i)
+ (f
/. (i
+ 1)));
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
now
per cases ;
case
A7: ((k
+ 1)
+ 1)
<= (
len f);
(k
+ 1)
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then
consider g be
FinSequence of (
REAL n) such that
A8: (k
+ 1)
= (
len g) and
A9: (f
. 1)
= (g
. 1) and
A10: for i be
Nat st 1
<= i & i
< (k
+ 1) holds (g
. (i
+ 1))
= ((g
/. i)
+ (f
/. (i
+ 1))) by
A6,
A7,
XXREAL_0: 2;
reconsider g2 = (g
^
<*((g
/. (k
+ 1))
+ (f
/. ((k
+ 1)
+ 1)))*>) as
FinSequence of (
REAL n);
A11: (
Seg (
len g))
= (
dom g) by
FINSEQ_1:def 3;
A12: (
len g2)
= ((
len g)
+ (
len
<*((g
/. (k
+ 1))
+ (f
/. ((k
+ 1)
+ 1)))*>)) by
FINSEQ_1: 22
.= ((k
+ 1)
+ 1) by
A8,
FINSEQ_1: 40;
A13: for i be
Nat st 1
<= i & i
< ((k
+ 1)
+ 1) holds (g2
. (i
+ 1))
= ((g2
/. i)
+ (f
/. (i
+ 1)))
proof
let i be
Nat;
assume that
A14: 1
<= i and
A15: i
< ((k
+ 1)
+ 1);
A16: i
<= (k
+ 1) by
A15,
NAT_1: 13;
per cases by
A16,
XXREAL_0: 1;
suppose
A17: i
< (k
+ 1);
A18: 1
<= (i
+ 1) by
NAT_1: 12;
(i
+ 1)
<= (k
+ 1) by
A17,
NAT_1: 13;
then (i
+ 1)
in (
Seg (
len g)) by
A8,
A18,
FINSEQ_1: 1;
then
A19: (g2
. (i
+ 1))
= (g
. (i
+ 1)) by
A11,
FINSEQ_1:def 7;
i
in (
Seg (
len g)) by
A8,
A14,
A16,
FINSEQ_1: 1;
then
A20: (g2
. i)
= (g
. i) by
A11,
FINSEQ_1:def 7;
A21: (g
/. i)
= (g
. i) by
A8,
A14,
A17,
FINSEQ_4: 15;
(g2
/. i)
= (g2
. i) by
A12,
A14,
A15,
FINSEQ_4: 15;
hence thesis by
A10,
A14,
A17,
A19,
A20,
A21;
end;
suppose
A22: i
= (k
+ 1);
A23: (g2
/. i)
= (g2
. i) by
A12,
A14,
A15,
FINSEQ_4: 15;
i
in (
Seg (
len g)) by
A8,
A14,
A16,
FINSEQ_1: 1;
then
A24: (g
. i)
= (g2
. i) by
A11,
FINSEQ_1:def 7;
(g
/. i)
= (g
. i) by
A8,
A14,
A16,
FINSEQ_4: 15;
hence thesis by
A8,
A22,
A24,
A23,
FINSEQ_1: 42;
end;
end;
1
<= (k
+ 1) by
NAT_1: 11;
then 1
in (
Seg (
len g)) by
A8,
FINSEQ_1: 1;
then (g2
. 1)
= (f
. 1) by
A9,
A11,
FINSEQ_1:def 7;
hence
P[(k
+ 1)] by
A12,
A13;
end;
case ((k
+ 1)
+ 1)
> (
len f);
hence
P[(k
+ 1)];
end;
end;
hence
P[(k
+ 1)];
end;
((
len f)
-' 1)
= ((
len f)
- 1) by
A2,
XREAL_1: 233;
then
A25: (((
len f)
-' 1)
+ 1)
= (
len f);
(
len q)
= 1 by
FINSEQ_1: 40;
then
A26:
P[
0 ] by
A3,
A4;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A26,
A5);
hence thesis by
A25;
end;
suppose
A27: (
len f)
<=
0 ;
take f;
thus (
len f)
= (
len f) & (f
. 1)
= (f
. 1);
let i be
Nat;
thus thesis by
A27;
end;
end;
uniqueness
proof
let g1,g2 be
FinSequence of (
REAL n);
assume that
A28: (
len f)
= (
len g1) and
A29: (f
. 1)
= (g1
. 1) and
A30: for i be
Nat st 1
<= i & i
< (
len f) holds (g1
. (i
+ 1))
= ((g1
/. i)
+ (f
/. (i
+ 1)));
defpred
P[
Nat] means 1
<= $1 & $1
<= (
len f) implies (g1
. $1)
= (g2
. $1);
assume that
A31: (
len f)
= (
len g2) and
A32: (f
. 1)
= (g2
. 1) and
A33: for i be
Nat st 1
<= i & i
< (
len f) holds (g2
. (i
+ 1))
= ((g2
/. i)
+ (f
/. (i
+ 1)));
A34: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A35:
P[k];
1
<= (k
+ 1) & (k
+ 1)
<= (
len f) implies (g1
. (k
+ 1))
= (g2
. (k
+ 1))
proof
assume that 1
<= (k
+ 1) and
A36: (k
+ 1)
<= (
len f);
A37: k
< (k
+ 1) by
XREAL_1: 29;
then
A38: k
< (
len f) by
A36,
XXREAL_0: 2;
per cases ;
suppose
A39: 1
<= k;
then
A40: (g2
. (k
+ 1))
= ((g2
/. k)
+ (f
/. (k
+ 1))) by
A33,
A38;
A41: k
<= (
len g2) by
A31,
A36,
A37,
XXREAL_0: 2;
A42: (g1
/. k)
= (g1
. k) by
A28,
A38,
A39,
FINSEQ_4: 15;
(g1
. (k
+ 1))
= ((g1
/. k)
+ (f
/. (k
+ 1))) by
A30,
A38,
A39;
hence thesis by
A35,
A36,
A37,
A39,
A40,
A42,
A41,
FINSEQ_4: 15,
XXREAL_0: 2;
end;
suppose 1
> k;
then (
0
+ 1)
> k;
then k
=
0 by
NAT_1: 13;
hence (g1
. (k
+ 1))
= (g2
. (k
+ 1)) by
A29,
A32;
end;
end;
hence
P[(k
+ 1)];
end;
A43:
P[
0 ];
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A43,
A34);
hence g1
= g2 by
A28,
A31,
FINSEQ_1: 14;
end;
end
definition
let n be
Nat;
let f be
FinSequence of (
REAL n);
::
EUCLID_7:def11
func
Sum f ->
Element of (
REAL n) equals
:
Def11: ((
accum f)
. (
len f)) if (
len f)
>
0
otherwise (
0* n);
coherence
proof
A1: (
len f)
= (
len (
accum f)) by
Def10;
now
per cases ;
case (
len f)
>
0 ;
then (
0
+ 1)
<= (
len f) by
NAT_1: 13;
then (
len f)
in (
dom (
accum f)) by
A1,
FINSEQ_3: 25;
then ((
accum f)
. (
len f))
in (
rng (
accum f)) by
FUNCT_1:def 3;
hence ((
accum f)
. (
len f)) is
Element of (
REAL n);
end;
case (
len f)
<=
0 ;
hence thesis;
end;
end;
hence thesis;
end;
consistency ;
end
theorem ::
EUCLID_7:21
Th20: for F,F2 be
FinSequence of (
REAL n), h be
Permutation of (
dom F) st F2
= (F
(*) h) holds (
Sum F2)
= (
Sum F)
proof
let F,F2 be
FinSequence of (
REAL n), h be
Permutation of (
dom F);
assume
A1: F2
= (F
(*) h);
per cases ;
suppose
A2: (
len F)
>
0 ;
then
A3: (
0
+ 1)
<= (
len F) by
NAT_1: 13;
then
A4: 1
in (
Seg (
len F)) by
FINSEQ_1: 1;
then
A5: 1
in (
dom F) by
FINSEQ_1:def 3;
then
A6: (
dom h)
= (
dom F) by
FUNCT_2:def 1;
then (
rng h)
= (
dom h) by
FUNCT_2:def 3;
then
A7: (
dom F2)
= (
dom h) by
A1,
RELAT_1: 27;
then
A8: (
Seg (
len F2))
= (
dom F) by
A6,
FINSEQ_1:def 3;
set gF = (
accum F);
A9: (
len F)
= (
len gF) by
Def10;
A10: for i be
Nat st 1
<= i & i
< (
len F) holds (gF
. (i
+ 1))
= ((gF
/. i)
+ (F
/. (i
+ 1))) by
Def10;
defpred
P[
Nat] means for h2 be
Permutation of (
dom F) st $1
< (
len F) & (for i be
Nat st ($1
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i) holds (gF
. ($1
+ 1))
= ((
accum (F
(*) h2))
. ($1
+ 1));
A11: (
dom F)
= (
Seg (
len F)) by
FINSEQ_1:def 3;
1
in (
Seg (
len F2)) by
A5,
A6,
A7,
FINSEQ_1:def 3;
then
A12: 1
<= (
len F2) by
FINSEQ_1: 1;
A13: ((
len F)
-' 1)
= ((
len F)
- 1) by
A3,
XREAL_1: 233;
then
A14: for i be
Nat st (((
len F)
-' 1)
+ 1)
< i & i
<= (
len F) holds (h
. i)
= i;
A15: (
dom F)
<>
{} by
A4,
FINSEQ_1:def 3;
A16: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A17:
P[k];
for h2 be
Permutation of (
dom F) st (k
+ 1)
< (
len F) & (for i be
Nat st ((k
+ 1)
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i) holds (gF
. ((k
+ 1)
+ 1))
= ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1))
proof
A18: 1
<= (k
+ 1) by
NAT_1: 11;
A19: (k
+ 1)
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
let h2 be
Permutation of (
dom F);
assume that
A20: (k
+ 1)
< (
len F) and
A21: for i be
Nat st ((k
+ 1)
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i;
A22: ((k
+ 1)
+ 1)
<= (
len F) by
A20,
NAT_1: 13;
k
< (k
+ 1) by
XREAL_1: 29;
then
A23: k
< (
len F) by
A20,
XXREAL_0: 2;
A24: 1
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then
A25: ((k
+ 1)
+ 1)
in (
Seg (
len F)) by
A22,
FINSEQ_1: 1;
then
A26: ((k
+ 1)
+ 1)
in (
dom F) by
FINSEQ_1:def 3;
A27: (
dom h2)
= (
dom F) by
A5,
FUNCT_2:def 1;
then
A28: (
rng h2)
= (
dom h2) by
FUNCT_2:def 3;
then
A29: (
dom (F
(*) h2))
= (
dom h2) by
RELAT_1: 27
.= (
dom F) by
A5,
FUNCT_2:def 1;
then
A30: (
Seg (
len (F
(*) h2)))
= (
dom F) by
FINSEQ_1:def 3;
then
A31: (
Seg (
len (F
(*) h2)))
= (
Seg (
len F)) by
FINSEQ_1:def 3;
A32: (
len (F
(*) h2))
= (
len F) by
A30,
FINSEQ_1:def 3;
h2 is
FinSequence by
A11,
A27,
FINSEQ_1:def 2;
then
reconsider h2r = h2 as
FinSequence of
NAT by
A27,
A28,
FINSEQ_1:def 4;
(
dom h2)
= (
Seg (
len F)) by
A2,
A11,
FUNCT_2:def 1;
then
A33: (
len h2r)
= (
len F) by
FINSEQ_1:def 3;
consider x be
object such that
A34: x
in (
dom h2) and
A35: ((k
+ 1)
+ 1)
= (h2
. x) by
A11,
A27,
A28,
A25,
FUNCT_1:def 3;
reconsider nx = x as
Element of
NAT by
A27,
A34;
A36: nx
<= (
len F) by
A11,
A34,
FINSEQ_1: 1;
reconsider h2b = (
Swap (h2r,nx,((k
+ 1)
+ 1))) as
FinSequence of
NAT ;
A37: (
rng h2b)
= (
rng h2) by
FINSEQ_7: 22
.= (
dom F) by
FUNCT_2:def 3;
A38: (
len h2b)
= (
len h2r) by
FINSEQ_7: 18;
then (
dom h2b)
= (
Seg (
len h2r)) by
FINSEQ_1:def 3;
then
A39: (
dom h2b)
= (
dom h2) by
FINSEQ_1:def 3
.= (
dom F) by
A15,
FUNCT_2:def 1;
then
reconsider h2d = h2b as
Function of (
dom F), (
dom F) by
A37,
FUNCT_2: 1;
A40: h2d is
one-to-one by
INT_5: 39;
h2d is
onto by
A37,
FUNCT_2:def 3;
then
reconsider h4 = h2d as
Permutation of (
dom F) by
A40;
A41: 1
<= nx by
A11,
A34,
FINSEQ_1: 1;
then
A42: (h4
. nx)
= (h2r
. ((k
+ 1)
+ 1)) by
A22,
A24,
A36,
A33,
Th9;
A43: (
dom h4)
= (
dom F) by
A5,
FUNCT_2:def 1;
(
rng h4)
c= (
dom F);
then
A44: (
dom (F
(*) h4))
= (
dom h4) by
RELAT_1: 27
.= (
dom F) by
A5,
FUNCT_2:def 1;
then (
Seg (
len (F
(*) h4)))
= (
dom F) by
FINSEQ_1:def 3;
then
A45: (
len (F
(*) h4))
= (
len F) by
FINSEQ_1:def 3;
per cases ;
suppose
A46: nx
<= (k
+ 1);
A47: ((k
+ 1)
+ 1)
<= (
len h2r) by
A20,
A33,
NAT_1: 13;
A48: nx
<= (
len h2r) by
A11,
A34,
A33,
FINSEQ_1: 1;
A49: (F
. ((k
+ 1)
+ 1))
= ((F
(*) h2)
. nx) by
A34,
A35,
FUNCT_1: 13;
A50: (
len (
accum (F
(*) h4)))
= (
len (F
(*) h4)) by
Def10;
A51: 1
<= nx by
A11,
A34,
FINSEQ_1: 1;
A52: 1
<= ((k
+ 1)
+ 1) by
NAT_1: 11;
then
A53: (h2b
. ((k
+ 1)
+ 1))
= ((k
+ 1)
+ 1) by
A35,
A51,
A48,
A47,
Th9;
A54: for i be
Nat st (k
+ 1)
< i & i
<= (
len F) holds (h4
. i)
= i
proof
let i be
Nat;
assume that
A55: (k
+ 1)
< i and
A56: i
<= (
len F);
A57: ((k
+ 1)
+ 1)
<= i by
A55,
NAT_1: 13;
per cases ;
suppose
A58: ((k
+ 1)
+ 1)
< i;
1
<= (k
+ 1) by
NAT_1: 11;
then
A59: 1
< i by
A55,
XXREAL_0: 2;
then (h4
. i)
= (h2b
/. i) by
A33,
A38,
A56,
FINSEQ_4: 15
.= (h2r
/. i) by
A33,
A46,
A55,
A56,
A58,
A59,
FINSEQ_7: 30
.= (h2
. i) by
A33,
A56,
A59,
FINSEQ_4: 15
.= i by
A21,
A56,
A58;
hence (h4
. i)
= i;
end;
suppose ((k
+ 1)
+ 1)
>= i;
then i
= ((k
+ 1)
+ 1) by
A57,
XXREAL_0: 1;
hence (h4
. i)
= i by
A35,
A51,
A48,
A52,
A47,
Th9;
end;
end;
A60: (F
/. ((k
+ 1)
+ 1))
= (F
. ((k
+ 1)
+ 1)) by
A22,
A24,
FINSEQ_4: 15;
A61: 1
<= (k
+ 1) by
NAT_1: 11;
A62: (
len (
accum (F
(*) h2)))
= (
len (F
(*) h2)) by
Def10;
A63: nx
< ((k
+ 1)
+ 1) by
A19,
A46,
XXREAL_0: 2;
A64: for i4 be
Nat st 1
<= i4 & i4
< nx holds ((
accum (F
(*) h2))
. i4)
= ((
accum (F
(*) h4))
. i4)
proof
defpred
T[
Nat] means (1
+ $1)
< nx implies ((
accum (F
(*) h2))
. (1
+ $1))
= ((
accum (F
(*) h4))
. (1
+ $1));
let i4 be
Nat;
assume that
A65: 1
<= i4 and
A66: i4
< nx;
A67: (1
+ (i4
-' 1))
= ((i4
- 1)
+ 1) by
A65,
XREAL_1: 233
.= i4;
A68: for k3 be
Nat st
T[k3] holds
T[(k3
+ 1)]
proof
let k3 be
Nat;
A69: 1
<= (k3
+ 1) by
NAT_1: 11;
assume
A70:
T[k3];
(1
+ (k3
+ 1))
< nx implies ((
accum (F
(*) h2))
. (1
+ (k3
+ 1)))
= ((
accum (F
(*) h4))
. (1
+ (k3
+ 1)))
proof
assume
A71: (1
+ (k3
+ 1))
< nx;
then
A72: (1
+ (k3
+ 1))
< (
len F) by
A36,
XXREAL_0: 2;
then
A73: (h2r
/. ((k3
+ 1)
+ 1))
= (h2r
. ((k3
+ 1)
+ 1)) by
A33,
FINSEQ_4: 15,
NAT_1: 11;
A74: ((k3
+ 1)
+ 1)
< ((k
+ 1)
+ 1) by
A63,
A71,
XXREAL_0: 2;
A75: (h2b
/. ((k3
+ 1)
+ 1))
= (h2b
. ((k3
+ 1)
+ 1)) by
A33,
A38,
A72,
FINSEQ_4: 15,
NAT_1: 11;
1
<= (1
+ (k3
+ 1)) by
NAT_1: 11;
then ((k3
+ 1)
+ 1)
in (
Seg (
len F)) by
A72,
FINSEQ_1: 1;
then
A76: ((k3
+ 1)
+ 1)
in (
dom (F
(*) h2)) by
A29,
FINSEQ_1:def 3;
then
A77: ((F
(*) h2)
. ((k3
+ 1)
+ 1))
= (F
. (h2
. ((k3
+ 1)
+ 1))) by
FUNCT_1: 12
.= (F
. (h4
. ((k3
+ 1)
+ 1))) by
A33,
A71,
A72,
A74,
A73,
A75,
FINSEQ_7: 30,
NAT_1: 11
.= ((F
(*) h4)
. ((k3
+ 1)
+ 1)) by
A44,
A29,
A76,
FUNCT_1: 12;
A78: ((F
(*) h4)
/. ((k3
+ 1)
+ 1))
= ((F
(*) h4)
. ((k3
+ 1)
+ 1)) by
A45,
A72,
FINSEQ_4: 15,
NAT_1: 11;
A79: (k3
+ 1)
< (1
+ (k3
+ 1)) by
XREAL_1: 29;
then
A80: (k3
+ 1)
< nx by
A71,
XXREAL_0: 2;
then
A81: (k3
+ 1)
< (
len F) by
A36,
XXREAL_0: 2;
(k3
+ 1)
< (
len (F
(*) h2)) by
A36,
A32,
A80,
XXREAL_0: 2;
then
A82: ((
accum (F
(*) h4))
/. (1
+ k3))
= ((
accum (F
(*) h4))
. (1
+ k3)) by
A45,
A32,
A50,
FINSEQ_4: 15,
NAT_1: 11;
((
accum (F
(*) h2))
/. (1
+ k3))
= ((
accum (F
(*) h2))
. (1
+ k3)) by
A32,
A62,
A81,
FINSEQ_4: 15,
NAT_1: 11;
then ((
accum (F
(*) h2))
. (1
+ (k3
+ 1)))
= (((
accum (F
(*) h4))
/. (k3
+ 1))
+ ((F
(*) h2)
/. ((k3
+ 1)
+ 1))) by
A32,
A70,
A69,
A71,
A79,
A81,
A82,
Def10,
XXREAL_0: 2
.= (((
accum (F
(*) h4))
/. (k3
+ 1))
+ ((F
(*) h4)
/. ((k3
+ 1)
+ 1))) by
A32,
A72,
A77,
A78,
FINSEQ_4: 15,
NAT_1: 11;
hence ((
accum (F
(*) h2))
. (1
+ (k3
+ 1)))
= ((
accum (F
(*) h4))
. (1
+ (k3
+ 1))) by
A45,
A69,
A81,
Def10;
end;
hence
T[(k3
+ 1)];
end;
0
< (k
+ 1);
then
A83: 1
<> ((k
+ 1)
+ 1);
A84: (h2b
/. 1)
= (h2b
. 1) by
A3,
A33,
A38,
FINSEQ_4: 15;
A85: (h2r
/. 1)
= (h2r
. 1) by
A3,
A33,
FINSEQ_4: 15;
((F
(*) h2)
. 1)
= (F
. (h2
. 1)) by
A5,
A29,
FUNCT_1: 12
.= (F
. (h4
. 1)) by
A3,
A33,
A65,
A66,
A83,
A85,
A84,
FINSEQ_7: 30
.= ((F
(*) h4)
. 1) by
A5,
A44,
FUNCT_1: 12;
then ((
accum (F
(*) h2))
. (1
+
0 ))
= ((F
(*) h4)
. 1) by
Def10
.= ((
accum (F
(*) h4))
. (1
+
0 )) by
Def10;
then
A86:
T[
0 ];
for k3 be
Nat holds
T[k3] from
NAT_1:sch 2(
A86,
A68);
hence ((
accum (F
(*) h2))
. i4)
= ((
accum (F
(*) h4))
. i4) by
A66,
A67;
end;
A87: for i be
Nat st nx
<= i & i
< ((k
+ 1)
+ 1) holds ((
accum (F
(*) h2))
. i)
= ((((
accum (F
(*) h4))
/. i)
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
proof
defpred
S[
Nat] means nx
<= (nx
+ $1) & (nx
+ $1)
< ((k
+ 1)
+ 1) implies ((
accum (F
(*) h2))
. (nx
+ $1))
= ((((
accum (F
(*) h4))
/. (nx
+ $1))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx));
let i be
Nat;
assume that
A88: nx
<= i and
A89: i
< ((k
+ 1)
+ 1);
A90: (
len (
accum (F
(*) h4)))
= (
len (F
(*) h4)) by
Def10;
A91: (
len (
accum (F
(*) h2)))
= (
len (F
(*) h2)) by
Def10;
A92: for k3 be
Nat st
S[k3] holds
S[(k3
+ 1)]
proof
let k3 be
Nat;
assume
A93:
S[k3];
nx
<= (nx
+ (k3
+ 1)) & (nx
+ (k3
+ 1))
< ((k
+ 1)
+ 1) implies ((
accum (F
(*) h2))
. (nx
+ (k3
+ 1)))
= ((((
accum (F
(*) h4))
/. (nx
+ (k3
+ 1)))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
proof
reconsider f3 = ((F
(*) h2)
/. ((nx
+ k3)
+ 1)) as
Element of (
REAL n);
A94: nx
<= (nx
+ k3) by
NAT_1: 11;
reconsider f2 = ((F
(*) h2)
/. ((k
+ 1)
+ 1)) as
Element of (
REAL n);
A95: (nx
+ k3)
< ((nx
+ k3)
+ 1) by
XREAL_1: 29;
reconsider f4 = ((F
(*) h2)
/. nx) as
Element of (
REAL n);
reconsider f1 = ((
accum (F
(*) h4))
/. (nx
+ k3)) as
Element of (
REAL n);
assume that nx
<= (nx
+ (k3
+ 1)) and
A96: (nx
+ (k3
+ 1))
< ((k
+ 1)
+ 1);
A97: (nx
+ (k3
+ 1))
< (
len (
accum (F
(*) h4))) by
A22,
A45,
A90,
A96,
XXREAL_0: 2;
A98: (nx
+ k3)
< ((nx
+ k3)
+ 1) by
XREAL_1: 29;
then (nx
+ k3)
< ((k
+ 1)
+ 1) by
A96,
XXREAL_0: 2;
then
A99: (nx
+ k3)
< (
len (F
(*) h2)) by
A33,
A32,
A47,
XXREAL_0: 2;
then
A100: ((nx
+ k3)
+ 1)
<= (
len (F
(*) h2)) by
NAT_1: 13;
(nx
+ k3)
>= (
0
+ 1) by
A41,
NAT_1: 13;
then
A101: 1
< (nx
+ (k3
+ 1)) by
A98,
XXREAL_0: 2;
then
A102: ((F
(*) h4)
/. ((nx
+ k3)
+ 1))
= ((F
(*) h4)
. ((nx
+ k3)
+ 1)) by
A45,
A32,
A100,
FINSEQ_4: 15;
((nx
+ k3)
+ 1)
in (
Seg (
len (F
(*) h2))) by
A101,
A100,
FINSEQ_1: 1;
then
A103: ((nx
+ k3)
+ 1)
in (
dom (F
(*) h2)) by
FINSEQ_1:def 3;
then
A104: ((F
(*) h4)
. ((nx
+ k3)
+ 1))
= (F
. (h4
. ((nx
+ k3)
+ 1))) by
A44,
A29,
FUNCT_1: 12
.= (F
. (h2b
/. ((nx
+ k3)
+ 1))) by
A33,
A38,
A32,
A101,
A100,
FINSEQ_4: 15
.= (F
. (h2r
/. ((nx
+ k3)
+ 1))) by
A33,
A32,
A96,
A101,
A100,
A95,
A94,
FINSEQ_7: 30
.= (F
. (h2r
. ((nx
+ k3)
+ 1))) by
A33,
A32,
A101,
A100,
FINSEQ_4: 15
.= ((F
(*) h2)
. ((nx
+ k3)
+ 1)) by
A103,
FUNCT_1: 12;
nx
<= (nx
+ k3) by
NAT_1: 11;
then
A105: 1
<= (nx
+ k3) by
A51,
XXREAL_0: 2;
then ((
accum (F
(*) h4))
. (nx
+ (k3
+ 1)))
= (((
accum (F
(*) h4))
/. (nx
+ k3))
+ ((F
(*) h4)
/. ((nx
+ k3)
+ 1))) by
A45,
A32,
A99,
Def10
.= (((
accum (F
(*) h4))
/. (nx
+ k3))
+ ((F
(*) h2)
/. ((nx
+ k3)
+ 1))) by
A101,
A100,
A104,
A102,
FINSEQ_4: 15;
then
A106: ((((
accum (F
(*) h4))
/. (nx
+ (k3
+ 1)))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
= (((f1
+ f3)
- f2)
+ f4) by
A101,
A97,
FINSEQ_4: 15
.= (((f1
- f2)
+ f3)
+ f4) by
RFUNCT_1: 8
.= (((f1
- f2)
+ f4)
+ f3) by
RFUNCT_1: 8;
((
accum (F
(*) h2))
. (nx
+ (k3
+ 1)))
= (((
accum (F
(*) h2))
/. (nx
+ k3))
+ ((F
(*) h2)
/. ((nx
+ k3)
+ 1))) by
A105,
A99,
Def10
.= (((((
accum (F
(*) h4))
/. (nx
+ k3))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
+ ((F
(*) h2)
/. ((nx
+ k3)
+ 1))) by
A91,
A93,
A96,
A98,
A105,
A99,
FINSEQ_4: 15,
NAT_1: 11,
XXREAL_0: 2;
hence ((
accum (F
(*) h2))
. (nx
+ (k3
+ 1)))
= ((((
accum (F
(*) h4))
/. (nx
+ (k3
+ 1)))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx)) by
A106;
end;
hence
S[(k3
+ 1)];
end;
now
per cases by
A41,
XXREAL_0: 1;
case
A107: 1
= nx;
((
accum (F
(*) h4))
/. (nx
+
0 ))
= ((
accum (F
(*) h4))
. 1) by
A3,
A45,
A90,
A107,
FINSEQ_4: 15
.= ((F
(*) h4)
. 1) by
Def10
.= (F
. (h2
. ((k
+ 1)
+ 1))) by
A5,
A43,
A42,
A107,
FUNCT_1: 13
.= ((F
(*) h2)
. ((k
+ 1)
+ 1)) by
A27,
A26,
FUNCT_1: 13
.= ((F
(*) h2)
/. ((k
+ 1)
+ 1)) by
A22,
A24,
A32,
FINSEQ_4: 15;
then
A108: ((((
accum (F
(*) h4))
/. (nx
+
0 ))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
= ((
0* n)
+ ((F
(*) h2)
/. nx)) by
EUCLIDLP: 2
.= ((F
(*) h2)
/. nx) by
EUCLID_4: 1;
((
accum (F
(*) h2))
. (nx
+
0 ))
= ((F
(*) h2)
. 1) by
A107,
Def10
.= ((F
(*) h2)
/. nx) by
A36,
A32,
A107,
FINSEQ_4: 15;
hence ((
accum (F
(*) h2))
. (nx
+
0 ))
= ((((
accum (F
(*) h4))
/. (nx
+
0 ))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx)) by
A108;
end;
case
A109: 1
< nx;
A110: ((F
(*) h4)
/. nx)
= ((F
(*) h4)
. nx) by
A41,
A36,
A45,
FINSEQ_4: 15
.= (F
. (h2
. ((k
+ 1)
+ 1))) by
A34,
A43,
A42,
FUNCT_1: 13
.= ((F
(*) h2)
. ((k
+ 1)
+ 1)) by
A27,
A26,
FUNCT_1: 13
.= ((F
(*) h2)
/. ((k
+ 1)
+ 1)) by
A22,
A24,
A32,
FINSEQ_4: 15;
A111: (nx
-' 1)
= (nx
- 1) by
A109,
XREAL_1: 233;
then (nx
-' 1)
>
0 by
A109,
XREAL_1: 50;
then
A112: (nx
-' 1)
>= (
0
+ 1) by
NAT_1: 13;
nx
< (nx
+ 1) by
XREAL_1: 29;
then
A113: (nx
- 1)
< ((nx
+ 1)
- 1) by
XREAL_1: 9;
then
A114: (nx
-' 1)
< (
len (F
(*) h2)) by
A33,
A32,
A48,
A111,
XXREAL_0: 2;
then
A115: ((
accum (F
(*) h2))
/. (nx
-' 1))
= ((
accum (F
(*) h2))
. (nx
-' 1)) by
A91,
A112,
FINSEQ_4: 15;
((
accum (F
(*) h4))
. (nx
+
0 ))
= (((
accum (F
(*) h4))
/. (nx
-' 1))
+ ((F
(*) h4)
/. ((nx
-' 1)
+ 1))) by
A45,
A32,
A111,
A112,
A114,
Def10
.= (((
accum (F
(*) h4))
/. (nx
-' 1))
+ ((F
(*) h2)
/. ((k
+ 1)
+ 1))) by
A111,
A110;
then
A116: (((
accum (F
(*) h4))
/. (nx
+
0 ))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
= ((((
accum (F
(*) h4))
/. (nx
-' 1))
+ ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1))) by
A33,
A45,
A51,
A48,
A90,
FINSEQ_4: 15
.= ((
accum (F
(*) h4))
/. (nx
-' 1)) by
RVSUM_1: 42;
A117: ((
accum (F
(*) h4))
/. (nx
-' 1))
= ((
accum (F
(*) h4))
. (nx
-' 1)) by
A45,
A32,
A90,
A112,
A114,
FINSEQ_4: 15;
((
accum (F
(*) h2))
. (nx
+
0 ))
= (((
accum (F
(*) h2))
/. (nx
-' 1))
+ ((F
(*) h2)
/. ((nx
-' 1)
+ 1))) by
A111,
A112,
A114,
Def10
.= (((
accum (F
(*) h4))
/. (nx
-' 1))
+ ((F
(*) h2)
/. nx)) by
A64,
A111,
A112,
A113,
A115,
A117;
hence ((
accum (F
(*) h2))
. (nx
+
0 ))
= ((((
accum (F
(*) h4))
/. (nx
+
0 ))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx)) by
A116;
end;
end;
then
A118:
S[
0 ];
A119: for k3 be
Nat holds
S[k3] from
NAT_1:sch 2(
A118,
A92);
A120: (i
-' nx)
= (i
- nx) by
A88,
XREAL_1: 233;
then nx
<= (nx
+ (i
-' nx)) by
A88;
hence ((
accum (F
(*) h2))
. i)
= ((((
accum (F
(*) h4))
/. i)
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx)) by
A89,
A119,
A120;
end;
A121: ((F
(*) h2)
/. nx)
= ((F
(*) h2)
. nx) by
A41,
A36,
A32,
FINSEQ_4: 15;
A122: ((k
+ 1)
+ 1)
in (
dom h4) by
A25,
A39,
FINSEQ_1:def 3;
A123: ((F
(*) h4)
/. ((k
+ 1)
+ 1))
= ((F
(*) h4)
. ((k
+ 1)
+ 1)) by
A22,
A24,
A45,
FINSEQ_4: 15;
A124: for i be
Nat st ((k
+ 1)
+ 1)
<= i & i
<= (
len F) holds ((
accum (F
(*) h2))
. i)
= ((
accum (F
(*) h4))
. i)
proof
defpred
Y[
Nat] means (((k
+ 1)
+ 1)
+ $1)
<= (
len F) implies ((
accum (F
(*) h2))
. (((k
+ 1)
+ 1)
+ $1))
= ((
accum (F
(*) h4))
. (((k
+ 1)
+ 1)
+ $1));
let i be
Nat;
assume that
A125: ((k
+ 1)
+ 1)
<= i and
A126: i
<= (
len F);
A127: for k3 be
Nat st
Y[k3] holds
Y[(k3
+ 1)]
proof
let k3 be
Nat;
assume
A128:
Y[k3];
(((k
+ 1)
+ 1)
+ (k3
+ 1))
<= (
len F) implies ((
accum (F
(*) h2))
. (((k
+ 1)
+ 1)
+ (k3
+ 1)))
= ((
accum (F
(*) h4))
. (((k
+ 1)
+ 1)
+ (k3
+ 1)))
proof
A129: ((k
+ 1)
+ 1)
<= (((k
+ 1)
+ 1)
+ k3) by
NAT_1: 11;
A130: (((k
+ 1)
+ 1)
+ k3)
< ((((k
+ 1)
+ 1)
+ k3)
+ 1) by
XREAL_1: 29;
then
A131: ((k
+ 1)
+ 1)
< ((((k
+ 1)
+ 1)
+ k3)
+ 1) by
A129,
XXREAL_0: 2;
assume
A132: (((k
+ 1)
+ 1)
+ (k3
+ 1))
<= (
len F);
then
A133: ((F
(*) h4)
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))
= ((F
(*) h4)
. ((((k
+ 1)
+ 1)
+ k3)
+ 1)) by
A45,
FINSEQ_4: 15,
NAT_1: 11;
1
<= ((((k
+ 1)
+ 1)
+ k3)
+ 1) by
NAT_1: 11;
then ((((k
+ 1)
+ 1)
+ k3)
+ 1)
in (
Seg (
len (F
(*) h2))) by
A31,
A132,
FINSEQ_1: 1;
then
A134: ((((k
+ 1)
+ 1)
+ k3)
+ 1)
in (
dom (F
(*) h2)) by
FINSEQ_1:def 3;
then
A135: ((F
(*) h4)
. ((((k
+ 1)
+ 1)
+ k3)
+ 1))
= (F
. (h4
. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A44,
A29,
FUNCT_1: 12
.= (F
. (h2b
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A33,
A38,
A132,
FINSEQ_4: 15,
NAT_1: 11
.= (F
. (h2r
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A33,
A63,
A132,
A131,
FINSEQ_7: 30,
NAT_1: 11
.= (F
. (h2r
. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A33,
A132,
FINSEQ_4: 15,
NAT_1: 11
.= ((F
(*) h2)
. ((((k
+ 1)
+ 1)
+ k3)
+ 1)) by
A134,
FUNCT_1: 12;
1
<= ((k
+ 1)
+ 1) by
NAT_1: 11;
then
A136: 1
<= (((k
+ 1)
+ 1)
+ k3) by
A129,
XXREAL_0: 2;
A137: (((k
+ 1)
+ 1)
+ k3)
< (
len F) by
A132,
A130,
XXREAL_0: 2;
then
A138: ((
accum (F
(*) h4))
/. (((k
+ 1)
+ 1)
+ k3))
= ((
accum (F
(*) h4))
. (((k
+ 1)
+ 1)
+ k3)) by
A45,
A50,
A136,
FINSEQ_4: 15;
((
accum (F
(*) h2))
. (((k
+ 1)
+ 1)
+ (k3
+ 1)))
= (((
accum (F
(*) h2))
/. (((k
+ 1)
+ 1)
+ k3))
+ ((F
(*) h2)
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A32,
A137,
A136,
Def10
.= (((
accum (F
(*) h4))
/. (((k
+ 1)
+ 1)
+ k3))
+ ((F
(*) h2)
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A32,
A62,
A128,
A137,
A136,
A138,
FINSEQ_4: 15
.= (((
accum (F
(*) h4))
/. (((k
+ 1)
+ 1)
+ k3))
+ ((F
(*) h4)
/. ((((k
+ 1)
+ 1)
+ k3)
+ 1))) by
A32,
A132,
A135,
A133,
FINSEQ_4: 15,
NAT_1: 11;
hence ((
accum (F
(*) h2))
. (((k
+ 1)
+ 1)
+ (k3
+ 1)))
= ((
accum (F
(*) h4))
. (((k
+ 1)
+ 1)
+ (k3
+ 1))) by
A45,
A137,
A136,
Def10;
end;
hence
Y[(k3
+ 1)];
end;
A139: (
len (
accum (F
(*) h2)))
= (
len (F
(*) h2)) by
Def10;
A140: ((
accum (F
(*) h2))
. (k
+ 1))
= ((((
accum (F
(*) h4))
/. (k
+ 1))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx)) by
A19,
A46,
A87;
A141: 1
<= (k
+ 1) by
NAT_1: 11;
(k
+ 1)
< (
len (F
(*) h2)) by
A20,
A30,
FINSEQ_1:def 3;
then ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1))
= (((
accum (F
(*) h2))
/. (k
+ 1))
+ ((F
(*) h2)
/. ((k
+ 1)
+ 1))) by
A141,
Def10
.= (((((
accum (F
(*) h4))
/. (k
+ 1))
- ((F
(*) h2)
/. ((k
+ 1)
+ 1)))
+ ((F
(*) h2)
/. nx))
+ ((F
(*) h2)
/. ((k
+ 1)
+ 1))) by
A20,
A32,
A139,
A140,
FINSEQ_4: 15,
NAT_1: 11
.= (((
accum (F
(*) h4))
/. (k
+ 1))
+ ((F
(*) h2)
/. nx)) by
Th4
.= (((
accum (F
(*) h4))
/. (k
+ 1))
+ ((F
(*) h4)
/. ((k
+ 1)
+ 1))) by
A122,
A53,
A49,
A123,
A121,
FUNCT_1: 13;
then
A142:
Y[
0 ] by
A20,
A18,
A45,
Def10;
A143: for k3 be
Nat holds
Y[k3] from
NAT_1:sch 2(
A142,
A127);
(((k
+ 1)
+ 1)
+ (i
-' ((k
+ 1)
+ 1)))
= ((i
- ((k
+ 1)
+ 1))
+ ((k
+ 1)
+ 1)) by
A125,
XREAL_1: 233
.= i;
hence ((
accum (F
(*) h2))
. i)
= ((
accum (F
(*) h4))
. i) by
A126,
A143;
end;
A144: (gF
/. (k
+ 1))
= (gF
. (k
+ 1)) by
A9,
A20,
FINSEQ_4: 15,
NAT_1: 11;
A145: (
dom h4)
= (
dom F) by
A5,
FUNCT_2:def 1;
(
rng h4)
c= (
dom F);
then
A146: (
dom (F
(*) h4))
= (
dom h4) by
RELAT_1: 27;
then (
Seg (
len (F
(*) h4)))
= (
dom h4) by
FINSEQ_1:def 3;
then
A147: (k
+ 1)
< (
len (F
(*) h4)) by
A20,
A145,
FINSEQ_1:def 3;
(
Seg (
len (F
(*) h4)))
= (
dom F) by
A145,
A146,
FINSEQ_1:def 3;
then (
len (F
(*) h4))
= (
len F) by
FINSEQ_1:def 3;
then
A148: ((F
(*) h4)
/. ((k
+ 1)
+ 1))
= ((F
(*) h4)
. ((k
+ 1)
+ 1)) by
A22,
A24,
FINSEQ_4: 15;
(
len (
accum (F
(*) h4)))
= (
len (F
(*) h4)) by
Def10;
then
A149: ((
accum (F
(*) h4))
/. (k
+ 1))
= ((
accum (F
(*) h4))
. (k
+ 1)) by
A147,
FINSEQ_4: 15,
NAT_1: 11;
(gF
. ((k
+ 1)
+ 1))
= ((gF
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A10,
A20,
NAT_1: 11
.= (((
accum (F
(*) h4))
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A17,
A23,
A54,
A149,
A144
.= (((
accum (F
(*) h4))
/. (k
+ 1))
+ ((F
(*) h4)
/. ((k
+ 1)
+ 1))) by
A122,
A53,
A148,
A60,
FUNCT_1: 13
.= ((
accum (F
(*) h4))
. ((k
+ 1)
+ 1)) by
A61,
A147,
Def10;
hence (gF
. ((k
+ 1)
+ 1))
= ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1)) by
A22,
A124;
end;
suppose nx
> (k
+ 1);
then
A150: ((k
+ 1)
+ 1)
<= nx by
NAT_1: 13;
per cases by
A150,
XXREAL_0: 1;
suppose
A151: ((k
+ 1)
+ 1)
= nx;
A152: for i be
Nat st (k
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i
proof
let i be
Nat;
assume that
A153: (k
+ 1)
< i and
A154: i
<= (
len F);
A155: ((k
+ 1)
+ 1)
<= i by
A153,
NAT_1: 13;
per cases by
A155,
XXREAL_0: 1;
suppose ((k
+ 1)
+ 1)
< i;
hence (h2
. i)
= i by
A21,
A154;
end;
suppose ((k
+ 1)
+ 1)
= i;
hence (h2
. i)
= i by
A35,
A151;
end;
end;
A156: (k
+ 1)
< (
len (F
(*) h2)) by
A20,
A30,
FINSEQ_1:def 3;
(
len (
accum (F
(*) h2)))
= (
len (F
(*) h2)) by
Def10;
then
A157: ((
accum (F
(*) h2))
/. (k
+ 1))
= ((
accum (F
(*) h2))
. (k
+ 1)) by
A20,
A32,
FINSEQ_4: 15,
NAT_1: 11;
A158: 1
<= (k
+ 1) by
NAT_1: 11;
A159: (F
. ((k
+ 1)
+ 1))
= ((F
(*) h2)
. nx) by
A34,
A35,
FUNCT_1: 13;
A160: (gF
/. (k
+ 1))
= (gF
. (k
+ 1)) by
A9,
A20,
FINSEQ_4: 15,
NAT_1: 11;
A161: (F
/. ((k
+ 1)
+ 1))
= (F
. ((k
+ 1)
+ 1)) by
A22,
A24,
FINSEQ_4: 15;
(gF
. ((k
+ 1)
+ 1))
= ((gF
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A10,
A20,
NAT_1: 11
.= (((
accum (F
(*) h2))
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A17,
A23,
A152,
A157,
A160
.= (((
accum (F
(*) h2))
/. (k
+ 1))
+ ((F
(*) h2)
/. ((k
+ 1)
+ 1))) by
A22,
A24,
A32,
A151,
A161,
A159,
FINSEQ_4: 15
.= ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1)) by
A158,
A156,
Def10;
hence (gF
. ((k
+ 1)
+ 1))
= ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1));
end;
suppose ((k
+ 1)
+ 1)
< nx;
hence (gF
. ((k
+ 1)
+ 1))
= ((
accum (F
(*) h2))
. ((k
+ 1)
+ 1)) by
A21,
A35,
A36;
end;
end;
end;
hence
P[(k
+ 1)];
end;
for h2 be
Permutation of (
dom F) st for i be
Nat st (
0
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i holds (gF
. (
0
+ 1))
= ((
accum (F
(*) h2))
. (
0
+ 1))
proof
let h2 be
Permutation of (
dom F);
A162: (
rng h2)
= (
dom F) by
FUNCT_2:def 3;
A163: (
dom h2)
= (
dom F) by
A5,
FUNCT_2:def 1;
assume
A164: for i be
Nat st (
0
+ 1)
< i & i
<= (
len F) holds (h2
. i)
= i;
A165:
now
assume
A166: (h2
. 1)
<> 1;
consider x be
object such that
A167: x
in (
dom h2) and
A168: (h2
. x)
= 1 by
A5,
A162,
FUNCT_1:def 3;
reconsider nx = x as
Element of
NAT by
A163,
A167;
1
<= nx by
A11,
A167,
FINSEQ_1: 1;
then nx
= 1 or 1
< nx & nx
<= (
len F) by
A11,
A167,
FINSEQ_1: 1,
XXREAL_0: 1;
hence contradiction by
A164,
A166,
A168;
end;
((
accum (F
(*) h2))
. 1)
= ((F
(*) h2)
. 1) by
Def10
.= (F
. 1) by
A5,
A163,
A165,
FUNCT_1: 13;
hence (gF
. (
0
+ 1))
= ((
accum (F
(*) h2))
. (
0
+ 1)) by
Def10;
end;
then
A169:
P[
0 ];
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A169,
A16);
then (gF
. (
len F))
= ((
accum (F
(*) h))
. (
len F)) by
A13,
A14,
XREAL_1: 44
.= ((
accum F2)
. (
len F2)) by
A1,
A8,
FINSEQ_1:def 3
.= (
Sum F2) by
A12,
Def11;
hence (
Sum F2)
= (
Sum F) by
A2,
Def11;
end;
suppose
A170: (
len F)
<=
0 ;
(
rng h)
c= (
dom F);
then
A171: (
dom (F
(*) h))
= (
dom h) by
RELAT_1: 27;
(
Seg (
len F))
=
{} by
A170;
then (
dom F)
=
{} by
FINSEQ_1:def 3;
then (
Seg (
len F))
= (
dom F2) by
A1,
A171,
FINSEQ_1:def 3;
then
A172: (
len F)
= (
len F2) by
FINSEQ_1:def 3;
(
Sum F)
= (
0* n) by
A170,
Def11;
hence (
Sum F2)
= (
Sum F) by
A170,
A172,
Def11;
end;
end;
theorem ::
EUCLID_7:22
Th21: for k be
Element of
NAT holds (
Sum (k
|-> (
0* n)))
= (
0* n)
proof
let k be
Element of
NAT ;
set g = (k
|-> (
0* n));
A1: for i be
Nat st i
in (
dom g) holds (g
. i)
= (
0* n)
proof
let i be
Nat;
assume i
in (
dom g);
then i
in (
Seg k) by
FUNCOP_1: 13;
hence thesis by
FINSEQ_2: 57;
end;
per cases ;
suppose
A2: (
len g)
>
0 ;
set g3 = (
accum g);
A3: (
len g)
= (
len g3) by
Def10;
A4: (g
. 1)
= (g3
. 1) by
Def10;
defpred
P[
Nat] means $1
< (
len g) implies (g3
. ($1
+ 1))
= (
0* n);
A5: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A6:
P[k];
A7: k
< (k
+ 1) by
XREAL_1: 29;
per cases ;
suppose
A8: (k
+ 1)
< (
len g);
then
A9: ((k
+ 1)
+ 1)
<= (
len g) by
NAT_1: 13;
per cases ;
suppose
A10: 1
<= (k
+ 1);
A11: 1
<= ((k
+ 1)
+ 1) by
XREAL_1: 29;
then ((k
+ 1)
+ 1)
in (
Seg (
len g)) by
A9,
FINSEQ_1: 1;
then ((k
+ 1)
+ 1)
in (
dom g) by
FINSEQ_1:def 3;
then
A12: (g
. ((k
+ 1)
+ 1))
= (
0* n) by
A1;
A13: (g3
/. (k
+ 1))
= (g3
. (k
+ 1)) by
A3,
A8,
A10,
FINSEQ_4: 15;
(g3
. ((k
+ 1)
+ 1))
= ((g3
/. (k
+ 1))
+ (g
/. ((k
+ 1)
+ 1))) by
Def10,
A8,
A10;
then (g3
. ((k
+ 1)
+ 1))
= ((
0* n)
+ (
0* n)) by
A6,
A7,
A8,
A9,
A13,
A11,
A12,
FINSEQ_4: 15,
XXREAL_0: 2
.= (
0* n) by
EUCLID_4: 1;
hence
P[(k
+ 1)];
end;
suppose 1
> (k
+ 1);
hence
P[(k
+ 1)] by
NAT_1: 14;
end;
end;
suppose (k
+ 1)
>= (
len g);
hence
P[(k
+ 1)];
end;
end;
A14: (
0
+ 1)
<= (
len g) by
A2,
NAT_1: 13;
then 1
in (
Seg (
len g)) by
FINSEQ_1: 1;
then 1
in (
dom g) by
FINSEQ_1:def 3;
then
A15:
P[
0 ] by
A1,
A4;
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A15,
A5);
then
A16:
P[((
len g3)
-' 1)];
((
len g3)
-' 1)
= ((
len g3)
- 1) by
A3,
A14,
XREAL_1: 233;
hence (
Sum g)
= (
0* n) by
A3,
Def11,
A16,
XREAL_1: 44;
end;
suppose (
len g)
<=
0 ;
hence (
Sum g)
= (
0* n) by
Def11;
end;
end;
theorem ::
EUCLID_7:23
Th22: for g be
FinSequence of (
REAL n), h be
FinSequence of
NAT , F be
FinSequence of (
REAL n) st h is
increasing & (
rng h)
c= (
dom g) & F
= (g
* h) & (for i be
Element of
NAT st i
in (
dom g) & not i
in (
rng h) holds (g
. i)
= (
0* n)) holds (
Sum g)
= (
Sum F)
proof
let g be
FinSequence of (
REAL n), h be
FinSequence of
NAT , F be
FinSequence of (
REAL n);
assume that
A1: h is
increasing and
A2: (
rng h)
c= (
dom g) and
A3: F
= (g
* h) and
A4: for i be
Element of
NAT st i
in (
dom g) & not i
in (
rng h) holds (g
. i)
= (
0* n);
A5: (
dom (h qua
Relation
* g qua
Relation))
= (
dom h) by
A2,
RELAT_1: 27;
(
dom (g
* h))
= (
dom h) by
A2,
RELAT_1: 27;
then
A6: (
dom F)
= (
Seg (
len h)) by
A3,
FINSEQ_1:def 3;
then
A7: (
len F)
= (
len h) by
FINSEQ_1:def 3;
(
dom h)
c= (
dom g)
proof
let x be
object;
assume
A8: x
in (
dom h);
then
reconsider nx = x as
Element of
NAT ;
A9: (h
. x)
in (
rng h) by
A8,
FUNCT_1:def 3;
then
reconsider nhx = (h
. x) as
Element of
NAT ;
A10: nx
in (
Seg (
len h)) by
A8,
FINSEQ_1:def 3;
then
A11: 1
<= nx by
FINSEQ_1: 1;
A12: nx
<= (
len h) by
A10,
FINSEQ_1: 1;
then 1
<= (
len h) by
A11,
XXREAL_0: 2;
then 1
in (
Seg (
len h)) by
FINSEQ_1: 1;
then 1
in (
dom h) by
FINSEQ_1:def 3;
then (h
. 1)
in (
rng h) by
FUNCT_1:def 3;
then (h
. 1)
in (
dom g) by
A2;
then (h
. 1)
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then 1
<= (h
. 1) by
FINSEQ_1: 1;
then
A13: nx
<= nhx by
A1,
A12,
Th7;
(h
. x)
in (
dom g) by
A2,
A9;
then (h
. x)
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then nhx
<= (
len g) by
FINSEQ_1: 1;
then nx
<= (
len g) by
A13,
XXREAL_0: 2;
then nx
in (
Seg (
len g)) by
A11,
FINSEQ_1: 1;
hence x
in (
dom g) by
FINSEQ_1:def 3;
end;
then (
dom h)
c= (
Seg (
len g)) by
FINSEQ_1:def 3;
then (
Seg (
len h))
c= (
Seg (
len g)) by
FINSEQ_1:def 3;
then
A14: (
len h)
<= (
len g) by
FINSEQ_1: 5;
per cases ;
suppose
A15: (
len F)
>
0 ;
then
A16: (
0
+ 1)
<= (
len F) by
NAT_1: 13;
then 1
in (
Seg (
len F)) by
FINSEQ_1: 1;
then
A17: 1
in (
dom F) by
FINSEQ_1:def 3;
then
A18: 1
in (
Seg (
len h)) by
A3,
A5,
FINSEQ_1:def 3;
then
A19: 1
<= (
len h) by
FINSEQ_1: 1;
then (
len h)
in (
Seg (
len h)) by
FINSEQ_1: 1;
then (
len h)
in (
dom h) by
FINSEQ_1:def 3;
then (h
. (
len h))
in (
rng h) by
FUNCT_1:def 3;
then (h
. (
len h))
in (
dom g) by
A2;
then
A20: (h
. (
len h))
in (
Seg (
len g)) by
FINSEQ_1:def 3;
reconsider j = (h
. 1) as
Nat;
(
dom (h qua
Relation
* g qua
Relation))
= (
dom h) by
A2,
RELAT_1: 27;
then
A21: (
Seg (
len F))
= (
dom h) by
A3,
FINSEQ_1:def 3;
then
A22: (
len F)
= (
len h) by
FINSEQ_1:def 3;
A23: (h
. 1)
in (
rng h) by
A3,
A5,
A17,
FUNCT_1:def 3;
then
A24: (h
. 1)
in (
dom g) by
A2;
then
A25: (h
. 1)
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then
A26: 1
<= (h
. 1) by
FINSEQ_1: 1;
then
A27: (j
-' 1)
= (j
- 1) by
XREAL_1: 233;
(
Seg (
len g))
<>
{} by
A2,
A23,
FINSEQ_1:def 3;
then
0
< (
len g);
then
A28: (
0
+ 1)
<= (
len g) by
NAT_1: 13;
then
A29: ((
len g)
-' 1)
= ((
len g)
- 1) by
XREAL_1: 233;
then
A30: (h
. (
len h))
<= (((
len g)
-' 1)
+ 1) by
A20,
FINSEQ_1: 1;
A31: 1
<= j by
A25,
FINSEQ_1: 1;
set g4 = (
accum g);
A32: (
len g)
= (
len g4) by
Def10;
A33: (g
. 1)
= (g4
. 1) by
Def10;
A34: for i be
Nat st 1
<= i & i
< (
len g) holds (g4
. (i
+ 1))
= ((g4
/. i)
+ (g
/. (i
+ 1))) by
Def10;
A35: (
Sum g)
= (g4
. (
len g)) by
A7,
A14,
A15,
Def11;
set g2 = (
accum F);
A36: (
len F)
= (
len g2) by
Def10;
A37: (F
. 1)
= (g2
. 1) by
Def10;
A38: for i be
Nat st 1
<= i & i
< (
len F) holds (g2
. (i
+ 1))
= ((g2
/. i)
+ (F
/. (i
+ 1))) by
Def10;
A39: (
Sum F)
= (g2
. (
len F)) by
A15,
Def11;
defpred
Q[
Nat] means 1
<= ($1
+ 1) & ($1
+ 1)
< j implies (g4
. ($1
+ 1))
= (
0* n);
A40: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
A41:
Q[k];
1
<= ((k
+ 1)
+ 1) & ((k
+ 1)
+ 1)
< j implies (g4
. ((k
+ 1)
+ 1))
= (
0* n)
proof
assume that
A42: 1
<= ((k
+ 1)
+ 1) and
A43: ((k
+ 1)
+ 1)
< j;
per cases by
A42,
XXREAL_0: 1;
suppose
A44: 1
< ((k
+ 1)
+ 1);
j
in (
Seg (
len g)) by
A24,
FINSEQ_1:def 3;
then
A45: j
<= (
len g) by
FINSEQ_1: 1;
then ((k
+ 1)
+ 1)
< (
len g) by
A43,
XXREAL_0: 2;
then
A46: (g
/. ((k
+ 1)
+ 1))
= (g
. ((k
+ 1)
+ 1)) by
A44,
FINSEQ_4: 15;
A47:
now
assume ((k
+ 1)
+ 1)
in (
rng h);
then
consider x0 be
object such that
A48: x0
in (
dom h) and
A49: (h
. x0)
= ((k
+ 1)
+ 1) by
FUNCT_1:def 3;
reconsider nx0 = x0 as
Element of
NAT by
A48;
A50: x0
in (
Seg (
len h)) by
A48,
FINSEQ_1:def 3;
then
A51: nx0
<= (
len h) by
FINSEQ_1: 1;
1
<= nx0 by
A50,
FINSEQ_1: 1;
hence contradiction by
A1,
A43,
A49,
A51,
Th6;
end;
A52: (k
+ 1)
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then (k
+ 1)
< j by
A43,
XXREAL_0: 2;
then
A53: (k
+ 1)
< (
len g) by
A45,
XXREAL_0: 2;
then
A54: (g4
/. (k
+ 1))
= (g4
. (k
+ 1)) by
A32,
FINSEQ_4: 15,
NAT_1: 11;
((k
+ 1)
+ 1)
<= (
len g) by
A53,
NAT_1: 13;
then ((k
+ 1)
+ 1)
in (
Seg (
len g)) by
A42,
FINSEQ_1: 1;
then ((k
+ 1)
+ 1)
in (
dom g) by
FINSEQ_1:def 3;
then
A55: (g
. ((k
+ 1)
+ 1))
= (
0* n) by
A4,
A47;
(g4
. ((k
+ 1)
+ 1))
= ((g4
/. (k
+ 1))
+ (g
/. ((k
+ 1)
+ 1))) by
A34,
A53,
NAT_1: 11;
hence (g4
. ((k
+ 1)
+ 1))
= (
0* n) by
A41,
A43,
A52,
A55,
A54,
A46,
EUCLID_4: 1,
NAT_1: 11,
XXREAL_0: 2;
end;
suppose 1
= ((k
+ 1)
+ 1);
hence (g4
. ((k
+ 1)
+ 1))
= (
0* n);
end;
end;
hence
Q[(k
+ 1)];
end;
defpred
P[
Nat] means 1
<= ($1
+ 1) & ($1
+ 1)
<= (
len g2) implies (g4
. (h
. ($1
+ 1)))
= (g2
. ($1
+ 1));
A56: 1
in (
Seg (
len g)) by
A28,
FINSEQ_1: 1;
1
< j implies 1
in (
dom g) & not 1
in (
rng h)
proof
assume
A57: 1
< j;
thus 1
in (
dom g) by
A56,
FINSEQ_1:def 3;
hereby
assume 1
in (
rng h);
then
consider x0 be
object such that
A58: x0
in (
dom h) and
A59: (h
. x0)
= 1 by
FUNCT_1:def 3;
reconsider nx0 = x0 as
Element of
NAT by
A58;
A60: x0
in (
Seg (
len h)) by
A58,
FINSEQ_1:def 3;
then
A61: nx0
<= (
len h) by
FINSEQ_1: 1;
1
<= nx0 by
A60,
FINSEQ_1: 1;
hence contradiction by
A1,
A57,
A59,
A61,
Th6;
end;
end;
then
A62:
Q[
0 ] by
A4,
A33;
A63: for k be
Nat holds
Q[k] from
NAT_1:sch 2(
A62,
A40);
A64:
now
per cases by
A26,
XXREAL_0: 1;
case 1
< j;
then (1
+ 1)
<= j by
NAT_1: 13;
then
A65: ((1
+ 1)
- 1)
<= (j
- 1) by
XREAL_1: 9;
then 1
<= (j
-' 1) by
A31,
XREAL_1: 233;
then
A66: (((j
-' 1)
-' 1)
+ 1)
= (((j
-' 1)
- 1)
+ 1) by
XREAL_1: 233
.= (j
-' 1);
A67: (g
. j)
= (g2
. 1) by
A3,
A5,
A37,
A17,
FUNCT_1: 13;
A68:
Q[((j
-' 1)
-' 1)] by
A63;
A69: j
in (
Seg (
len g)) by
A24,
FINSEQ_1:def 3;
then
A70: j
<= (
len g) by
FINSEQ_1: 1;
1
<= j by
A69,
FINSEQ_1: 1;
then
A71: (g
/. j)
= (g
. j) by
A70,
FINSEQ_4: 15;
A72: (g2
/. 1)
= (g2
. 1) by
A36,
A16,
FINSEQ_4: 15;
A73: (j
-' 1)
< ((j
-' 1)
+ 1) by
XREAL_1: 29;
then
A74: (j
-' 1)
< (
len g) by
A27,
A70,
XXREAL_0: 2;
then (g4
/. (j
-' 1))
= (g4
. (j
-' 1)) by
A32,
A27,
A65,
FINSEQ_4: 15;
then (g4
. j)
= ((
0* n)
+ (g2
/. 1)) by
Def10,
A27,
A65,
A73,
A74,
A68,
A66,
A71,
A72,
A67
.= (g2
/. 1) by
EUCLID_4: 1;
hence
P[
0 ] by
FINSEQ_4: 15;
end;
case j
= 1;
hence
P[
0 ] by
A3,
A5,
A37,
A33,
A17,
FUNCT_1: 13;
end;
end;
A75: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A76:
P[k];
1
<= ((k
+ 1)
+ 1) & ((k
+ 1)
+ 1)
<= (
len g2) implies (g4
. (h
. ((k
+ 1)
+ 1)))
= (g2
. ((k
+ 1)
+ 1))
proof
defpred
R[
Nat] means (h
. (k
+ 1))
<= ($1
+ 1) & ($1
+ 1)
< (h
. ((k
+ 1)
+ 1)) implies (g4
. ($1
+ 1))
= (g2
. (k
+ 1));
assume that
A77: 1
<= ((k
+ 1)
+ 1) and
A78: ((k
+ 1)
+ 1)
<= (
len g2);
A79: ((k
+ 1)
+ 1)
in (
Seg (
len g2)) by
A77,
A78,
FINSEQ_1: 1;
then (h
. ((k
+ 1)
+ 1))
in (
rng h) by
A36,
A21,
FUNCT_1:def 3;
then (h
. ((k
+ 1)
+ 1))
in (
dom g) by
A2;
then
A80: (h
. ((k
+ 1)
+ 1))
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then
A81: 1
<= (h
. ((k
+ 1)
+ 1)) by
FINSEQ_1: 1;
A82: (k
+ 1)
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then
A83: (k
+ 1)
< (
len g2) by
A78,
XXREAL_0: 2;
now
per cases ;
case
A84: 1
<= k;
k
< (k
+ 1) by
XREAL_1: 29;
then
A85: k
< (
len h) by
A36,
A22,
A83,
XXREAL_0: 2;
then k
<= (h
. k) by
A1,
A26,
Th7;
then
A86: 1
<= (h
. k) by
A84,
XXREAL_0: 2;
(h
. k)
< (h
. (k
+ 1)) by
A1,
A84,
A85;
hence
R[
0 ] by
A86,
XXREAL_0: 2;
end;
case k
< 1;
hence
R[
0 ] by
A15,
A36,
A64,
NAT_1: 14;
end;
end;
then
A87:
R[
0 ];
A88: (h
. ((k
+ 1)
+ 1))
<= (
len g) by
A80,
FINSEQ_1: 1;
1
<= (k
+ 1) by
NAT_1: 11;
then
A89: (h
. (k
+ 1))
< (h
. ((k
+ 1)
+ 1)) by
A1,
A36,
A22,
A83;
A90: for k2 be
Nat st
R[k2] holds
R[(k2
+ 1)]
proof
let k2 be
Nat;
assume
A91:
R[k2];
(h
. (k
+ 1))
<= ((k2
+ 1)
+ 1) & ((k2
+ 1)
+ 1)
< (h
. ((k
+ 1)
+ 1)) implies (g4
. ((k2
+ 1)
+ 1))
= (g2
. (k
+ 1))
proof
assume that
A92: (h
. (k
+ 1))
<= ((k2
+ 1)
+ 1) and
A93: ((k2
+ 1)
+ 1)
< (h
. ((k
+ 1)
+ 1));
per cases by
A92,
XXREAL_0: 1;
suppose
A94: (h
. (k
+ 1))
< ((k2
+ 1)
+ 1);
A95:
now
assume ((k2
+ 1)
+ 1)
in (
rng h);
then
consider x0 be
object such that
A96: x0
in (
dom h) and
A97: (h
. x0)
= ((k2
+ 1)
+ 1) by
FUNCT_1:def 3;
reconsider nx0 = x0 as
Element of
NAT by
A96;
A98: x0
in (
Seg (
len h)) by
A96,
FINSEQ_1:def 3;
then nx0
<= (
len h) by
FINSEQ_1: 1;
then
A99: nx0
>= ((k
+ 1)
+ 1) implies (h
. nx0)
>= (h
. ((k
+ 1)
+ 1)) by
A1,
A77,
Th6;
1
<= nx0 by
A98,
FINSEQ_1: 1;
then nx0
<= (k
+ 1) implies (h
. nx0)
<= (h
. (k
+ 1)) by
A1,
A36,
A22,
A83,
Th6;
hence contradiction by
A93,
A94,
A97,
A99,
NAT_1: 13;
end;
(h
. ((k
+ 1)
+ 1))
in (
rng h) by
A36,
A21,
A79,
FUNCT_1:def 3;
then (h
. ((k
+ 1)
+ 1))
in (
dom g) by
A2;
then (h
. ((k
+ 1)
+ 1))
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then (h
. ((k
+ 1)
+ 1))
<= (
len g) by
FINSEQ_1: 1;
then
A100: ((k2
+ 1)
+ 1)
< (
len g) by
A93,
XXREAL_0: 2;
A101: 1
< ((k2
+ 1)
+ 1) by
XREAL_1: 29;
then
A102: (g
/. ((k2
+ 1)
+ 1))
= (g
. ((k2
+ 1)
+ 1)) by
A100,
FINSEQ_4: 15;
A103: (k2
+ 1)
< ((k2
+ 1)
+ 1) by
XREAL_1: 29;
then
A104: (k2
+ 1)
< (
len g) by
A100,
XXREAL_0: 2;
((k2
+ 1)
+ 1)
in (
Seg (
len g)) by
A100,
A101,
FINSEQ_1: 1;
then ((k2
+ 1)
+ 1)
in (
dom g) by
FINSEQ_1:def 3;
then
A105: (g
. ((k2
+ 1)
+ 1))
= (
0* n) by
A4,
A95;
(k2
+ 1)
< (
len g) by
A103,
A100,
XXREAL_0: 2;
then (g4
. ((k2
+ 1)
+ 1))
= ((g4
/. (k2
+ 1))
+ (g
/. ((k2
+ 1)
+ 1))) by
A34,
NAT_1: 11
.= (g4
/. (k2
+ 1)) by
A105,
A102,
EUCLID_4: 1
.= (g4
. (k2
+ 1)) by
A32,
A104,
FINSEQ_4: 15,
NAT_1: 11;
hence (g4
. ((k2
+ 1)
+ 1))
= (g2
. (k
+ 1)) by
A91,
A93,
A94,
NAT_1: 13;
end;
suppose (h
. (k
+ 1))
= ((k2
+ 1)
+ 1);
hence (g4
. ((k2
+ 1)
+ 1))
= (g2
. (k
+ 1)) by
A76,
A78,
A82,
NAT_1: 11,
XXREAL_0: 2;
end;
end;
hence
R[(k2
+ 1)];
end;
A106: for k2 be
Nat holds
R[k2] from
NAT_1:sch 2(
A87,
A90);
then
A107: (h
. (k
+ 1))
<= ((((h
. ((k
+ 1)
+ 1))
-' 1)
-' 1)
+ 1) & ((((h
. ((k
+ 1)
+ 1))
-' 1)
-' 1)
+ 1)
< (h
. ((k
+ 1)
+ 1)) implies (g4
. ((((h
. ((k
+ 1)
+ 1))
-' 1)
-' 1)
+ 1))
= (g2
. (k
+ 1));
now
per cases ;
case
A108: 1
<= k;
k
< (k
+ 1) by
XREAL_1: 29;
then
A109: k
< (
len h) by
A36,
A22,
A83,
XXREAL_0: 2;
then k
<= (h
. k) by
A1,
A26,
Th7;
then
A110: 1
<= (h
. k) by
A108,
XXREAL_0: 2;
A111: (1
+ (h
. (k
+ 1)))
<= (h
. ((k
+ 1)
+ 1)) by
A89,
NAT_1: 13;
then
A112: (((h
. (k
+ 1))
+ 1)
- 1)
<= ((h
. ((k
+ 1)
+ 1))
- 1) by
XREAL_1: 9;
(h
. k)
< (h
. (k
+ 1)) by
A1,
A108,
A109;
then 1
< (h
. (k
+ 1)) by
A110,
XXREAL_0: 2;
then (1
+ 1)
< ((h
. (k
+ 1))
+ 1) by
XREAL_1: 6;
then (1
+ 1)
< (h
. ((k
+ 1)
+ 1)) by
A111,
XXREAL_0: 2;
then
A113: ((1
+ 1)
- 1)
< ((h
. ((k
+ 1)
+ 1))
- 1) by
XREAL_1: 9;
then
A114: ((h
. ((k
+ 1)
+ 1))
-' 1)
= ((h
. ((k
+ 1)
+ 1))
- 1) by
XREAL_0:def 2;
then
A115: (((h
. ((k
+ 1)
+ 1))
-' 1)
-' 1)
= (((h
. ((k
+ 1)
+ 1))
- 1)
- 1) by
A113,
XREAL_1: 233;
A116: (g
/. (h
. ((k
+ 1)
+ 1)))
= (g
. (h
. ((k
+ 1)
+ 1))) by
A81,
A88,
FINSEQ_4: 15;
A117: (F
. ((k
+ 1)
+ 1))
= (g
. (h
. ((k
+ 1)
+ 1))) by
A3,
A36,
A21,
A79,
FUNCT_1: 13;
A118: (k
+ 1)
<= (
len g2) by
A78,
A82,
XXREAL_0: 2;
(h
. ((k
+ 1)
+ 1))
< ((h
. ((k
+ 1)
+ 1))
+ 1) by
XREAL_1: 29;
then
A119: ((h
. ((k
+ 1)
+ 1))
- 1)
< (((h
. ((k
+ 1)
+ 1))
+ 1)
- 1) by
XREAL_1: 9;
then
A120: ((h
. ((k
+ 1)
+ 1))
-' 1)
< (
len g) by
A88,
A114,
XXREAL_0: 2;
then
A121: (g4
/. ((h
. ((k
+ 1)
+ 1))
-' 1))
= (g4
. ((h
. ((k
+ 1)
+ 1))
-' 1)) by
A32,
A113,
A114,
FINSEQ_4: 15;
(g2
. ((k
+ 1)
+ 1))
= ((g2
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A36,
A38,
A83,
NAT_1: 11
.= ((g4
/. ((h
. ((k
+ 1)
+ 1))
-' 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A107,
A114,
A115,
A112,
A119,
A121,
A118,
FINSEQ_4: 15,
NAT_1: 11
.= ((g4
/. ((h
. ((k
+ 1)
+ 1))
-' 1))
+ (g
/. (h
. ((k
+ 1)
+ 1)))) by
A36,
A77,
A78,
A117,
A116,
FINSEQ_4: 15
.= (g4
. (((h
. ((k
+ 1)
+ 1))
-' 1)
+ 1)) by
Def10,
A113,
A114,
A120;
hence (g4
. (h
. ((k
+ 1)
+ 1)))
= (g2
. ((k
+ 1)
+ 1)) by
A114;
end;
case k
< 1;
then
A122: k
=
0 by
NAT_1: 14;
then 1
< (
len h) by
A36,
A22,
A78,
NAT_1: 13;
then
A123: (h
. 1)
< (h
. (1
+ 1)) by
A1;
then
A124: ((h
. (1
+ 1))
-' 1)
= ((h
. (1
+ 1))
- 1) by
A31,
XREAL_1: 233,
XXREAL_0: 2;
then
A125: (((h
. ((k
+ 1)
+ 1))
-' 1)
+ 1)
= (h
. ((k
+ 1)
+ 1)) by
A122;
A126: (F
/. 1)
= (F
. 1) by
A19,
A22,
FINSEQ_4: 15;
(k
+ 1)
< (
len F) by
A36,
A78,
NAT_1: 13;
then
A127: (g2
. ((k
+ 1)
+ 1))
= ((g2
/. (k
+ 1))
+ (F
/. ((k
+ 1)
+ 1))) by
A38,
NAT_1: 11
.= ((F
/. 1)
+ (F
/. (1
+ 1))) by
A36,
A37,
A19,
A22,
A122,
A126,
FINSEQ_4: 15;
A128: (F
/. (1
+ 1))
= (F
. (1
+ 1)) by
A36,
A78,
A122,
FINSEQ_4: 15;
((h
. 1)
+ 1)
<= (h
. (1
+ 1)) by
A123,
NAT_1: 13;
then (((h
. 1)
+ 1)
- 1)
<= ((h
. (1
+ 1))
- 1) by
XREAL_1: 9;
then
A129: (h
. 1)
<= ((h
. (1
+ 1))
-' 1) by
A26,
A123,
XREAL_1: 233,
XXREAL_0: 2;
(h
. (1
+ 1))
< ((h
. (1
+ 1))
+ 1) by
XREAL_1: 29;
then
A130: ((h
. (1
+ 1))
- 1)
< (((h
. (1
+ 1))
+ 1)
- 1) by
XREAL_1: 9;
then
A131: ((h
. (1
+ 1))
-' 1)
< (h
. (1
+ 1)) by
A26,
A123,
XREAL_1: 233,
XXREAL_0: 2;
A132: 1
< (h
. (1
+ 1)) by
A26,
A123,
XXREAL_0: 2;
then (1
+ 1)
<= (h
. (1
+ 1)) by
NAT_1: 13;
then
A133: ((1
+ 1)
- 1)
<= ((h
. (1
+ 1))
- 1) by
XREAL_1: 9;
then (((h
. (1
+ 1))
-' 1)
-' 1)
= (((h
. (1
+ 1))
-' 1)
- 1) by
A124,
XREAL_1: 233;
then
A134: ((((h
. (1
+ 1))
-' 1)
-' 1)
+ 1)
= ((h
. (1
+ 1))
-' 1);
(1
+ 1)
in (
Seg (
len h)) by
A36,
A22,
A78,
A122,
FINSEQ_1: 1;
then (1
+ 1)
in (
dom h) by
FINSEQ_1:def 3;
then (h
. (1
+ 1))
in (
rng h) by
FUNCT_1:def 3;
then (h
. (1
+ 1))
in (
dom g) by
A2;
then
A135: (h
. (1
+ 1))
in (
Seg (
len g)) by
FINSEQ_1:def 3;
then (h
. (1
+ 1))
<= (
len g) by
FINSEQ_1: 1;
then
A136: (g
/. (h
. ((k
+ 1)
+ 1)))
= (g
. (h
. (1
+ 1))) by
A122,
A132,
FINSEQ_4: 15
.= (F
. (1
+ 1)) by
A3,
A36,
A21,
A79,
A122,
FUNCT_1: 13;
(h
. (1
+ 1))
<= (
len g) by
A135,
FINSEQ_1: 1;
then ((h
. (1
+ 1))
-' 1)
< (
len g4) by
A32,
A124,
A130,
XXREAL_0: 2;
then
A137: (g4
/. ((h
. ((k
+ 1)
+ 1))
-' 1))
= (g4
. ((h
. (1
+ 1))
-' 1)) by
A122,
A124,
A133,
FINSEQ_4: 15
.= (F
. 1) by
A37,
A106,
A122,
A134,
A131,
A129;
(h
. (1
+ 1))
<= (
len g) by
A135,
FINSEQ_1: 1;
then
A138: ((h
. ((k
+ 1)
+ 1))
-' 1)
< (
len g) by
A122,
A125,
NAT_1: 13;
1
<= ((h
. ((k
+ 1)
+ 1))
-' 1) by
A122,
A132,
A125,
NAT_1: 13;
then (g4
. (((h
. ((k
+ 1)
+ 1))
-' 1)
+ 1))
= ((g4
/. ((h
. ((k
+ 1)
+ 1))
-' 1))
+ (g
/. (h
. ((k
+ 1)
+ 1)))) by
Def10,
A122,
A124,
A138
.= ((F
/. 1)
+ (F
/. (1
+ 1))) by
A19,
A22,
A128,
A137,
A136,
FINSEQ_4: 15;
hence (g4
. (h
. ((k
+ 1)
+ 1)))
= (g2
. ((k
+ 1)
+ 1)) by
A122,
A124,
A127;
end;
end;
hence (g4
. (h
. ((k
+ 1)
+ 1)))
= (g2
. ((k
+ 1)
+ 1));
end;
hence
P[(k
+ 1)];
end;
defpred
R[
Nat] means (h
. (
len h))
<= ($1
+ 1) & ($1
+ 1)
<= (
len g) implies (g4
. ($1
+ 1))
= (g2
. (
len g2));
A139:
P[
0 ] by
A64;
A140: for k be
Nat holds
P[k] from
NAT_1:sch 2(
A139,
A75);
A141: for k be
Nat st
R[k] holds
R[(k
+ 1)]
proof
let k be
Nat;
assume
A142:
R[k];
(h
. (
len h))
<= ((k
+ 1)
+ 1) & ((k
+ 1)
+ 1)
<= (
len g) implies (g4
. ((k
+ 1)
+ 1))
= (g2
. (
len g2))
proof
assume that
A143: (h
. (
len h))
<= ((k
+ 1)
+ 1) and
A144: ((k
+ 1)
+ 1)
<= (
len g);
per cases by
A143,
XXREAL_0: 1;
suppose
A145: (h
. (
len h))
< ((k
+ 1)
+ 1);
A146:
now
assume ((k
+ 1)
+ 1)
in (
rng h);
then
consider x be
object such that
A147: x
in (
dom h) and
A148: (h
. x)
= ((k
+ 1)
+ 1) by
FUNCT_1:def 3;
reconsider nx = x as
Element of
NAT by
A147;
A149: x
in (
Seg (
len h)) by
A147,
FINSEQ_1:def 3;
then
A150: nx
<= (
len h) by
FINSEQ_1: 1;
1
<= nx by
A149,
FINSEQ_1: 1;
hence contradiction by
A1,
A145,
A148,
A150,
Th6;
end;
1
<= ((k
+ 1)
+ 1) by
NAT_1: 11;
then ((k
+ 1)
+ 1)
in (
Seg (
len g)) by
A144,
FINSEQ_1: 1;
then
A151: ((k
+ 1)
+ 1)
in (
dom g) by
FINSEQ_1:def 3;
(k
+ 1)
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then
A152: (k
+ 1)
< (
len g) by
A144,
XXREAL_0: 2;
then
A153: (g4
. ((k
+ 1)
+ 1))
= ((g4
/. (k
+ 1))
+ (g
/. ((k
+ 1)
+ 1))) by
A34,
NAT_1: 11;
1
<= (k
+ 1) by
NAT_1: 11;
then
A154: (g4
/. (k
+ 1))
= (g2
. (
len g2)) by
A32,
A142,
A145,
A152,
FINSEQ_4: 15,
NAT_1: 13;
1
< ((k
+ 1)
+ 1) by
XREAL_1: 29;
then (g
/. ((k
+ 1)
+ 1))
= (g
. ((k
+ 1)
+ 1)) by
A144,
FINSEQ_4: 15
.= (
0* n) by
A4,
A146,
A151;
hence (g4
. ((k
+ 1)
+ 1))
= (g2
. (
len g2)) by
A154,
A153,
EUCLID_4: 1;
end;
suppose
A155: (h
. (
len h))
= ((k
+ 1)
+ 1);
A156: ((
len h)
-' 1)
= ((
len h)
- 1) by
A19,
XREAL_1: 233;
1
<= (((
len h)
-' 1)
+ 1) & (((
len h)
-' 1)
+ 1)
<= (
len g2) implies (g4
. (h
. (((
len h)
-' 1)
+ 1)))
= (g2
. (((
len h)
-' 1)
+ 1)) by
A140;
hence (g4
. ((k
+ 1)
+ 1))
= (g2
. (
len g2)) by
A36,
A18,
A21,
A155,
A156,
FINSEQ_1: 1,
FINSEQ_1:def 3;
end;
end;
hence
R[(k
+ 1)];
end;
A157: 1
<= (h
. (
len h)) by
A20,
FINSEQ_1: 1;
(h
. (
len h))
<= 1 & 1
<= (
len g) implies (g4
. (
0
+ 1))
= (g2
. (
len g2))
proof
assume that
A158: (h
. (
len h))
<= 1 and 1
<= (
len g);
(h
. (
len h))
= 1 by
A157,
A158,
XXREAL_0: 1;
then (
len h)
<= 1 by
A1,
A26,
Th7;
then
A159: (
len h)
= 1 by
A19,
XXREAL_0: 1;
(g2
. 1)
= (g
. (h
. 1)) by
A3,
A5,
A37,
A17,
FUNCT_1: 13
.= (g
. 1) by
A157,
A158,
A159,
XXREAL_0: 1;
hence (g4
. (
0
+ 1))
= (g2
. (
len g2)) by
A6,
A36,
A33,
A159,
FINSEQ_1:def 3;
end;
then
A160:
R[
0 ];
for k be
Nat holds
R[k] from
NAT_1:sch 2(
A160,
A141);
hence (
Sum g)
= (
Sum F) by
A36,
A39,
A35,
A29,
A30;
end;
suppose
A161: (
len F)
<=
0 ;
then
A162: (
Sum F)
= (
0* n) by
Def11;
A163: (
dom g)
= (
Seg (
len g)) by
FINSEQ_1:def 3;
(
Seg (
len F))
=
{} by
A161;
then (
dom F)
=
{} by
FINSEQ_1:def 3;
then
A164: (
rng h)
=
{} by
A3,
A5,
RELAT_1: 42;
A165:
now
let z be
object;
assume
A166: z
in (
dom g);
hence (g
. z)
= (
0* n) by
A4,
A164
.= (((
len g)
|-> (
0* n))
. z) by
A163,
A166,
FINSEQ_2: 57;
end;
(
Seg (
len g))
= (
dom ((
len g)
|-> (
0* n))) by
FUNCOP_1: 13;
then g
= ((
len g)
|-> (
0* n)) by
A163,
A165,
FUNCT_1: 2;
hence (
Sum g)
= (
Sum F) by
A162,
Th21;
end;
end;
theorem ::
EUCLID_7:24
Th23: for g be
FinSequence of (
REAL n), h be
FinSequence of
NAT , F be
FinSequence of (
REAL n) st h is
one-to-one & (
rng h)
c= (
dom g) & F
= (g
* h) & (for i be
Element of
NAT st i
in (
dom g) & not i
in (
rng h) holds (g
. i)
= (
0* n)) holds (
Sum g)
= (
Sum F)
proof
let g be
FinSequence of (
REAL n), h be
FinSequence of
NAT , F be
FinSequence of (
REAL n);
assume that
A1: h is
one-to-one and
A2: (
rng h)
c= (
dom g) and
A3: F
= (g
* h) and
A4: for i be
Element of
NAT st i
in (
dom g) & not i
in (
rng h) holds (g
. i)
= (
0* n);
consider h3 be
Permutation of (
dom h), h2 be
FinSequence of
NAT such that
A5: h2
= (h
* h3) and
A6: h2 is
increasing and
A7: (
dom h)
= (
dom h2) and
A8: (
rng h)
= (
rng h2) by
A1,
Th13;
(
dom (g
* h))
= (
dom h) by
A2,
RELAT_1: 27;
then
reconsider h33 = h3 as
Permutation of (
dom F) by
A3;
reconsider F22 = (g
* h2) as
Function;
(
dom F22)
= (
dom h) by
A2,
A7,
A8,
RELAT_1: 27
.= (
Seg (
len h)) by
FINSEQ_1:def 3;
then
reconsider F23 = F22 as
FinSequence by
FINSEQ_1:def 2;
(
rng F22)
c= (
rng g) by
RELAT_1: 26;
then (
rng F23)
c= (
REAL n) by
XBOOLE_1: 1;
then
reconsider F2 = F23 as
FinSequence of (
REAL n) by
FINSEQ_1:def 4;
F2
= (F
(*) h33) by
A3,
A5,
RELAT_1: 36;
then (
Sum F)
= (
Sum F2) by
Th20;
hence (
Sum g)
= (
Sum F) by
A2,
A4,
A6,
A8,
Th22;
end;
begin
definition
let n,i be
Nat;
:: original:
Base_FinSeq
redefine
func
Base_FinSeq (n,i) ->
Element of (
REAL n) ;
coherence
proof
(
len (
Base_FinSeq (n,i)))
= n by
MATRIXR2: 74;
hence thesis by
FINSEQ_2: 92;
end;
end
theorem ::
EUCLID_7:25
Th24: for i1,i2 be
Nat st 1
<= i1 & i1
<= n & (
Base_FinSeq (n,i1))
= (
Base_FinSeq (n,i2)) holds i1
= i2
proof
let i1,i2 be
Nat;
assume that
A1: 1
<= i1 and
A2: i1
<= n and
A3: (
Base_FinSeq (n,i1))
= (
Base_FinSeq (n,i2));
((
Base_FinSeq (n,i1))
. i1)
= 1 by
A1,
A2,
MATRIXR2: 75;
hence thesis by
A1,
A2,
A3,
MATRIXR2: 76;
end;
theorem ::
EUCLID_7:26
Th25: (
sqr (
Base_FinSeq (n,i)))
= (
Base_FinSeq (n,i))
proof
A1: (
dom (
sqrreal
* (
Base_FinSeq (n,i))))
= ((
Base_FinSeq (n,i))
" (
dom
sqrreal )) by
RELAT_1: 147;
A2: (
rng (
Base_FinSeq (n,i)))
c=
REAL ;
A3: for x be
object st x
in (
dom (
Base_FinSeq (n,i))) holds ((
sqrreal
* (
Base_FinSeq (n,i)) qua
Function)
. x)
= ((
Base_FinSeq (n,i))
. x)
proof
let x be
object;
assume
A4: x
in (
dom (
Base_FinSeq (n,i)));
then
reconsider nx = x as
Element of
NAT ;
A5: ((
sqrreal
* (
Base_FinSeq (n,i)) qua
Function)
. x)
= (
sqrreal
. ((
Base_FinSeq (n,i))
. x)) by
A4,
FUNCT_1: 13;
A6: x
in (
Seg (
len (
Base_FinSeq (n,i)))) by
A4,
FINSEQ_1:def 3;
then
A7: 1
<= nx by
FINSEQ_1: 1;
(
len (
Base_FinSeq (n,i)))
= n by
MATRIXR2: 74;
then
A8: nx
<= n by
A6,
FINSEQ_1: 1;
per cases ;
suppose
A9: nx
= i;
hence ((
sqrreal
* (
Base_FinSeq (n,i)) qua
Function)
. x)
= (
sqrreal
. 1) by
A7,
A8,
A5,
MATRIXR2: 75
.= (1
^2 ) by
RVSUM_1:def 2
.= ((
Base_FinSeq (n,i))
. x) by
A7,
A8,
A9,
MATRIXR2: 75;
end;
suppose
A10: nx
<> i;
hence ((
sqrreal
* (
Base_FinSeq (n,i)) qua
Function)
. x)
= (
sqrreal
.
0 ) by
A7,
A8,
A5,
MATRIXR2: 76
.= (
0
^2 ) by
RVSUM_1:def 2
.= ((
Base_FinSeq (n,i))
. x) by
A7,
A8,
A10,
MATRIXR2: 76;
end;
end;
((
Base_FinSeq (n,i))
" (
dom
sqrreal ))
= ((
Base_FinSeq (n,i))
"
REAL ) by
FUNCT_2:def 1
.= (
dom (
Base_FinSeq (n,i))) by
A2,
Th1;
hence thesis by
A1,
A3,
FUNCT_1: 2;
end;
Lm3:
0
in
REAL by
XREAL_0:def 1;
theorem ::
EUCLID_7:27
Th26: 1
<= i & i
<= n implies (
Sum (
Base_FinSeq (n,i)))
= 1
proof
assume that
A1: 1
<= i and
A2: i
<= n;
defpred
P[
object,
object] means ( not ($1
in i) implies $2
= 1) & ($1
in i implies $2
=
0 );
set F = (
Base_FinSeq (n,i));
A3: for x be
object st x
in
NAT holds ex y be
object st y
in
REAL &
P[x, y]
proof
let x be
object;
assume x
in
NAT ;
then
reconsider nx = x as
Element of
NAT ;
per cases ;
suppose nx
>= i;
then
A4: not nx
in (
Segm i) by
NAT_1: 44;
take y = 1;
thus y
in
REAL by
XREAL_0:def 1;
thus
P[x, y] by
A4;
end;
suppose nx
< i;
then nx
in (
Segm i) by
NAT_1: 44;
hence ex y be
object st y
in
REAL &
P[x, y] by
Lm3;
end;
end;
consider f0 be
sequence of
REAL such that
A5: for x be
object st x
in
NAT holds
P[x, (f0
. x)] from
FUNCT_2:sch 1(
A3);
A6: (
len (
Base_FinSeq (n,i)))
= n by
MATRIXR2: 74;
A7: for n2 be
Nat st
0
<> n2 & n2
< (
len F) holds (f0
. (n2
+ 1))
= (
addreal
. ((f0
. n2),(F
. (n2
+ 1))))
proof
let n2 be
Nat;
assume that
0
<> n2 and
A8: n2
< (
len F);
A9: (n2
+ 1)
<= n by
A6,
A8,
NAT_1: 13;
A10: n2
in
NAT by
ORDINAL1:def 12;
per cases ;
suppose
A11: (n2
+ 1)
< i;
then n2
< i by
NAT_1: 13;
then n2
in (
Segm i) by
NAT_1: 44;
then
A12: (f0
. n2)
=
0 by
A5,
A10;
A13: 1
<= (n2
+ 1) by
NAT_1: 11;
A14: (n2
+ 1)
in (
Segm i) by
A11,
NAT_1: 44;
(n2
+ 1)
<= (
len F) by
A8,
NAT_1: 13;
then (
addreal
. ((f0
. n2),(F
. (n2
+ 1))))
= (
addreal
. (
0 ,
0 )) by
A6,
A11,
A12,
A13,
MATRIXR2: 76
.= (
0
+
0 ) by
BINOP_2:def 9;
hence (f0
. (n2
+ 1))
= (
addreal
. ((f0
. n2),(F
. (n2
+ 1)))) by
A5,
A14;
end;
suppose
A15: i
<= (n2
+ 1);
per cases by
A15,
XXREAL_0: 1;
suppose
A16: i
< (n2
+ 1);
then i
<= n2 by
NAT_1: 13;
then
A17: not n2
in (
Segm i) by
NAT_1: 44;
A18: not (n2
+ 1)
in (
Segm i) by
A16,
NAT_1: 44;
1
<= (n2
+ 1) by
A1,
A16,
XXREAL_0: 2;
then (F
. (n2
+ 1))
=
0 by
A9,
A16,
MATRIXR2: 76;
then (
addreal
. ((f0
. n2),(F
. (n2
+ 1))))
= (
addreal
. (1,
0 )) by
A5,
A17,
A10
.= (1
+
0 ) by
BINOP_2:def 9;
hence (f0
. (n2
+ 1))
= (
addreal
. ((f0
. n2),(F
. (n2
+ 1)))) by
A5,
A18;
end;
suppose
A19: i
= (n2
+ 1);
n2
< (n2
+ 1) by
XREAL_1: 29;
then n2
in (
Segm i) by
A19,
NAT_1: 44;
then (f0
. n2)
=
0 by
A5,
A10;
then
A20: (
addreal
. ((f0
. n2),(F
. (n2
+ 1))))
= (
addreal
. (
0 ,1)) by
A1,
A2,
A19,
MATRIXR2: 75
.= (
0
+ 1) by
BINOP_2:def 9;
not (n2
+ 1)
in i by
A19;
hence (f0
. (n2
+ 1))
= (
addreal
. ((f0
. n2),(F
. (n2
+ 1)))) by
A5,
A20;
end;
end;
end;
A21: (f0
. 1)
= (F
. 1)
proof
per cases ;
suppose
A22: 1
< i;
then 1
in (
Segm i) by
NAT_1: 44;
then
A23: (f0
. 1)
=
0 by
A5;
1
<= n by
A1,
A2,
XXREAL_0: 2;
hence (f0
. 1)
= (F
. 1) by
A22,
A23,
MATRIXR2: 76;
end;
suppose
A24: 1
>= i;
then not 1
in (
Segm i) by
NAT_1: 44;
then
A25: (f0
. 1)
= 1 by
A5;
1
= i by
A1,
A24,
XXREAL_0: 1;
hence (f0
. 1)
= (F
. 1) by
A2,
A25,
MATRIXR2: 75;
end;
end;
A26: (f0
. (
len F))
= 1
proof
per cases ;
suppose (
len F)
< i;
hence (f0
. (
len F))
= 1 by
A2,
MATRIXR2: 74;
end;
suppose (
len F)
>= i;
then not (
len F)
in (
Segm i) by
NAT_1: 44;
hence (f0
. (
len F))
= 1 by
A5;
end;
end;
(
len (
Base_FinSeq (n,i)))
>= 1 by
A1,
A2,
A6,
XXREAL_0: 2;
hence thesis by
A21,
A26,
A7,
FINSOP_1:def 1;
end;
theorem ::
EUCLID_7:28
Th27: 1
<= i & i
<= n implies
|.(
Base_FinSeq (n,i)).|
= 1
proof
assume that
A1: 1
<= i and
A2: i
<= n;
(
sqrt (
Sum (
Base_FinSeq (n,i))))
= 1 by
A1,
A2,
Th26,
SQUARE_1: 18;
hence thesis by
Th25;
end;
theorem ::
EUCLID_7:29
Th28: 1
<= i & i
<= n & i
<> j implies
|((
Base_FinSeq (n,i)), (
Base_FinSeq (n,j)))|
=
0
proof
assume that
A1: 1
<= i and
A2: i
<= n and
A3: i
<> j;
set x1 = (
Base_FinSeq (n,i)), x2 = (
Base_FinSeq (n,j));
A4: (
dom (
0* n))
= (
Seg (
len (n
|->
0 qua
Real))) by
FINSEQ_1:def 3
.= (
Seg n) by
CARD_1:def 7;
A5: (
dom x2)
= (
Seg (
len x2)) by
FINSEQ_1:def 3
.= (
Seg n) by
MATRIXR2: 74;
A6: (
dom x1)
= (
Seg (
len x1)) by
FINSEQ_1:def 3
.= (
Seg n) by
MATRIXR2: 74;
A7: (
dom
<:x1, x2:>)
= ((
dom x1)
/\ (
dom x2)) by
FUNCT_3:def 7;
(
dom
multreal )
=
[:
REAL ,
REAL :] by
FUNCT_2:def 1;
then
A8: (
dom (
multreal
*
<:x1, x2:>))
= (
<:x1, x2:>
"
[:
REAL ,
REAL :]) by
RELAT_1: 147
.= (
Seg n) by
A7,
A6,
A5,
RELSET_1: 22;
for x be
object st x
in (
dom (
0* n)) holds ((
multreal
*
<:x1, x2:>)
. x)
= ((
0* n)
. x)
proof
let x be
object;
assume
A9: x
in (
dom (
0* n));
then
reconsider nx = x as
Element of
NAT ;
A10: ((
multreal
*
<:x1, x2:>)
. x)
= (
multreal
. (
<:x1, x2:>
. x)) by
A4,
A8,
A9,
FUNCT_1: 12
.= (
multreal
.
[(x1
. nx), (x2
. nx)]) by
A4,
A7,
A6,
A5,
A9,
FUNCT_3:def 7;
A11: nx
<= n by
A4,
A9,
FINSEQ_1: 1;
A12: 1
<= nx by
A4,
A9,
FINSEQ_1: 1;
per cases ;
suppose
A13: nx
= i;
then
A14: (x1
. nx)
= 1 by
A1,
A2,
MATRIXR2: 75;
A15: (x2
. nx)
=
0 by
A1,
A2,
A3,
A13,
MATRIXR2: 76;
(
multreal
.
[(x1
. nx), (x2
. nx)])
= (
multreal
. ((x1
. nx),(x2
. nx)))
.= (1
*
0 ) by
A15,
A14,
BINOP_2:def 11
.=
0 ;
hence ((
multreal
*
<:x1, x2:>)
. x)
= ((
0* n)
. x) by
A10;
end;
suppose
A16: nx
<> i;
reconsider r = (x2
. nx) as
Element of
REAL by
XREAL_0:def 1;
A17: (x1
. nx)
=
0 by
A12,
A11,
A16,
MATRIXR2: 76;
(
multreal
.
[(x1
. nx), (x2
. nx)])
= (
multreal
. ((x1
. nx),(x2
. nx)))
.= (
0
* r) by
A17,
BINOP_2:def 11
.=
0 ;
hence ((
multreal
*
<:x1, x2:>)
. x)
= ((
0* n)
. x) by
A10;
end;
end;
then (
multreal
*
<:x1, x2:>)
= (
0* n) by
A4,
A8,
FUNCT_1: 2;
then (
multreal
.: (x1,x2))
= (
0* n) by
FUNCOP_1:def 3;
hence
|((
Base_FinSeq (n,i)), (
Base_FinSeq (n,j)))|
=
0 by
RVSUM_1: 81;
end;
theorem ::
EUCLID_7:30
Th29: for x be
Element of (
REAL n) st 1
<= i & i
<= n holds
|(x, (
Base_FinSeq (n,i)))|
= (x
. i)
proof
let x be
Element of (
REAL n);
assume that
A1: 1
<= i and
A2: i
<= n;
set x2 = (
Base_FinSeq (n,i));
A3: (
len x)
= n by
CARD_1:def 7;
A4: (
len x2)
= n by
MATRIXR2: 74;
then
A5: (
len (
mlt (x,x2)))
= n by
A3,
MATRPROB: 30;
A6: for j be
Nat st 1
<= j & j
<= n holds ((
mlt (x,x2))
. j)
= (((x
/. i)
* (
Base_FinSeq (n,i)))
. j)
proof
let j be
Nat;
assume that
A7: 1
<= j and
A8: j
<= n;
reconsider j0 = j as
Element of
NAT by
ORDINAL1:def 12;
A9:
now
per cases ;
case i
= j;
hence (((x
/. i)
* (
Base_FinSeq (n,i)))
. j)
= ((x
/. j)
* ((
Base_FinSeq (n,i))
. j)) by
RVSUM_1: 44;
end;
case i
<> j;
then
A10: ((
Base_FinSeq (n,i))
. j0)
=
0 by
A7,
A8,
MATRIXR2: 76;
(((x
/. i)
* (
Base_FinSeq (n,i)))
. j)
= ((x
/. i)
* ((
Base_FinSeq (n,i))
. j)) by
RVSUM_1: 44
.= ((x
/. j)
* ((
Base_FinSeq (n,i))
. j)) by
A10;
hence (((x
/. i)
* (
Base_FinSeq (n,i)))
. j)
= ((x
/. j)
* ((
Base_FinSeq (n,i))
. j));
end;
end;
((
mlt (x,x2))
. j)
= ((x
. j)
* (x2
. j)) by
RVSUM_1: 59;
hence ((
mlt (x,x2))
. j)
= (((x
/. i)
* (
Base_FinSeq (n,i)))
. j) by
A3,
A7,
A8,
A9,
FINSEQ_4: 15;
end;
(
len ((x
/. i)
* (
Base_FinSeq (n,i))))
= n by
A4,
RVSUM_1: 117;
then (
mlt (x,x2))
= ((x
/. i)
* (
Base_FinSeq (n,i))) by
A5,
A6,
FINSEQ_1: 14;
then (
Sum (
mlt (x,x2)))
= ((x
/. i)
* (
Sum (
Base_FinSeq (n,i)))) by
RVSUM_1: 87
.= ((x
/. i)
* 1) by
A1,
A2,
Th26
.= (x
. i) by
A1,
A2,
A3,
FINSEQ_4: 15;
hence
|(x, (
Base_FinSeq (n,i)))|
= (x
. i);
end;
definition
let n be
Nat;
let x0 be
Element of (
REAL n);
::
EUCLID_7:def12
func
ProjFinSeq x0 ->
FinSequence of (
REAL n) means
:
Def12: (
len it )
= n & for i be
Nat st 1
<= i & i
<= n holds (it
. i)
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i)));
existence
proof
defpred
P[
set,
set] means for i be
Nat st i
= $1 & 1
<= i & i
<= n holds $2
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i)));
A1: for k be
Nat st k
in (
Seg n) holds ex x be
Element of (
REAL n) st
P[k, x]
proof
reconsider n0 = n as
Element of
NAT by
ORDINAL1:def 12;
let k be
Nat;
assume k
in (
Seg n);
reconsider k0 = k as
Element of
NAT by
ORDINAL1:def 12;
reconsider x00 = (
|(x0, (
Base_FinSeq (n0,k0)))|
* (
Base_FinSeq (n0,k0))) as
Element of (
REAL n);
for i be
Nat st i
= k & 1
<= i & i
<= n holds x00
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i)));
hence ex x be
Element of (
REAL n) st
P[k, x];
end;
consider p be
FinSequence of (
REAL n) such that
A2: (
dom p)
= (
Seg n) & for k be
Nat st k
in (
Seg n) holds
P[k, (p
. k)] from
FINSEQ_1:sch 5(
A1);
A3: for i be
Nat st 1
<= i & i
<= n holds (p
. i)
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i))) by
FINSEQ_1: 1,
A2;
(
Seg n)
= (
Seg (
len p)) by
A2,
FINSEQ_1:def 3;
hence thesis by
A3,
FINSEQ_1: 6;
end;
uniqueness
proof
let r1,r2 be
FinSequence of (
REAL n);
assume that
A4: (
len r1)
= n and
A5: for i be
Nat st 1
<= i & i
<= n holds (r1
. i)
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i))) and
A6: (
len r2)
= n and
A7: for i be
Nat st 1
<= i & i
<= n holds (r2
. i)
= (
|(x0, (
Base_FinSeq (n,i)))|
* (
Base_FinSeq (n,i)));
for k be
Nat st 1
<= k & k
<= n holds (r1
. k)
= (r2
. k)
proof
reconsider n22 = n as
Element of
NAT by
ORDINAL1:def 12;
let k be
Nat;
assume that
A8: 1
<= k and
A9: k
<= n;
reconsider k0 = k as
Element of
NAT by
ORDINAL1:def 12;
(r1
. k0)
= (
|(x0, (
Base_FinSeq (n22,k0)))|
* (
Base_FinSeq (n22,k0))) by
A5,
A8,
A9;
hence (r1
. k)
= (r2
. k) by
A7,
A8,
A9;
end;
hence r1
= r2 by
A4,
A6,
FINSEQ_1: 14;
end;
end
theorem ::
EUCLID_7:31
Th30: for x0 be
Element of (
REAL n) holds x0
= (
Sum (
ProjFinSeq x0))
proof
let x0 be
Element of (
REAL n);
set f = (
ProjFinSeq x0);
reconsider n2 = n as
Element of
NAT by
ORDINAL1:def 12;
now
per cases ;
case
A1: (
len f)
>
0 ;
set g2 = (
accum f);
A2: (
len f)
= (
len g2) by
Def10;
A3: (f
. 1)
= (g2
. 1) by
Def10;
defpred
P[
Nat] means for i be
Nat st 1
<= i & i
<= (
len f) & 1
<= $1 & $1
<= (
len f) holds (i
<= $1 implies ((g2
/. $1)
. i)
= (x0
. i)) & (i
> $1 implies ((g2
/. $1)
. i)
=
0 );
A4: (
len f)
= n by
Def12;
A5: (
0
+ 1)
<= (
len f) by
A1,
NAT_1: 13;
A6: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
reconsider k2 = k as
Element of
NAT by
ORDINAL1:def 12;
assume
A7:
P[k];
for i be
Nat st 1
<= i & i
<= (
len f) & 1
<= (k
+ 1) & (k
+ 1)
<= (
len f) holds (i
<= (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
= (x0
. i)) & (i
> (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
=
0 )
proof
let i be
Nat;
assume that
A8: 1
<= i and
A9: i
<= (
len f) and
A10: 1
<= (k
+ 1) and
A11: (k
+ 1)
<= (
len f);
(g2
/. (k
+ 1)) is
Element of (
REAL n);
then
reconsider r = (g2
. (k
+ 1)) as
Element of (
REAL n) by
A2,
A10,
A11,
FINSEQ_4: 15;
reconsider i2 = i as
Element of
NAT by
ORDINAL1:def 12;
A12: (g2
/. (k
+ 1))
= (g2
. (k
+ 1)) by
A2,
A10,
A11,
FINSEQ_4: 15;
now
per cases ;
case
A13: 1
<= k;
reconsider r3 = (f
/. (k
+ 1)) as
Element of (
REAL n);
reconsider r2 = (g2
/. k) as
Element of (
REAL n);
A14: ((
ProjFinSeq x0)
/. (k
+ 1))
= ((
ProjFinSeq x0)
. (k
+ 1)) by
A10,
A11,
FINSEQ_4: 15
.= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
* (
Base_FinSeq (n2,(k
+ 1)))) by
A4,
A10,
A11,
Def12;
A15: k
< (k
+ 1) by
XREAL_1: 29;
then
A16: k
< (
len f) by
A11,
XXREAL_0: 2;
then r
= ((g2
/. k)
+ (f
/. (k
+ 1))) by
Def10,
A13;
then
A17: (r
. i)
= ((r2
. i)
+ (r3
. i)) by
RVSUM_1: 11;
A18:
now
assume
A19: i
<= (k
+ 1);
per cases by
A19,
XXREAL_0: 1;
suppose
A20: i
< (k
+ 1);
then
A21: i
<= k by
NAT_1: 13;
((f
/. (k
+ 1))
. i)
= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
* ((
Base_FinSeq (n2,(k2
+ 1)))
. i2)) by
A14,
RVSUM_1: 44
.= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
*
0 ) by
A4,
A8,
A9,
A20,
MATRIXR2: 76;
hence ((g2
/. (k
+ 1))
. i)
= (x0
. i) by
A7,
A8,
A9,
A12,
A13,
A16,
A17,
A21;
end;
suppose
A22: i
= (k
+ 1);
then
A23: ((g2
/. k)
. i)
=
0 by
A7,
A8,
A9,
A13,
A15,
A16;
((f
/. (k
+ 1))
. i)
= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
* ((
Base_FinSeq (n2,(k2
+ 1)))
. i2)) by
A14,
RVSUM_1: 44
.= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
* 1) by
A4,
A8,
A9,
A22,
MATRIXR2: 75
.= (x0
. (k
+ 1)) by
A4,
A10,
A11,
Th29;
hence ((g2
/. (k
+ 1))
. i)
= (x0
. i) by
A2,
A8,
A9,
A17,
A22,
A23,
FINSEQ_4: 15;
end;
end;
now
assume
A24: i
> (k
+ 1);
then
A25: i
> k by
A15,
XXREAL_0: 2;
((f
/. (k
+ 1))
. i)
= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
* ((
Base_FinSeq (n2,(k2
+ 1)))
. i2)) by
A14,
RVSUM_1: 44
.= (
|(x0, (
Base_FinSeq (n2,(k
+ 1))))|
*
0 ) by
A4,
A8,
A9,
A24,
MATRIXR2: 76
.=
0 ;
hence ((g2
/. (k
+ 1))
. i)
=
0 by
A7,
A8,
A9,
A12,
A13,
A16,
A17,
A25;
end;
hence (i
<= (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
= (x0
. i)) & (i
> (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
=
0 ) by
A18;
end;
case k
< 1;
then
A26: (k
+ 1)
<= (
0
+ 1) by
NAT_1: 13;
then
A27: k
=
0 by
XREAL_1: 6;
A28:
now
assume
A29: i
> (
0
+ 1);
(g2
/. 1)
= (f
. 1) by
A5,
A2,
A3,
FINSEQ_4: 15;
then ((g2
/. 1)
. i)
= ((
|(x0, (
Base_FinSeq (n2,1)))|
* (
Base_FinSeq (n2,1)))
. i) by
A5,
A4,
Def12
.= (
|(x0, (
Base_FinSeq (n2,1)))|
* ((
Base_FinSeq (n2,1))
. i2)) by
RVSUM_1: 44
.= (
|(x0, (
Base_FinSeq (n2,1)))|
*
0 ) by
A4,
A9,
A29,
MATRIXR2: 76
.=
0 ;
hence ((g2
/. (k
+ 1))
. i)
=
0 by
A27;
end;
A30:
now
assume i
<= (
0
+ 1);
then
A31: i
= 1 by
A8,
XXREAL_0: 1;
(g2
/. 1)
= (f
. 1) by
A5,
A2,
A3,
FINSEQ_4: 15;
then ((g2
/. 1)
. 1)
= ((
|(x0, (
Base_FinSeq (n2,1)))|
* (
Base_FinSeq (n2,1)))
. 1) by
A5,
A4,
Def12
.= (
|(x0, (
Base_FinSeq (n2,1)))|
* ((
Base_FinSeq (n2,1))
. 1)) by
RVSUM_1: 44
.= (
|(x0, (
Base_FinSeq (n2,1)))|
* 1) by
A5,
A4,
MATRIXR2: 75
.= (x0
. 1) by
A5,
A4,
Th29;
hence ((g2
/. (
0
+ 1))
. i)
= (x0
. i) by
A31;
end;
k
<=
0 by
A26,
XREAL_1: 6;
hence (i
<= (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
= (x0
. i)) & (i
> (k
+ 1) implies ((g2
/. (k
+ 1))
. i)
=
0 ) by
A30,
A28;
end;
end;
hence thesis;
end;
hence
P[(k
+ 1)];
end;
reconsider r4 = (g2
/. (
len f)) as
Element of (
REAL n);
A32: (
len x0)
= n by
CARD_1:def 7;
then
A33: (
len x0)
= (
len r4) by
CARD_1:def 7;
A34:
P[
0 ];
A35: for k be
Nat holds
P[k] from
NAT_1:sch 2(
A34,
A6);
for i be
Nat st 1
<= i & i
<= (
len r4) holds ((g2
/. (
len f))
. i)
= (x0
. i)
proof
let i be
Nat;
assume that
A36: 1
<= i and
A37: i
<= (
len r4);
A38: i
<= (
len f) by
A4,
A37,
CARD_1:def 7;
1
<= (
len f) by
A4,
A32,
A33,
A36,
A37,
XXREAL_0: 2;
hence ((g2
/. (
len f))
. i)
= (x0
. i) by
A35,
A36,
A38;
end;
then x0
= (g2
/. (
len f)) by
A33,
FINSEQ_1: 14;
hence x0
= (g2
. (
len f)) by
A5,
A2,
FINSEQ_4: 15;
end;
case (
len f)
<=
0 ;
then
A39: n
=
0 by
Def12;
then x0
= (
<*>
REAL );
hence x0
= (
0* n) by
A39;
end;
end;
hence x0
= (
Sum (
ProjFinSeq x0)) by
Def11;
end;
definition
let n be
Nat;
::
EUCLID_7:def13
func
RN_Base n ->
Subset of (
REAL n) equals { (
Base_FinSeq (n,i)) where i be
Element of
NAT : 1
<= i & i
<= n };
coherence
proof
{ (
Base_FinSeq (n,i)) where i be
Element of
NAT : 1
<= i & i
<= n }
c= (
REAL n)
proof
let x be
object;
assume x
in { (
Base_FinSeq (n,i)) where i be
Element of
NAT : 1
<= i & i
<= n };
then ex i be
Element of
NAT st x
= (
Base_FinSeq (n,i)) & 1
<= i & i
<= n;
hence x
in (
REAL n);
end;
hence thesis;
end;
end
theorem ::
EUCLID_7:32
Th31: for n be non
zero
Nat holds (
RN_Base n)
<>
{}
proof
let n be non
zero
Nat;
(
0
+ 1)
<= n by
NAT_1: 13;
then (
Base_FinSeq (n,1))
in { (
Base_FinSeq (n,i)) where i be
Element of
NAT : 1
<= i & i
<= n };
hence thesis;
end;
registration
cluster (
RN_Base
0 ) ->
empty;
coherence
proof
assume not thesis;
then
consider x be
object such that
A1: x
in (
RN_Base
0 ) by
XBOOLE_0:def 1;
ex i be
Element of
NAT st x
= (
Base_FinSeq (
0 ,i)) & 1
<= i & i
<=
0 by
A1;
hence thesis;
end;
end
registration
let n be non
zero
Nat;
cluster (
RN_Base n) -> non
empty;
coherence by
Th31;
end
registration
let n;
cluster (
RN_Base n) ->
orthogonal_basis;
coherence
proof
set B = (
RN_Base n);
A1: { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
c= B
proof
let y be
object;
assume y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
then ex x be
Element of (
REAL n) st y
= x & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i));
hence y
in B;
end;
B
c= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
proof
let y be
object;
assume y
in B;
then ex i2 be
Element of
NAT st y
= (
Base_FinSeq (n,i2)) & 1
<= i2 & i2
<= n;
hence y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
end;
then
A2: B
= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) } by
A1,
XBOOLE_0:def 10;
thus B is
R-orthogonal
proof
let x,y be
real-valued
FinSequence;
assume that
A3: x
in B and
A4: y
in B and
A5: x
<> y;
A6: ex y0 be
Element of (
REAL n) st y
= y0 & ex i be
Element of
NAT st 1
<= i & i
<= n & y0
= (
Base_FinSeq (n,i)) by
A2,
A4;
ex x0 be
Element of (
REAL n) st x
= x0 & ex i be
Element of
NAT st 1
<= i & i
<= n & x0
= (
Base_FinSeq (n,i)) by
A2,
A3;
hence
|(x, y)|
=
0 by
A5,
A6,
Th28;
end;
thus B is
R-normal
proof
let x be
real-valued
FinSequence;
assume x
in B;
then ex x0 be
Element of (
REAL n) st x
= x0 & ex i be
Element of
NAT st 1
<= i & i
<= n & x0
= (
Base_FinSeq (n,i)) by
A2;
hence
|.x.|
= 1 by
Th27;
end;
let B2 be
R-orthonormal
Subset of (
REAL n);
assume
A7: B
c= B2;
now
assume not B2
c= B;
then
consider x be
object such that
A8: x
in B2 and
A9: not x
in B;
reconsider rx = x as
Element of (
REAL n) by
A8;
A10:
now
assume rx
<> (
0* n);
then
consider i be
Element of
NAT such that
A11: 1
<= i and
A12: i
<= n and
A13: (rx
. i)
<>
0 by
JORDAN2C: 46;
(
Base_FinSeq (n,i))
in B by
A11,
A12;
then
|(rx, (
Base_FinSeq (n,i)))|
=
0 by
A7,
A8,
A9,
Def3;
hence contradiction by
A11,
A12,
A13,
Th29;
end;
|.(
0* n).|
=
0 by
EUCLID: 7;
hence contradiction by
A8,
A10,
Def4;
end;
hence thesis by
A7,
XBOOLE_0:def 10;
end;
end
registration
let n;
cluster
orthogonal_basis for
Subset of (
REAL n);
existence
proof
take (
RN_Base n);
thus thesis;
end;
end
definition
let n;
mode
Orthogonal_Basis of n is
orthogonal_basis
Subset of (
REAL n);
end
registration
let n be non
zero
Nat;
cluster -> non
empty for
Orthogonal_Basis of n;
coherence
proof
let B be
Orthogonal_Basis of n;
assume
A1: B is
empty;
then B
c= (
RN_Base n);
hence contradiction by
A1,
Def6;
end;
end
begin
registration
let n be
Element of
NAT ;
cluster (
REAL-US n) ->
constituted-FinSeqs;
coherence
proof
let a be
Element of (
REAL-US n);
reconsider a as
Element of (
REAL n) by
REAL_NS1:def 6;
a is
FinSequence of
REAL ;
hence thesis;
end;
end
registration
let n be
Element of
NAT ;
cluster ->
real-valued for
Element of (
REAL-US n);
coherence
proof
let a be
Element of (
REAL-US n);
reconsider a as
Element of (
REAL n) by
REAL_NS1:def 6;
a is
FinSequence of
REAL ;
hence thesis;
end;
end
registration
let n be
Element of
NAT ;
let x,y be
VECTOR of (
REAL-US n), a,b be
real-valued
Function;
identify a
+ b with x
+ y when x = a, y = b;
compatibility
proof
assume that
A1: x
= a and
A2: y
= b;
reconsider a1 = a, b1 = b as
Element of (
REAL n) by
A1,
A2,
REAL_NS1:def 6;
thus (x
+ y)
= ((
Euclid_add n)
. (a1,b1)) by
A1,
A2,
REAL_NS1:def 6
.= (a
+ b) by
REAL_NS1:def 1;
end;
end
registration
let n be
Element of
NAT , x be
VECTOR of (
REAL-US n), y be
real-valued
Function, a,b be
Element of
REAL ;
identify b
(#) y with a
* x when x = y, a = b;
compatibility
proof
assume that
A1: x
= y and
A2: a
= b;
reconsider y1 = y as
Element of (
REAL n) by
A1,
REAL_NS1:def 6;
thus (a
* x)
= ((
Euclid_mult n)
. (b,y1)) by
A1,
A2,
REAL_NS1:def 6
.= (b
(#) y) by
REAL_NS1:def 2;
end;
end
registration
let n be
Element of
NAT ;
let x be
VECTOR of (
REAL-US n), a be
real-valued
Function;
identify
- a with
- x when x = a;
compatibility
proof
x is
Element of (
REAL n) by
REAL_NS1:def 6;
then
reconsider xn = x as
Element of (
REAL-NS n) by
REAL_NS1:def 4;
assume
A1: x
= a;
then
reconsider a1 = a as
Element of (
REAL n) by
REAL_NS1:def 6;
thus (
- x)
= (
- xn) by
REAL_NS1: 13
.= (
- a1) by
A1,
REAL_NS1: 4
.= (
- a);
end;
end
registration
let n be
Element of
NAT ;
let x,y be
VECTOR of (
REAL-US n), a,b be
real-valued
Function;
identify a
- b with x
- y when x = a, y = b;
compatibility ;
end
theorem ::
EUCLID_7:33
Th32: for n,j be
Element of
NAT , F be
FinSequence of the
carrier of (
REAL-US n), Bn be
Subset of (
REAL-US n), v0 be
Element of (
REAL-US n), l be
Linear_Combination of Bn st F is
one-to-one & Bn is
R-orthogonal & (
rng F)
= (
Carrier l) & v0
in Bn & j
in (
dom (l
(#) F)) & v0
= (F
. j) holds ((
Euclid_scalar n)
. (v0,(
Sum (l
(#) F))))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0)))
proof
let n,j be
Element of
NAT , F be
FinSequence of the
carrier of (
REAL-US n), Bn be
Subset of (
REAL-US n), v0 be
Element of (
REAL-US n), l be
Linear_Combination of Bn;
assume that
A1: F is
one-to-one and
A2: Bn is
R-orthogonal and
A3: (
rng F)
= (
Carrier l) and
A4: v0
in Bn and
A5: j
in (
dom (l
(#) F)) and
A6: v0
= (F
. j);
A7: (
len (l
(#) F))
= (
len F) by
RLVECT_2:def 7;
then
A8: (
dom (l
(#) F))
= (
Seg (
len F)) by
FINSEQ_1:def 3
.= (
dom F) by
FINSEQ_1:def 3;
reconsider F2 = (l
(#) F) as
FinSequence of the
carrier of (
REAL-US n);
reconsider rv0 = v0 as
Element of (
REAL n) by
REAL_NS1:def 6;
A9: (
Carrier l)
c= Bn by
RLVECT_2:def 6;
A10: (
dom (l
(#) F))
= (
Seg (
len (l
(#) F))) by
FINSEQ_1:def 3;
then
A11: j
<= (
len F) by
A5,
A7,
FINSEQ_1: 1;
consider f be
sequence of the
carrier of (
REAL-US n) such that
A12: (
Sum F2)
= (f
. (
len F2)) and
A13: (f
.
0 )
= (
0. (
REAL-US n)) and
A14: for j2 be
Nat holds for v be
Element of (
REAL-US n) st j2
< (
len F2) & v
= (F2
. (j2
+ 1)) holds (f
. (j2
+ 1))
= ((f
. j2)
+ v) by
RLVECT_1:def 12;
defpred
Q[
Nat] means $1
>= j & $1
<= (
len F) implies ((
Euclid_scalar n)
. (v0,(f
. $1)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0)));
defpred
P[
Nat] means $1
< j implies ((
Euclid_scalar n)
. (v0,(f
. $1)))
=
0 ;
(
0. (
REAL-US n))
= (
0* n) by
REAL_NS1:def 6;
then ((
Euclid_scalar n)
. (v0,(f
.
0 )))
=
|(rv0, (
0* n))| by
A13,
REAL_NS1:def 5
.=
0 by
EUCLID_4: 18;
then
A15:
P[
0 ];
A16: j
in (
Seg (
len F)) by
A5,
A7,
FINSEQ_1:def 3;
then
A17: j
<= (
len F2) by
A7,
FINSEQ_1: 1;
A18: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A19:
P[k];
now
per cases ;
case
A20: k
< (
len F2);
A21: 1
<= (k
+ 1) by
NAT_1: 11;
(k
+ 1)
<= (
len F2) by
A20,
NAT_1: 13;
then (k
+ 1)
in (
Seg (
len F2)) by
A21,
FINSEQ_1: 1;
then (k
+ 1)
in (
dom F2) by
FINSEQ_1:def 3;
then (F2
. (k
+ 1))
in (
rng F2) by
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
REAL-US n);
A22: (f
. (k
+ 1))
= ((f
. k)
+ v) by
A14,
A20;
reconsider rv = v as
Element of (
REAL n) by
REAL_NS1:def 6;
reconsider fk = (f
. k) as
Element of (
REAL n) by
REAL_NS1:def 6;
per cases ;
suppose
A23: (k
+ 1)
< j;
A24: 1
<= (k
+ 1) by
NAT_1: 11;
(k
+ 1)
< (
len F) by
A11,
A23,
XXREAL_0: 2;
then (k
+ 1)
in (
Seg (
len F)) by
A24,
FINSEQ_1: 1;
then
A25: (k
+ 1)
in (
dom F) by
FINSEQ_1:def 3;
then
A26: (F
/. (k
+ 1))
= (F
. (k
+ 1)) by
PARTFUN1:def 6;
then
A27: rv0
<> (F
/. (k
+ 1)) by
A1,
A5,
A6,
A8,
A23,
A25,
FUNCT_1:def 4;
reconsider fk1 = (F
/. (k
+ 1)) as
Element of (
REAL n) by
REAL_NS1:def 6;
A28: k
< (k
+ 1) by
XREAL_1: 29;
A29:
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
A30: (F
/. (k
+ 1))
in (
rng F) by
A25,
A26,
FUNCT_1:def 3;
v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A8,
A25,
RLVECT_2:def 7;
then
|(rv0, rv)|
= ((l
. (F
/. (k
+ 1)))
*
|(rv0, fk1)|) by
EUCLID_4: 22
.= ((l
. (F
/. (k
+ 1)))
*
0 ) by
A2,
A3,
A4,
A9,
A30,
A27
.=
0 ;
then
|(rv0, (fk
+ rv))|
=
0 by
A19,
A23,
A28,
A29,
REAL_NS1:def 5,
XXREAL_0: 2;
hence
P[(k
+ 1)] by
A22,
REAL_NS1:def 5;
end;
suppose (k
+ 1)
>= j;
hence
P[(k
+ 1)];
end;
end;
case
A31: k
>= (
len F2);
(k
+ 1)
> k by
XREAL_1: 29;
then (k
+ 1)
> (
len F2) by
A31,
XXREAL_0: 2;
hence
P[(k
+ 1)] by
A17,
XXREAL_0: 2;
end;
end;
hence
P[(k
+ 1)];
end;
A32: for i be
Nat holds
P[i] from
NAT_1:sch 2(
A15,
A18);
A33: for i be
Nat st i
< j holds ((
Euclid_scalar n)
. (v0,(f
. i)))
=
0 by
A32;
A34: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
A35:
Q[k];
per cases ;
suppose (k
+ 1)
< j;
hence
Q[(k
+ 1)];
end;
suppose
A36: (k
+ 1)
>= j;
per cases by
A36,
XXREAL_0: 1;
suppose
A37: (k
+ 1)
> j;
per cases ;
suppose
A38: (k
+ 1)
<= (
len F2);
1
<= (k
+ 1) by
NAT_1: 11;
then
A39: (k
+ 1)
in (
Seg (
len F2)) by
A38,
FINSEQ_1: 1;
then
A40: (k
+ 1)
in (
dom F) by
A7,
FINSEQ_1:def 3;
then
A41: (F
/. (k
+ 1))
= (F
. (k
+ 1)) by
PARTFUN1:def 6;
then
A42: (F
/. (k
+ 1))
in (
rng F) by
A40,
FUNCT_1:def 3;
(k
+ 1)
in (
dom F2) by
A39,
FINSEQ_1:def 3;
then (F2
. (k
+ 1))
in (
rng F2) by
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
REAL-US n);
reconsider fk1 = (F
/. (k
+ 1)) as
Element of (
REAL n) by
REAL_NS1:def 6;
reconsider fk = (f
. k) as
Element of (
REAL n) by
REAL_NS1:def 6;
k
< (k
+ 1) by
XREAL_1: 29;
then
A43: k
< (
len F2) by
A38,
XXREAL_0: 2;
then
A44: (f
. (k
+ 1))
= ((f
. k)
+ v) by
A14;
A45: rv0
<> (F
/. (k
+ 1)) by
A1,
A5,
A6,
A8,
A37,
A40,
A41,
FUNCT_1:def 4;
reconsider rv = v as
Element of (
REAL n) by
REAL_NS1:def 6;
v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A8,
A40,
RLVECT_2:def 7;
then
A46:
|(rv0, rv)|
= ((l
. (F
/. (k
+ 1)))
*
|(rv0, fk1)|) by
EUCLID_4: 22
.= ((l
. (F
/. (k
+ 1)))
*
0 ) by
A2,
A3,
A4,
A9,
A42,
A45
.=
0 ;
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
then
|(rv0, (fk
+ rv))|
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A35,
A37,
A43,
A46,
NAT_1: 13,
REAL_NS1:def 5,
RLVECT_2:def 7;
hence
Q[(k
+ 1)] by
A44,
REAL_NS1:def 5;
end;
suppose (k
+ 1)
> (
len F2);
hence
Q[(k
+ 1)] by
RLVECT_2:def 7;
end;
end;
suppose
A47: (k
+ 1)
= j;
then (F2
. (k
+ 1))
in (
rng F2) by
A5,
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
REAL-US n);
reconsider rv = v as
Element of (
REAL n) by
REAL_NS1:def 6;
A48: v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A5,
A47,
RLVECT_2:def 7;
(k
+ 1)
in (
dom F) by
A5,
A10,
A7,
A47,
FINSEQ_1:def 3;
then
A49:
|(rv0, rv)|
=
|(rv0, ((l
. (F
/. j))
* rv0))| by
A6,
A47,
A48,
PARTFUN1:def 6
.= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
REAL_NS1:def 5;
k
< (k
+ 1) by
XREAL_1: 29;
then k
< (
len F2) by
A7,
A11,
A47,
XXREAL_0: 2;
then
A50: (f
. (k
+ 1))
= ((f
. k)
+ v) by
A14;
reconsider fk = (f
. k) as
Element of (
REAL n) by
REAL_NS1:def 6;
((
Euclid_scalar n)
. (v0,(f
. k)))
=
0 by
A33,
A47,
XREAL_1: 29;
then
A51:
|(rv0, fk)|
=
0 by
REAL_NS1:def 5;
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
hence
Q[(k
+ 1)] by
A50,
A51,
A49,
REAL_NS1:def 5;
end;
end;
end;
A52:
Q[
0 ] by
A16,
FINSEQ_1: 1;
A53: for i be
Nat holds
Q[i] from
NAT_1:sch 2(
A52,
A34);
for i be
Nat st i
>= j & i
<= (
len F) holds ((
Euclid_scalar n)
. (v0,(f
. i)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A53;
hence thesis by
A12,
A7,
A11;
end;
theorem ::
EUCLID_7:34
Th33: for n be
Element of
NAT , f be
FinSequence of (
REAL n), g be
FinSequence of the
carrier of (
REAL-US n) st f
= g holds (
Sum f)
= (
Sum g)
proof
let n be
Element of
NAT , f be
FinSequence of (
REAL n), g be
FinSequence of the
carrier of (
REAL-US n);
set V = (
REAL-US n);
assume
A1: f
= g;
now
per cases ;
case
A2: (
len f)
>
0 ;
set g2 = (
accum f);
A3: (
len f)
= (
len g2) by
Def10;
A4: (f
. 1)
= (g2
. 1) by
Def10;
A5: (
Sum f)
= (g2
. (
len f)) by
A2,
Def11;
deffunc
F(
set) = (
IFIN ($1,((
len f)
+ 1),(
IFEQ ($1,
0 ,(
0. V),(g2
/. $1))),(
0. V)));
A6: for x be
set st x
in
NAT holds
F(x)
in the
carrier of V
proof
let x be
set;
assume x
in
NAT ;
then
reconsider nx = x as
Element of
NAT ;
per cases ;
suppose nx
in ((
len f)
+ 1);
then
A7:
F(x)
= (
IFEQ (x,
0 ,(
0. V),(g2
/. nx))) by
MATRIX_7:def 1;
per cases ;
suppose x
=
0 ;
then
F(x)
= (
0. V) by
A7,
FUNCOP_1:def 8;
hence
F(x)
in the
carrier of V;
end;
suppose
A8: x
<>
0 ;
A9: the
carrier of V
= (
REAL n) by
REAL_NS1:def 6;
F(x)
= (g2
/. nx) by
A7,
A8,
FUNCOP_1:def 8;
hence
F(x)
in the
carrier of V by
A9;
end;
end;
suppose not nx
in ((
len f)
+ 1);
then
F(x)
= (
0. V) by
MATRIX_7:def 1;
hence
F(x)
in the
carrier of V;
end;
end;
consider f3 be
sequence of the
carrier of V such that
A10: for x be
set st x
in
NAT holds (f3
. x)
=
F(x) from
FUNCT_2:sch 11(
A6);
A11: for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f3
. (j
+ 1))
= ((f3
. j)
+ v)
proof
let j be
Nat;
A12: j
in
NAT by
ORDINAL1:def 12;
let v be
Element of V;
assume that
A13: j
< (
len g) and
A14: v
= (g
. (j
+ 1));
A15: (j
+ 1)
<= (
len f) by
A1,
A13,
NAT_1: 13;
per cases ;
suppose
A16: j
=
0 ;
then (j
+ 1)
< ((
len f)
+ 1) by
A2,
XREAL_1: 6;
then
A17: (j
+ 1)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A18:
0
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A19: (f3
. j)
= (
IFIN (j,((
len f)
+ 1),(
IFEQ (j,
0 ,(
0. V),(g2
/. j))),(
0. V))) by
A10,
A12
.= (
IFEQ (j,
0 ,(
0. V),(g2
/. j))) by
A16,
A18,
MATRIX_7:def 1
.= (
0. V) by
A16,
FUNCOP_1:def 8;
thus (f3
. (j
+ 1))
= (
IFIN ((j
+ 1),((
len f)
+ 1),(
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))),(
0. V))) by
A10
.= (
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))) by
A17,
MATRIX_7:def 1
.= (g2
/. 1) by
A16,
FUNCOP_1:def 8
.= (g
. (j
+ 1)) by
A1,
A3,
A4,
A15,
A16,
FINSEQ_4: 15
.= ((f3
. j)
+ v) by
A14,
A19,
RLVECT_1: 4;
end;
suppose
A20: j
<>
0 ;
(
len f)
< ((
len f)
+ 1) by
XREAL_1: 29;
then j
< ((
len f)
+ 1) by
A1,
A13,
XXREAL_0: 2;
then
A21: j
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A22: (f3
. j)
= (
IFIN (j,((
len f)
+ 1),(
IFEQ (j,
0 ,(
0. V),(g2
/. j))),(
0. V))) by
A10,
A12
.= (
IFEQ (j,
0 ,(
0. V),(g2
/. j))) by
A21,
MATRIX_7:def 1
.= (g2
/. j) by
A20,
FUNCOP_1:def 8;
A23: (
0
+ 1)
<= (j
+ 1) by
NAT_1: 13;
A24: (
0
+ 1)
<= j by
A20,
NAT_1: 13;
(j
+ 1)
< ((
len f)
+ 1) by
A1,
A13,
XREAL_1: 6;
then
A25: (j
+ 1)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
thus (f3
. (j
+ 1))
= (
IFIN ((j
+ 1),((
len f)
+ 1),(
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))),(
0. V))) by
A10
.= (
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))) by
A25,
MATRIX_7:def 1
.= (g2
/. (j
+ 1)) by
FUNCOP_1:def 8
.= (g2
. (j
+ 1)) by
A3,
A15,
A23,
FINSEQ_4: 15
.= ((g2
/. j)
+ (f
/. (j
+ 1))) by
A1,
Def10,
A13,
A24
.= ((f3
. j)
+ v) by
A1,
A14,
A15,
A23,
A22,
FINSEQ_4: 15;
end;
end;
(
len f)
< ((
len f)
+ 1) by
XREAL_1: 29;
then
A26: (
len f)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A27: (
0
+ 1)
<= (
len f) by
A2,
NAT_1: 13;
A28:
0
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A29: (f3
.
0 )
= (
IFIN (
0 ,((
len f)
+ 1),(
IFEQ (
0 ,
0 ,(
0. V),(g2
/.
0 ))),(
0. V))) by
A10
.= (
IFEQ (
0 ,
0 ,(
0. V),(g2
/.
0 ))) by
A28,
MATRIX_7:def 1
.= (
0. V) by
FUNCOP_1:def 8;
(f3
. (
len g))
=
F(len) by
A1,
A10
.= (
IFEQ ((
len f),
0 ,(
0. V),(g2
/. (
len f)))) by
A26,
MATRIX_7:def 1
.= (g2
/. (
len f)) by
A2,
FUNCOP_1:def 8
.= (
Sum f) by
A3,
A5,
A27,
FINSEQ_4: 15;
hence ex f2 be
sequence of the
carrier of V st (
Sum f)
= (f2
. (
len g)) & (f2
.
0 )
= (
0. V) & for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f2
. (j
+ 1))
= ((f2
. j)
+ v) by
A29,
A11;
end;
case
A30: (
len f)
<=
0 ;
set f3 = (
NAT
--> (
0. V));
A31: for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f3
. (j
+ 1))
= ((f3
. j)
+ v) by
A1,
A30;
A32: (f3
. (
len g))
= (
0. V) by
FUNCOP_1: 7
.= (
0* n) by
REAL_NS1:def 6;
A33: (f3
.
0 )
= (
0. V) by
FUNCOP_1: 7;
(
Sum f)
= (
0* n) by
A30,
Def11;
hence ex f2 be
sequence of the
carrier of V st (
Sum f)
= (f2
. (
len g)) & (f2
.
0 )
= (
0. V) & for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f2
. (j
+ 1))
= ((f2
. j)
+ v) by
A32,
A33,
A31;
end;
end;
hence (
Sum f)
= (
Sum g) by
RLVECT_1:def 12;
end;
registration
let A be
set;
cluster (
RealVectSpace A) ->
constituted-Functions;
coherence ;
end
registration
let n;
cluster (
RealVectSpace (
Seg n)) ->
constituted-FinSeqs;
coherence
proof
let a be
Element of (
RealVectSpace (
Seg n));
a is
Element of (
REAL n) by
FINSEQ_2: 93;
hence thesis;
end;
end
registration
let A be
set;
cluster ->
real-valued for
Element of (
RealVectSpace A);
coherence
proof
let a be
Element of (
RealVectSpace A);
a is
Function of A,
REAL or a is
empty by
FUNCT_2: 66;
hence thesis;
end;
end
registration
let A be
set;
let x,y be
VECTOR of (
RealVectSpace A), a,b be
real-valued
Function;
identify a
+ b with x
+ y when x = a, y = b;
compatibility
proof
A1: (
dom y)
= A by
FUNCT_2: 92;
assume that
A2: x
= a and
A3: y
= b;
(
dom (x
+ y))
= A by
FUNCT_2: 92;
then
A4: for c be
object st c
in (
dom (x
+ y)) holds ((x
+ y)
. c)
= ((a
. c)
+ (b
. c)) by
A2,
A3,
FUNCSDOM: 1;
(
dom x)
= A by
FUNCT_2: 92;
then (
dom (x
+ y))
= ((
dom a)
/\ (
dom b)) by
A2,
A3,
A1,
FUNCT_2: 92;
hence thesis by
A4,
VALUED_1:def 1;
end;
end
registration
let A be
set;
let x be
VECTOR of (
RealVectSpace A), y be
real-valued
Function, a,b be
Element of
REAL ;
identify b
(#) y with a
* x when x = y, a = b;
compatibility
proof
A1: (
dom x)
= A by
FUNCT_2: 92;
assume that
A2: x
= y and
A3: a
= b;
A4: (
dom (a
* x))
= A by
FUNCT_2: 92;
then for c be
object st c
in (
dom (a
* x)) holds ((a
* x)
. c)
= (b
* (y
. c)) by
A2,
A3,
FUNCSDOM: 4;
hence thesis by
A2,
A4,
A1,
VALUED_1:def 5;
end;
end
registration
let A be
set;
let x be
VECTOR of (
RealVectSpace A), a be
real-valued
Function;
identify
- a with
- x when x = a;
compatibility
proof
assume
A1: x
= a;
A2:
now
let c be
object;
assume c
in (
dom a);
reconsider jj = 1 as
Element of
REAL by
XREAL_0:def 1;
thus ((
- x)
. c)
= (((
- jj)
* x)
. c) by
RLVECT_1: 16
.= ((
- jj)
* (a
. c)) by
A1,
VALUED_1: 6
.= (
- (a
. c));
end;
(
dom (
- x))
= A by
FUNCT_2: 92;
hence thesis by
A1,
A2,
FUNCT_2: 92,
VALUED_1: 9;
end;
end
registration
let A be
set;
let x,y be
VECTOR of (
RealVectSpace A), a,b be
real-valued
Function;
identify a
- b with x
- y when x = a, y = b;
compatibility ;
end
theorem ::
EUCLID_7:35
Th34: for X be
Subspace of (
RealVectSpace (
Seg n)), x be
Element of (
REAL n), a be
Real st x
in the
carrier of X holds (a
* x)
in the
carrier of X
proof
let X be
Subspace of (
RealVectSpace (
Seg n)), x be
Element of (
REAL n), a be
Real;
reconsider x1 = x as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
assume x
in the
carrier of X;
then
A1: x
in X;
reconsider aa = a as
Element of
REAL by
XREAL_0:def 1;
(aa
* x)
= (aa
* x1);
then (a
* x)
in X by
A1,
RLSUB_1: 21;
hence (a
* x)
in the
carrier of X;
end;
theorem ::
EUCLID_7:36
Th35: for X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n) st x
in the
carrier of X & y
in the
carrier of X holds (x
+ y)
in the
carrier of X
proof
let X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n);
assume that
A1: x
in the
carrier of X and
A2: y
in the
carrier of X;
A3: y
in X by
A2;
reconsider x1 = x, y1 = y as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
A4: (x1
+ y1)
= (x
+ y);
x
in X by
A1;
then (x
+ y)
in X by
A3,
A4,
RLSUB_1: 20;
hence (x
+ y)
in the
carrier of X;
end;
theorem ::
EUCLID_7:37
for X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a,b be
Real st x
in the
carrier of X & y
in the
carrier of X holds ((a
* x)
+ (b
* y))
in the
carrier of X
proof
let X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a,b be
Real;
assume that
A1: x
in the
carrier of X and
A2: y
in the
carrier of X;
A3: (b
* y)
in the
carrier of X by
A2,
Th34;
(a
* x)
in the
carrier of X by
A1,
Th34;
hence ((a
* x)
+ (b
* y))
in the
carrier of X by
A3,
Th35;
end;
Lm4: for X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a be
Real st x
in the
carrier of X & y
in the
carrier of X holds ((a
* x)
+ y)
in the
carrier of X
proof
let X be
Subspace of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a be
Real;
assume that
A1: x
in the
carrier of X and
A2: y
in the
carrier of X;
(a
* x)
in the
carrier of X by
A1,
Th34;
hence ((a
* x)
+ y)
in the
carrier of X by
A2,
Th35;
end;
theorem ::
EUCLID_7:38
for u,v be
Element of (
REAL n) holds ((
Euclid_scalar n)
. (u,v))
=
|(u, v)| by
REAL_NS1:def 5;
theorem ::
EUCLID_7:39
Th38: for F be
FinSequence of the
carrier of (
RealVectSpace (
Seg n)), Bn be
Subset of (
RealVectSpace (
Seg n)), v0 be
Element of (
RealVectSpace (
Seg n)), l be
Linear_Combination of Bn st F is
one-to-one & Bn is
R-orthogonal & (
rng F)
= (
Carrier l) & v0
in Bn & j
in (
dom (l
(#) F)) & v0
= (F
. j) holds ((
Euclid_scalar n)
. (v0,(
Sum (l
(#) F))))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0)))
proof
let F be
FinSequence of the
carrier of (
RealVectSpace (
Seg n)), Bn be
Subset of (
RealVectSpace (
Seg n)), v0 be
Element of (
RealVectSpace (
Seg n)), l be
Linear_Combination of Bn;
assume that
A1: F is
one-to-one and
A2: Bn is
R-orthogonal and
A3: (
rng F)
= (
Carrier l) and
A4: v0
in Bn and
A5: j
in (
dom (l
(#) F)) and
A6: v0
= (F
. j);
A7: (
len (l
(#) F))
= (
len F) by
RLVECT_2:def 7;
then
A8: (
dom (l
(#) F))
= (
Seg (
len F)) by
FINSEQ_1:def 3
.= (
dom F) by
FINSEQ_1:def 3;
reconsider F2 = (l
(#) F) as
FinSequence of the
carrier of (
RealVectSpace (
Seg n));
reconsider rv0 = v0 as
Element of (
REAL n) by
FINSEQ_2: 93;
consider f be
sequence of the
carrier of (
RealVectSpace (
Seg n)) such that
A9: (
Sum F2)
= (f
. (
len F2)) and
A10: (f
.
0 )
= (
0. (
RealVectSpace (
Seg n))) and
A11: for j2 be
Nat holds for v be
Element of (
RealVectSpace (
Seg n)) st j2
< (
len F2) & v
= (F2
. (j2
+ 1)) holds (f
. (j2
+ 1))
= ((f
. j2)
+ v) by
RLVECT_1:def 12;
defpred
Q[
Nat] means $1
>= j & $1
<= (
len F) implies ((
Euclid_scalar n)
. (v0,(f
. $1)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0)));
defpred
P[
Nat] means $1
< j implies ((
Euclid_scalar n)
. (v0,(f
. $1)))
=
0 ;
((
Euclid_scalar n)
. (v0,(f
.
0 )))
=
|(rv0, (
0* n))| by
A10,
REAL_NS1:def 5
.=
0 by
EUCLID_4: 18;
then
A12:
P[
0 ];
A13: (
dom (l
(#) F))
= (
Seg (
len (l
(#) F))) by
FINSEQ_1:def 3;
then
A14: j
<= (
len F) by
A5,
A7,
FINSEQ_1: 1;
A15: (
Carrier l)
c= Bn by
RLVECT_2:def 6;
A16: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A17:
P[k];
now
per cases ;
case
A18: k
< (
len F2);
A19: 1
<= (k
+ 1) by
NAT_1: 11;
(k
+ 1)
<= (
len F2) by
A18,
NAT_1: 13;
then (k
+ 1)
in (
Seg (
len F2)) by
A19,
FINSEQ_1: 1;
then (k
+ 1)
in (
dom F2) by
FINSEQ_1:def 3;
then (F2
. (k
+ 1))
in (
rng F2) by
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
RealVectSpace (
Seg n));
reconsider rv = v as
Element of (
REAL n) by
FINSEQ_2: 93;
reconsider fk = (f
. k) as
Element of (
REAL n) by
FINSEQ_2: 93;
per cases ;
suppose
A20: (k
+ 1)
< j;
A21: 1
<= (k
+ 1) by
NAT_1: 11;
(k
+ 1)
< (
len F) by
A14,
A20,
XXREAL_0: 2;
then (k
+ 1)
in (
Seg (
len F)) by
A21,
FINSEQ_1: 1;
then
A22: (k
+ 1)
in (
dom F) by
FINSEQ_1:def 3;
then
A23: (F
/. (k
+ 1))
= (F
. (k
+ 1)) by
PARTFUN1:def 6;
then
A24: rv0
<> (F
/. (k
+ 1)) by
A1,
A5,
A6,
A8,
A20,
A22,
FUNCT_1:def 4;
k
< (k
+ 1) by
XREAL_1: 29;
then
A25:
|(rv0, fk)|
=
0 by
A17,
A20,
REAL_NS1:def 5,
XXREAL_0: 2;
reconsider fk1 = (F
/. (k
+ 1)) as
Element of (
REAL n) by
FINSEQ_2: 93;
A26:
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
A27: (F
/. (k
+ 1))
in (
rng F) by
A22,
A23,
FUNCT_1:def 3;
v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A8,
A22,
RLVECT_2:def 7;
then
|(rv0, rv)|
= ((l
. (F
/. (k
+ 1)))
*
|(rv0, fk1)|) by
EUCLID_4: 22
.= ((l
. (F
/. (k
+ 1)))
*
0 ) by
A2,
A3,
A4,
A15,
A27,
A24
.=
0 ;
then ((
Euclid_scalar n)
. (v0,((f
. k)
+ v)))
=
0 by
A25,
A26,
REAL_NS1:def 5;
hence
P[(k
+ 1)] by
A11,
A18;
end;
suppose (k
+ 1)
>= j;
hence
P[(k
+ 1)];
end;
end;
case
A28: k
>= (
len F2);
(k
+ 1)
> k by
XREAL_1: 29;
then (k
+ 1)
> (
len F2) by
A28,
XXREAL_0: 2;
hence
P[(k
+ 1)] by
A7,
A14,
XXREAL_0: 2;
end;
end;
hence
P[(k
+ 1)];
end;
A29: for i be
Nat holds
P[i] from
NAT_1:sch 2(
A12,
A16);
A30: for i be
Nat st i
< j holds ((
Euclid_scalar n)
. (v0,(f
. i)))
=
0 by
A29;
A31: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
A32:
Q[k];
per cases ;
suppose (k
+ 1)
< j;
hence
Q[(k
+ 1)];
end;
suppose
A33: (k
+ 1)
>= j;
per cases by
A33,
XXREAL_0: 1;
suppose
A34: (k
+ 1)
> j;
per cases ;
suppose
A35: (k
+ 1)
<= (
len F2);
1
<= (k
+ 1) by
NAT_1: 11;
then
A36: (k
+ 1)
in (
Seg (
len F2)) by
A35,
FINSEQ_1: 1;
then
A37: (k
+ 1)
in (
dom F) by
A7,
FINSEQ_1:def 3;
then
A38: (F
/. (k
+ 1))
= (F
. (k
+ 1)) by
PARTFUN1:def 6;
then
A39: (F
/. (k
+ 1))
in (
rng F) by
A37,
FUNCT_1:def 3;
(k
+ 1)
in (
dom F2) by
A36,
FINSEQ_1:def 3;
then (F2
. (k
+ 1))
in (
rng F2) by
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
RealVectSpace (
Seg n));
reconsider rv = v as
Element of (
REAL n) by
FINSEQ_2: 93;
reconsider fk1 = (F
/. (k
+ 1)) as
Element of (
REAL n) by
FINSEQ_2: 93;
reconsider fk = (f
. k) as
Element of (
REAL n) by
FINSEQ_2: 93;
A40:
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
A41: rv0
<> (F
/. (k
+ 1)) by
A1,
A5,
A6,
A8,
A34,
A37,
A38,
FUNCT_1:def 4;
v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A8,
A37,
RLVECT_2:def 7;
then
A42:
|(rv0, rv)|
= ((l
. (F
/. (k
+ 1)))
*
|(rv0, fk1)|) by
EUCLID_4: 22
.= ((l
. (F
/. (k
+ 1)))
*
0 ) by
A2,
A3,
A4,
A15,
A39,
A41
.=
0 ;
k
< (k
+ 1) by
XREAL_1: 29;
then
A43: k
< (
len F2) by
A35,
XXREAL_0: 2;
then
|(rv0, fk)|
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A32,
A34,
NAT_1: 13,
REAL_NS1:def 5,
RLVECT_2:def 7;
then ((
Euclid_scalar n)
. (v0,((f
. k)
+ v)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A40,
A42,
REAL_NS1:def 5;
hence
Q[(k
+ 1)] by
A11,
A43;
end;
suppose (k
+ 1)
> (
len F2);
hence
Q[(k
+ 1)] by
RLVECT_2:def 7;
end;
end;
suppose
A44: (k
+ 1)
= j;
then (F2
. (k
+ 1))
in (
rng F2) by
A5,
FUNCT_1:def 3;
then
reconsider v = (F2
. (k
+ 1)) as
Element of (
RealVectSpace (
Seg n));
A45: v
= ((l
. (F
/. (k
+ 1)))
* (F
/. (k
+ 1))) by
A5,
A44,
RLVECT_2:def 7;
reconsider rv = v as
Element of (
REAL n) by
FINSEQ_2: 93;
(k
+ 1)
in (
dom F) by
A5,
A13,
A7,
A44,
FINSEQ_1:def 3;
then
A46:
|(rv0, rv)|
=
|(rv0, ((l
. (F
/. j))
* rv0))| by
A6,
A44,
A45,
PARTFUN1:def 6
.= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
REAL_NS1:def 5;
reconsider fk = (f
. k) as
Element of (
REAL n) by
FINSEQ_2: 93;
((
Euclid_scalar n)
. (v0,(f
. k)))
=
0 by
A30,
A44,
XREAL_1: 29;
then
A47:
|(rv0, fk)|
=
0 by
REAL_NS1:def 5;
k
< (k
+ 1) by
XREAL_1: 29;
then
A48: k
< (
len F2) by
A7,
A14,
A44,
XXREAL_0: 2;
|(rv0, (fk
+ rv))|
= (
|(rv0, fk)|
+
|(rv0, rv)|) by
EUCLID_4: 28;
then ((
Euclid_scalar n)
. (v0,((f
. k)
+ v)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A47,
A46,
REAL_NS1:def 5;
hence
Q[(k
+ 1)] by
A11,
A48;
end;
end;
end;
A49:
Q[
0 ] by
A5,
A13,
FINSEQ_1: 1;
A50: for i be
Nat holds
Q[i] from
NAT_1:sch 2(
A49,
A31);
for i be
Nat st i
>= j & i
<= (
len F) holds ((
Euclid_scalar n)
. (v0,(f
. i)))
= ((
Euclid_scalar n)
. (v0,((l
. (F
/. j))
* v0))) by
A50;
hence thesis by
A9,
A7,
A14;
end;
registration
let n;
cluster
R-orthonormal ->
linearly-independent for
Subset of (
RealVectSpace (
Seg n));
coherence
proof
let Bn be
Subset of (
RealVectSpace (
Seg n));
assume
A1: Bn is
R-orthonormal;
let l be
Linear_Combination of Bn;
assume
A2: (
Sum l)
= (
0. (
RealVectSpace (
Seg n)));
set v0 = the
Element of (
Carrier l);
consider F be
FinSequence of the
carrier of (
RealVectSpace (
Seg n)) such that
A3: F is
one-to-one and
A4: (
rng F)
= (
Carrier l) and
A5: (
Sum l)
= (
Sum (l
(#) F)) by
RLVECT_2:def 8;
assume
A6: (
Carrier l)
<>
{} ;
then
A7: v0
in (
Carrier l);
then v0
in { v2 where v2 be
Element of (
RealVectSpace (
Seg n)) : (l
. v2)
<>
0 } by
RLVECT_2:def 4;
then
consider v be
Element of (
RealVectSpace (
Seg n)) such that
A8: v0
= v and (l
. v)
<>
0 ;
reconsider xv = v as
Element of (
REAL n) by
FINSEQ_2: 93;
A9: ((
Euclid_scalar n)
. (v,(
Sum (l
(#) F))))
=
|(xv, (
0* n))| by
A2,
A5,
REAL_NS1:def 5
.=
0 by
EUCLID_4: 18;
consider x0 be
object such that
A10: x0
in (
dom F) and
A11: v0
= (F
. x0) by
A6,
A4,
FUNCT_1:def 3;
reconsider nx0 = x0 as
Element of
NAT by
A10;
(F
. x0)
= (F
/. x0) by
A10,
PARTFUN1:def 6;
then (F
/. nx0)
in (
rng F) by
A10,
FUNCT_1:def 3;
then (F
/. nx0)
in { v3 where v3 be
Element of (
RealVectSpace (
Seg n)) : (l
. v3)
<>
0 } by
A4,
RLVECT_2:def 4;
then
A12: ex v3 be
Element of (
RealVectSpace (
Seg n)) st v3
= (F
/. nx0) & (l
. v3)
<>
0 ;
A13: (
dom F)
= (
Seg (
len F)) by
FINSEQ_1:def 3;
A14: (
Carrier l)
c= Bn by
RLVECT_2:def 6;
(
len (l
(#) F))
= (
len F) by
RLVECT_2:def 7;
then nx0
in (
dom (l
(#) F)) by
A10,
A13,
FINSEQ_1:def 3;
then ((
Euclid_scalar n)
. (v,(
Sum (l
(#) F))))
= ((
Euclid_scalar n)
. (v,((l
. (F
/. nx0))
* v))) by
A1,
A14,
A7,
A8,
A3,
A4,
A11,
Th38
.=
|(xv, ((l
. (F
/. nx0))
* xv))| by
REAL_NS1:def 5
.= ((l
. (F
/. nx0))
*
|(xv, xv)|) by
EUCLID_4: 22
.= ((l
. (F
/. nx0))
* (
|.xv.|
^2 )) by
EUCLID_2: 4
.= ((l
. (F
/. nx0))
* (1
^2 )) by
A1,
A14,
A7,
A8,
Def4
.= (l
. (F
/. nx0));
hence contradiction by
A12,
A9;
end;
end
registration
let n be
Element of
NAT ;
cluster
R-orthonormal ->
linearly-independent for
Subset of (
REAL-US n);
coherence
proof
let Bn be
Subset of (
REAL-US n);
assume
A1: Bn is
R-orthonormal;
let l be
Linear_Combination of Bn;
assume
A2: (
Sum l)
= (
0. (
REAL-US n));
set v0 = the
Element of (
Carrier l);
consider F be
FinSequence of the
carrier of (
REAL-US n) such that
A3: F is
one-to-one and
A4: (
rng F)
= (
Carrier l) and
A5: (
Sum l)
= (
Sum (l
(#) F)) by
RLVECT_2:def 8;
assume
A6: (
Carrier l)
<>
{} ;
then
A7: v0
in (
Carrier l);
then v0
in { v2 where v2 be
Element of (
REAL-US n) : (l
. v2)
<>
0 } by
RLVECT_2:def 4;
then
consider v be
Element of (
REAL-US n) such that
A8: v0
= v and (l
. v)
<>
0 ;
reconsider xv = v as
Element of (
REAL n) by
REAL_NS1:def 6;
(
0. (
REAL-US n))
= (
0* n) by
REAL_NS1:def 6;
then
A9: ((
Euclid_scalar n)
. (v,(
Sum (l
(#) F))))
=
|(xv, (
0* n))| by
A2,
A5,
REAL_NS1:def 5
.=
0 by
EUCLID_4: 18;
consider x0 be
object such that
A10: x0
in (
dom F) and
A11: v0
= (F
. x0) by
A6,
A4,
FUNCT_1:def 3;
reconsider nx0 = x0 as
Element of
NAT by
A10;
(F
. x0)
= (F
/. x0) by
A10,
PARTFUN1:def 6;
then (F
/. nx0)
in (
rng F) by
A10,
FUNCT_1:def 3;
then (F
/. nx0)
in { v3 where v3 be
Element of (
REAL-US n) : (l
. v3)
<>
0 } by
A4,
RLVECT_2:def 4;
then
A12: ex v3 be
Element of (
REAL-US n) st v3
= (F
/. nx0) & (l
. v3)
<>
0 ;
A13: (
dom F)
= (
Seg (
len F)) by
FINSEQ_1:def 3;
A14: (
Carrier l)
c= Bn by
RLVECT_2:def 6;
(
len (l
(#) F))
= (
len F) by
RLVECT_2:def 7;
then nx0
in (
dom (l
(#) F)) by
A10,
A13,
FINSEQ_1:def 3;
then ((
Euclid_scalar n)
. (v,(
Sum (l
(#) F))))
= ((
Euclid_scalar n)
. (v,((l
. (F
/. nx0))
* v))) by
A1,
A14,
A7,
A8,
A3,
A4,
A11,
Th32
.=
|(xv, ((l
. (F
/. nx0))
* xv))| by
REAL_NS1:def 5
.= ((l
. (F
/. nx0))
*
|(xv, xv)|) by
EUCLID_4: 22
.= ((l
. (F
/. nx0))
* (
|.xv.|
^2 )) by
EUCLID_2: 4
.= ((l
. (F
/. nx0))
* (1
^2 )) by
A1,
A14,
A7,
A8,
Def4
.= (l
. (F
/. nx0));
hence contradiction by
A12,
A9;
end;
end
theorem ::
EUCLID_7:40
Th39: for Bn be
Subset of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a be
Real st Bn is
linearly-independent & x
in Bn & y
in Bn & y
= (a
* x) holds x
= y
proof
let Bn be
Subset of (
RealVectSpace (
Seg n)), x,y be
Element of (
REAL n), a be
Real;
assume that
A1: Bn is
linearly-independent and
A2: x
in Bn and
A3: y
in Bn and
A4: y
= (a
* x);
reconsider x0 = x, y0 = y as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
reconsider A =
{y0, x0} as
Subset of (
RealVectSpace (
Seg n));
A
c= Bn by
A2,
A3,
TARSKI:def 2;
then
A5: A is
linearly-independent by
A1,
RLVECT_3: 5;
reconsider aa = a as
Element of
REAL by
XREAL_0:def 1;
y0
= (aa
* x0) by
A4;
hence x
= y by
A5,
RLVECT_3: 12;
end;
Lm5:
now
let n;
thus (
RN_Base n) is
finite & (
card (
RN_Base n))
= n
proof
per cases ;
suppose n is
zero;
hence thesis;
end;
suppose n is non
zero;
then
reconsider n as non
zero
Nat;
defpred
P[
object,
object] means for i be
Element of
NAT st i
= $1 holds $2
= (
Base_FinSeq (n,i));
A1: for x be
object st x
in (
Seg n) holds ex y be
object st
P[x, y]
proof
let x be
object;
assume x
in (
Seg n);
then
reconsider j = x as
Element of
NAT ;
for i be
Element of
NAT st i
= j holds (
Base_FinSeq (n,j))
= (
Base_FinSeq (n,i));
hence ex y be
object st
P[x, y];
end;
consider f be
Function such that
A2: (
dom f)
= (
Seg n) & for x2 be
object st x2
in (
Seg n) holds
P[x2, (f
. x2)] from
CLASSES1:sch 1(
A1);
A3: (
rng f)
c= (
RN_Base n)
proof
let y be
object;
assume y
in (
rng f);
then
consider x be
object such that
A4: x
in (
dom f) and
A5: y
= (f
. x) by
FUNCT_1:def 3;
reconsider nx = x as
Element of
NAT by
A2,
A4;
A6: nx
<= n by
A2,
A4,
FINSEQ_1: 1;
A7: (f
. x)
= (
Base_FinSeq (n,nx)) by
A2,
A4;
1
<= nx by
A2,
A4,
FINSEQ_1: 1;
hence y
in (
RN_Base n) by
A5,
A6,
A7;
end;
then
reconsider f2 = f as
Function of (
Seg n), (
RN_Base n) by
A2,
FUNCT_2: 2;
for x1,x2 be
object st x1
in (
dom f) & x2
in (
dom f) & (f
. x1)
= (f
. x2) holds x1
= x2
proof
let x1,x2 be
object;
assume that
A8: x1
in (
dom f) and
A9: x2
in (
dom f) and
A10: (f
. x1)
= (f
. x2);
reconsider nx1 = x1, nx2 = x2 as
Element of
NAT by
A2,
A8,
A9;
A11: nx1
<= n by
A2,
A8,
FINSEQ_1: 1;
A12: (f
. x2)
= (
Base_FinSeq (n,nx2)) by
A2,
A9;
A13: (f
. x1)
= (
Base_FinSeq (n,nx1)) by
A2,
A8;
1
<= nx1 by
A2,
A8,
FINSEQ_1: 1;
hence x1
= x2 by
A10,
A11,
A13,
A12,
Th24;
end;
then
A14: f2 is
one-to-one by
FUNCT_1:def 4;
(
RN_Base n)
c= (
rng f)
proof
let y be
object;
assume y
in (
RN_Base n);
then
consider i be
Element of
NAT such that
A15: y
= (
Base_FinSeq (n,i)) and
A16: 1
<= i and
A17: i
<= n;
A18: i
in (
Seg n) by
A16,
A17,
FINSEQ_1: 1;
then (f
. i)
= (
Base_FinSeq (n,i)) by
A2;
hence y
in (
rng f) by
A2,
A15,
A18,
FUNCT_1:def 3;
end;
then (
rng f)
= (
RN_Base n) by
A3,
XBOOLE_0:def 10;
then f2 is
onto by
FUNCT_2:def 3;
then (
card (
Seg n))
= (
card (
RN_Base n)) by
A14,
Lm1;
hence thesis by
FINSEQ_1: 57;
end;
end;
end;
begin
registration
let n;
cluster (
RN_Base n) ->
finite;
coherence by
Lm5;
end
theorem ::
EUCLID_7:41
(
card (
RN_Base n))
= n by
Lm5;
theorem ::
EUCLID_7:42
Th41: for f be
FinSequence of (
REAL n), g be
FinSequence of the
carrier of (
RealVectSpace (
Seg n)) st f
= g holds (
Sum f)
= (
Sum g)
proof
let f be
FinSequence of (
REAL n), g be
FinSequence of the
carrier of (
RealVectSpace (
Seg n));
assume
A1: f
= g;
set V = (
RealVectSpace (
Seg n));
A2: (
REAL n)
= the
carrier of V by
FINSEQ_2: 93;
now
per cases ;
case
A3: (
len f)
>
0 ;
set g2 = (
accum f);
A4: (
len f)
= (
len g2) by
Def10;
A5: (f
. 1)
= (g2
. 1) by
Def10;
A6: (
Sum f)
= (g2
. (
len f)) by
Def11,
A3;
deffunc
F(
set) = (
IFIN ($1,((
len f)
+ 1),(
IFEQ ($1,
0 ,(
0. V),(g2
/. $1))),(
0. V)));
A7: for x be
set st x
in
NAT holds
F(x)
in the
carrier of V
proof
let x be
set;
assume x
in
NAT ;
then
reconsider nx = x as
Element of
NAT ;
per cases ;
suppose nx
in ((
len f)
+ 1);
then
A8:
F(x)
= (
IFEQ (x,
0 ,(
0. V),(g2
/. nx))) by
MATRIX_7:def 1;
per cases ;
suppose x
=
0 ;
then
F(x)
= (
0. V) by
A8,
FUNCOP_1:def 8;
hence
F(x)
in the
carrier of V;
end;
suppose x
<>
0 ;
then
F(x)
= (g2
/. nx) by
A8,
FUNCOP_1:def 8;
hence
F(x)
in the
carrier of V by
A2;
end;
end;
suppose not nx
in ((
len f)
+ 1);
then
F(x)
= (
0. V) by
MATRIX_7:def 1;
hence
F(x)
in the
carrier of V;
end;
end;
consider f3 be
sequence of the
carrier of V such that
A9: for x be
set st x
in
NAT holds (f3
. x)
=
F(x) from
FUNCT_2:sch 11(
A7);
A10: for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f3
. (j
+ 1))
= ((f3
. j)
+ v)
proof
let j be
Nat;
A11: j
in
NAT by
ORDINAL1:def 12;
let v be
Element of V;
assume that
A12: j
< (
len g) and
A13: v
= (g
. (j
+ 1));
A14: (j
+ 1)
<= (
len f) by
A1,
A12,
NAT_1: 13;
per cases ;
suppose
A15: j
=
0 ;
then (j
+ 1)
< ((
len f)
+ 1) by
A3,
XREAL_1: 6;
then
A16: (j
+ 1)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A17:
0
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A18: (f3
. j)
= (
IFIN (j,((
len f)
+ 1),(
IFEQ (j,
0 ,(
0. V),(g2
/. j))),(
0. V))) by
A9,
A11
.= (
IFEQ (j,
0 ,(
0. V),(g2
/. j))) by
A15,
A17,
MATRIX_7:def 1
.= (
0. V) by
A15,
FUNCOP_1:def 8;
thus (f3
. (j
+ 1))
= (
IFIN ((j
+ 1),((
len f)
+ 1),(
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))),(
0. V))) by
A9
.= (
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))) by
A16,
MATRIX_7:def 1
.= (g2
/. 1) by
A15,
FUNCOP_1:def 8
.= (g
. (j
+ 1)) by
A1,
A4,
A5,
A14,
A15,
FINSEQ_4: 15
.= ((f3
. j)
+ v) by
A13,
A18,
RLVECT_1: 4;
end;
suppose
A19: j
<>
0 ;
(
len f)
< ((
len f)
+ 1) by
XREAL_1: 29;
then j
< ((
len f)
+ 1) by
A1,
A12,
XXREAL_0: 2;
then
A20: j
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A21: (f3
. j)
= (
IFIN (j,((
len f)
+ 1),(
IFEQ (j,
0 ,(
0. V),(g2
/. j))),(
0. V))) by
A9,
A11
.= (
IFEQ (j,
0 ,(
0. V),(g2
/. j))) by
A20,
MATRIX_7:def 1
.= (g2
/. j) by
A19,
FUNCOP_1:def 8;
A22: (
0
+ 1)
<= (j
+ 1) by
NAT_1: 13;
A23: (
0
+ 1)
<= j by
A19,
NAT_1: 13;
(j
+ 1)
< ((
len f)
+ 1) by
A1,
A12,
XREAL_1: 6;
then
A24: (j
+ 1)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
thus (f3
. (j
+ 1))
= (
IFIN ((j
+ 1),((
len f)
+ 1),(
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))),(
0. V))) by
A9
.= (
IFEQ ((j
+ 1),
0 ,(
0. V),(g2
/. (j
+ 1)))) by
A24,
MATRIX_7:def 1
.= (g2
/. (j
+ 1)) by
FUNCOP_1:def 8
.= (g2
. (j
+ 1)) by
A4,
A14,
A22,
FINSEQ_4: 15
.= ((g2
/. j)
+ (f
/. (j
+ 1))) by
A1,
Def10,
A12,
A23
.= ((f3
. j)
+ v) by
A1,
A13,
A14,
A22,
A21,
FINSEQ_4: 15;
end;
end;
(
len f)
< ((
len f)
+ 1) by
XREAL_1: 29;
then
A25: (
len f)
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A26: (
0
+ 1)
<= (
len f) by
A3,
NAT_1: 13;
A27:
0
in (
Segm ((
len f)
+ 1)) by
NAT_1: 44;
A28: (f3
.
0 )
= (
IFIN (
0 ,((
len f)
+ 1),(
IFEQ (
0 ,
0 ,(
0. V),(g2
/.
0 ))),(
0. V))) by
A9
.= (
IFEQ (
0 ,
0 ,(
0. V),(g2
/.
0 ))) by
A27,
MATRIX_7:def 1
.= (
0. V) by
FUNCOP_1:def 8;
(f3
. (
len g))
=
F(len) by
A1,
A9
.= (
IFEQ ((
len f),
0 ,(
0. V),(g2
/. (
len f)))) by
A25,
MATRIX_7:def 1
.= (g2
/. (
len f)) by
A3,
FUNCOP_1:def 8
.= (
Sum f) by
A4,
A6,
A26,
FINSEQ_4: 15;
hence ex f2 be
sequence of the
carrier of V st (
Sum f)
= (f2
. (
len g)) & (f2
.
0 )
= (
0. V) & for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f2
. (j
+ 1))
= ((f2
. j)
+ v) by
A28,
A10;
end;
case
A29: (
len f)
<=
0 ;
set f3 = (
NAT
--> (
0. V));
A30: for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f3
. (j
+ 1))
= ((f3
. j)
+ v) by
A1,
A29;
A31: (f3
. (
len g))
= (
0* n) by
FUNCOP_1: 7;
A32: (f3
.
0 )
= (
0. V) by
FUNCOP_1: 7;
(
Sum f)
= (
0* n) by
A29,
Def11;
hence ex f2 be
sequence of the
carrier of V st (
Sum f)
= (f2
. (
len g)) & (f2
.
0 )
= (
0. V) & for j be
Nat holds for v be
Element of V st j
< (
len g) & v
= (g
. (j
+ 1)) holds (f2
. (j
+ 1))
= ((f2
. j)
+ v) by
A31,
A32,
A30;
end;
end;
hence (
Sum f)
= (
Sum g) by
RLVECT_1:def 12;
end;
theorem ::
EUCLID_7:43
Th42: for x0 be
Element of (
RealVectSpace (
Seg n)), B be
Subset of (
RealVectSpace (
Seg n)) st B
= (
RN_Base n) holds ex l be
Linear_Combination of B st x0
= (
Sum l)
proof
let x0 be
Element of (
RealVectSpace (
Seg n)), B be
Subset of (
RealVectSpace (
Seg n));
reconsider x1 = x0 as
Element of (
REAL n) by
FINSEQ_2: 93;
A1: (
REAL n)
= the
carrier of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
assume
A2: B
= (
RN_Base n);
A3: { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
c= B
proof
let y be
object;
assume y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
then ex x be
Element of (
REAL n) st y
= x & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i));
hence y
in B by
A2;
end;
B
c= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
proof
let y be
object;
assume y
in B;
then ex i2 be
Element of
NAT st y
= (
Base_FinSeq (n,i2)) & 1
<= i2 & i2
<= n by
A2;
hence y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
end;
then
A4: B
= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) } by
A3,
XBOOLE_0:def 10;
A5: { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 }
c= B
proof
let x2 be
object;
assume x2
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 };
then ex x be
Element of (
REAL n) st x
= x2 & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 ;
hence x2
in B by
A4;
end;
then
reconsider B0 = { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 } as
Subset of (
RealVectSpace (
Seg n)) by
XBOOLE_1: 1;
A6: (
dom (
ProjFinSeq x1))
= (
Seg (
len (
ProjFinSeq x1))) by
FINSEQ_1:def 3
.= (
Seg n) by
Def12;
defpred
R[
object,
object] means $1
in B0 implies ex i be
Element of
NAT st $2
= i & 1
<= i & i
<= n & $1
= (
Base_FinSeq (n,i));
A7: for x be
object st x
in B0 holds ex y be
object st y
in (
Seg n) &
R[x, y]
proof
let x be
object;
assume x
in B0;
then
consider x2 be
Element of (
REAL n) such that
A8: x
= x2 and
A9: ex i be
Element of
NAT st 1
<= i & i
<= n & x2
= (
Base_FinSeq (n,i)) &
|(x1, x2)|
<>
0 ;
consider i0 be
Element of
NAT such that
A10: 1
<= i0 and
A11: i0
<= n and
A12: x2
= (
Base_FinSeq (n,i0)) and
|(x1, x2)|
<>
0 by
A9;
i0
in (
Seg n) by
A10,
A11,
FINSEQ_1: 1;
hence ex y be
object st y
in (
Seg n) &
R[x, y] by
A8,
A10,
A11,
A12;
end;
consider f be
Function of B0, (
Seg n) such that
A13: for x be
object st x
in B0 holds
R[x, (f
. x)] from
FUNCT_2:sch 1(
A7);
defpred
Q[
object,
object] means ($1
in B0 implies (for i be
Element of
NAT st 1
<= i & i
<= n & $1
= (
Base_FinSeq (n,i)) holds $2
=
|(x1, (
Base_FinSeq (n,i)))|)) & ( not $1
in B0 implies $2
=
0 );
A14: for x be
object st x
in the
carrier of (
RealVectSpace (
Seg n)) holds ex y be
object st y
in
REAL &
Q[x, y]
proof
let x be
object;
assume x
in the
carrier of (
RealVectSpace (
Seg n));
per cases ;
suppose
A15: x
in B0;
then
consider x2 be
Element of (
REAL n) such that
A16: x2
= x and ex i be
Element of
NAT st 1
<= i & i
<= n & x2
= (
Base_FinSeq (n,i)) &
|(x1, x2)|
<>
0 ;
reconsider y0 =
|(x1, x2)| as
Element of
REAL ;
for i2 be
Element of
NAT st 1
<= i2 & i2
<= n & x
= (
Base_FinSeq (n,i2)) holds y0
=
|(x1, (
Base_FinSeq (n,i2)))| by
A16;
hence ex y be
object st y
in
REAL &
Q[x, y] by
A15;
end;
suppose not x
in B0;
hence ex y be
object st y
in
REAL &
Q[x, y] by
Lm3;
end;
end;
consider l2 be
Function of the
carrier of (
RealVectSpace (
Seg n)),
REAL such that
A17: for x be
object st x
in the
carrier of (
RealVectSpace (
Seg n)) holds
Q[x, (l2
. x)] from
FUNCT_2:sch 1(
A14);
A18: l2 is
Element of (
Funcs (the
carrier of (
RealVectSpace (
Seg n)),
REAL )) by
FUNCT_2: 8;
for v be
Element of (
RealVectSpace (
Seg n)) st not v
in B0 holds (l2
. v)
=
0 by
A17;
then
reconsider l3 = l2 as
Linear_Combination of (
RealVectSpace (
Seg n)) by
A2,
A5,
A18,
RLVECT_2:def 3;
(
Carrier l3)
c= B0
proof
let x be
object;
assume x
in (
Carrier l3);
then x
in { v where v be
Element of (
RealVectSpace (
Seg n)) : (l3
. v)
<>
0 } by
RLVECT_2:def 4;
then ex v be
Element of (
RealVectSpace (
Seg n)) st x
= v & (l3
. v)
<>
0 ;
hence x
in B0 by
A17;
end;
then
reconsider l0 = l3 as
Linear_Combination of B0 by
RLVECT_2:def 6;
A19: (
Carrier l0)
c= B0 by
RLVECT_2:def 6;
then (
Carrier l0)
c= B by
A5;
then
reconsider l2 = l0 as
Linear_Combination of B by
RLVECT_2:def 6;
A20: B0
c= (
Carrier l0)
proof
let x be
object;
assume
A21: x
in B0;
then
consider x6 be
Element of (
REAL n) such that
A22: x
= x6 and
A23: ex i be
Element of
NAT st 1
<= i & i
<= n & x6
= (
Base_FinSeq (n,i)) &
|(x1, x6)|
<>
0 ;
reconsider x66 = x6 as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
consider i8 be
Element of
NAT such that 1
<= i8 and i8
<= n and
A24: x6
= (
Base_FinSeq (n,i8)) and
|(x1, x6)|
<>
0 by
A23;
(l0
. x66)
=
|(x1, (
Base_FinSeq (n,i8)))| by
A17,
A21,
A22,
A23,
A24;
then x
in { v where v be
Element of (
RealVectSpace (
Seg n)) : (l0
. v)
<>
0 } by
A22,
A23,
A24;
hence x
in (
Carrier l0) by
RLVECT_2:def 4;
end;
for x1,x2 be
object st x1
in (
dom f) & x2
in (
dom f) & (f
. x1)
= (f
. x2) holds x1
= x2
proof
let x1,x2 be
object;
assume that
A25: x1
in (
dom f) and
A26: x2
in (
dom f) and
A27: (f
. x1)
= (f
. x2);
A28: ex i2 be
Element of
NAT st (f
. x2)
= i2 & 1
<= i2 & i2
<= n & x2
= (
Base_FinSeq (n,i2)) by
A13,
A26;
ex i1 be
Element of
NAT st (f
. x1)
= i1 & 1
<= i1 & i1
<= n & x1
= (
Base_FinSeq (n,i1)) by
A13,
A25;
hence x1
= x2 by
A27,
A28;
end;
then
A29: f is
one-to-one by
FUNCT_1:def 4;
A30: (
Seg n)
=
{} implies B0
=
{}
proof
assume
A31: (
Seg n)
=
{} ;
now
set y = the
Element of B0;
assume B0
<>
{} ;
then y
in B0;
then ex x be
Element of (
REAL n) st x
= y & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 ;
hence contradiction by
A31;
end;
hence B0
=
{} ;
end;
A32: for i3 be
Element of
NAT st i3
in (
dom (
ProjFinSeq x1)) & not i3
in (
rng (
Sgm (
rng f))) holds ((
ProjFinSeq x1)
. i3)
= (
0* n)
proof
let i3 be
Element of
NAT ;
assume that
A33: i3
in (
dom (
ProjFinSeq x1)) and
A34: not i3
in (
rng (
Sgm (
rng f)));
A35: i3
in (
Seg (
len (
ProjFinSeq x1))) by
A33,
FINSEQ_1:def 3;
then
A36: 1
<= i3 by
FINSEQ_1: 1;
(
len (
ProjFinSeq x1))
= n by
Def12;
then
A37: i3
<= n by
A35,
FINSEQ_1: 1;
A38: not i3
in (
rng f) by
A34,
FINSEQ_1:def 13;
A39:
now
assume
|(x1, (
Base_FinSeq (n,i3)))|
<>
0 ;
then
A40: (
Base_FinSeq (n,i3))
in B0 by
A36,
A37;
then
consider i5 be
Element of
NAT such that
A41: (f
. (
Base_FinSeq (n,i3)))
= i5 and 1
<= i5 and i5
<= n and
A42: (
Base_FinSeq (n,i3))
= (
Base_FinSeq (n,i5)) by
A13;
A43: (
Base_FinSeq (n,i3))
in (
dom f) by
A30,
A40,
FUNCT_2:def 1;
i3
= i5 by
A36,
A37,
A42,
Th24;
hence contradiction by
A38,
A41,
A43,
FUNCT_1:def 3;
end;
((
ProjFinSeq x1)
. i3)
= (
|(x1, (
Base_FinSeq (n,i3)))|
* (
Base_FinSeq (n,i3))) by
A36,
A37,
Def12;
hence ((
ProjFinSeq x1)
. i3)
= (
0* n) by
A39,
EUCLID_4: 3;
end;
A44: (
dom (
Sgm (
rng f)) qua
Function)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
A45: (
rng ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function))
c= (
REAL n);
A46: (
rng (
Sgm (
rng f)))
= (
rng f) by
FINSEQ_1:def 13;
(
dom ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function))
= ((
Sgm (
rng f)) qua
Function
" (
dom (
ProjFinSeq x1))) by
RELAT_1: 147
.= (
dom (
Sgm (
rng f))) by
A46,
A6,
Th1;
then ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function) is
FinSequence by
A44,
FINSEQ_1:def 2;
then
reconsider F2 = ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function) as
FinSequence of the
carrier of (
RealVectSpace (
Seg n)) by
A1,
A45,
FINSEQ_1:def 4;
(
dom F2)
= ((
Sgm (
rng f)) qua
Function
" (
Seg n)) by
A6,
RELAT_1: 147
.= (
dom (
Sgm (
rng f))) by
A46,
Th1;
then
A47: (
dom F2)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
then
A48: (
Seg (
len F2))
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
reconsider F3 = F2 as
FinSequence of (
REAL n) by
FINSEQ_2: 93;
A49: x0
= (
Sum (
ProjFinSeq x1)) by
Th30
.= (
Sum F3) by
A46,
A6,
A32,
Th23,
FINSEQ_3: 92
.= (
Sum F2) by
Th41;
A50: (
rng ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function))
c= (
rng (f qua
Function
" ) qua
Function) by
RELAT_1: 26;
A51: (
dom (
Sgm (
rng f)) qua
Function)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
A52: (
len F2)
= (
len (
Sgm (
rng f))) by
A47,
FINSEQ_1:def 3;
A53: (
dom f)
= B0 by
A30,
FUNCT_2:def 1;
then (
rng (f qua
Function
" ))
= B0 by
A29,
FUNCT_1: 33;
then
A54: (
rng ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function))
c= the
carrier of (
RealVectSpace (
Seg n)) by
A50,
XBOOLE_1: 1;
(
dom ((f qua
Function
" ) qua
Function
* (
Sgm (
rng f)) qua
Function))
= ((
Sgm (
rng f)) qua
Function
" (
dom (f qua
Function
" ))) by
RELAT_1: 147
.= ((
Sgm (
rng f)) qua
Function
" (
rng f)) by
A29,
FUNCT_1: 33
.= (
dom (
Sgm (
rng f))) by
A46,
Th1;
then ((f qua
Function
" ) qua
Function
* (
Sgm (
rng f)) qua
Function) is
FinSequence by
A51,
FINSEQ_1:def 2;
then
reconsider F0 = ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function) as
FinSequence of the
carrier of (
RealVectSpace (
Seg n)) by
A54,
FINSEQ_1:def 4;
(
dom F0)
= ((
Sgm (
rng f)) qua
Function
" (
dom (f qua
Function
" ))) by
RELAT_1: 147
.= ((
Sgm (
rng f)) qua
Function
" (
rng f)) by
A29,
FUNCT_1: 33
.= (
dom (
Sgm (
rng f)) qua
Function) by
A46,
Th1;
then
A55: (
dom F0)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
(
dom (f qua
Function
" ))
= (
rng f) by
A29,
FUNCT_1: 33;
then (
rng F0)
= (
rng (f qua
Function
" )) by
A46,
RELAT_1: 28
.= (
dom f) by
A29,
FUNCT_1: 33;
then
A56: (
rng F0)
= (
Carrier l0) by
A53,
A19,
A20,
XBOOLE_0:def 10;
A57: for i be
Nat st i
in (
dom F2) holds (F2
. i)
= ((l0
. (F0
/. i))
* (F0
/. i))
proof
let i be
Nat;
A58: (
Sgm (
rng f)) is
one-to-one by
FINSEQ_3: 92;
assume i
in (
dom F2);
then
A59: i
in (
Seg (
len F2)) by
FINSEQ_1:def 3;
then
A60: i
in (
dom (
Sgm (
rng f))) by
A52,
FINSEQ_1:def 3;
then ((
Sgm (
rng f))
. i)
in (
rng (
Sgm (
rng f))) by
FUNCT_1:def 3;
then
reconsider i2 = ((
Sgm (
rng f))
. i) as
Element of
NAT ;
reconsider r = (
Base_FinSeq (n,i2)) as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
i2
in (
rng (
Sgm (
rng f))) by
A60,
FUNCT_1:def 3;
then
consider x2 be
object such that
A61: x2
in (
dom f) and
A62: (f
. x2)
= i2 by
A46,
FUNCT_1:def 3;
(
dom f)
= B0 by
A30,
FUNCT_2:def 1;
then
reconsider r2 = x2 as
Element of (
RealVectSpace (
Seg n)) by
A61;
A63: ex i22 be
Element of
NAT st (f
. r2)
= i22 & 1
<= i22 & i22
<= n & r2
= (
Base_FinSeq (n,i22)) by
A13,
A61;
then
consider i4 be
Element of
NAT such that
A64: (f
. r)
= i4 and 1
<= i4 and i4
<= n and
A65: r
= (
Base_FinSeq (n,i4)) by
A62;
A66: (
dom f)
= B0 by
A30,
FUNCT_2:def 1;
(F0
. i)
= ((f qua
Function
" )
. ((
Sgm (
rng f)) qua
Function
. i)) by
A60,
FUNCT_1: 13
.= (
Base_FinSeq (n,i2)) by
A29,
A61,
A62,
A63,
FUNCT_1: 32;
then (
Base_FinSeq (n,i2))
in (
rng F0) by
A55,
A48,
A59,
FUNCT_1:def 3;
then (
Base_FinSeq (n,i2))
in { v where v be
Element of (
RealVectSpace (
Seg n)) : (l0
. v)
<>
0 } by
A56,
RLVECT_2:def 4;
then
A67: ex v0 be
Element of (
RealVectSpace (
Seg n)) st (
Base_FinSeq (n,i2))
= v0 & (l0
. v0)
<>
0 ;
then (
Base_FinSeq (n,i2))
in B0 by
A17;
then
A68: ((f qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
= (
Base_FinSeq (n,i2)) by
A29,
A66,
FUNCT_1: 34;
then
A69: (((f qua
Function
" )
* (
Sgm (
rng f)))
. i)
= (
Base_FinSeq (n,i2)) by
A60,
A62,
A63,
FUNCT_1: 13;
A70: i2
in (
rng f) by
A46,
A60,
FUNCT_1:def 3;
then
A71: 1
<= i2 by
FINSEQ_1: 1;
A72: i2
<= n by
A70,
FINSEQ_1: 1;
then i4
= i2 by
A71,
A65,
Th24;
then
A73: (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
= i by
A60,
A64,
A58,
FUNCT_1: 32;
A74: (f
. (
Base_FinSeq (n,i2)))
in (
rng (
Sgm (
rng f))) by
A46,
A61,
A62,
A63,
FUNCT_1:def 3;
then
A75: ((f qua
Function
" )
. ((
Sgm (
rng f))
. (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))))
= (
Base_FinSeq (n,i2)) by
A58,
A68,
FUNCT_1: 35;
(
dom ((
Sgm (
rng f)) qua
Function
" ))
= (
rng (
Sgm (
rng f))) by
A58,
FUNCT_1: 33;
then (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
in (
rng ((
Sgm (
rng f)) qua
Function
" )) by
A74,
FUNCT_1:def 3;
then
A76: (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
in (
dom (
Sgm (
rng f))) by
A58,
FUNCT_1: 33;
(l0
. (F0
/. i))
= (l0
. (((f qua
Function
" )
* (
Sgm (
rng f)))
. i)) by
A55,
A52,
A59,
PARTFUN1:def 6
.= (l0
. (
Base_FinSeq (n,i2))) by
A73,
A76,
A75,
FUNCT_1: 13
.=
|(x1, (
Base_FinSeq (n,i2)))| by
A17,
A71,
A72,
A67;
then ((l0
. (F0
/. i))
* (F0
/. i))
= (
|(x1, (
Base_FinSeq (n,i2)))|
* (
Base_FinSeq (n,i2))) by
A55,
A52,
A59,
A69,
PARTFUN1:def 6
.= ((
ProjFinSeq x1)
. ((
Sgm (
rng f))
. i)) by
A71,
A72,
Def12
.= (((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function)
. i) by
A60,
FUNCT_1: 13;
hence (F2
. i)
= ((l0
. (F0
/. i))
* (F0
/. i));
end;
A77: (
Sgm (
rng f)) qua
Function is
one-to-one by
FINSEQ_3: 92;
(
len F2)
= (
len F0) by
A55,
A48,
FINSEQ_1:def 3;
then x1
= (
Sum (l0
(#) F0)) by
A49,
A57,
RLVECT_2:def 7;
then x1
= (
Sum l2) by
A29,
A77,
A56,
RLVECT_2:def 8;
hence ex l be
Linear_Combination of B st x0
= (
Sum l);
end;
theorem ::
EUCLID_7:44
Th43: for n be
Element of
NAT , x0 be
Element of (
REAL-US n), B be
Subset of (
REAL-US n) st B
= (
RN_Base n) holds ex l be
Linear_Combination of B st x0
= (
Sum l)
proof
let n be
Element of
NAT , x0 be
Element of (
REAL-US n), B be
Subset of (
REAL-US n);
reconsider x1 = x0 as
Element of (
REAL n) by
REAL_NS1:def 6;
A1: (
REAL n)
= the
carrier of (
REAL-US n) by
REAL_NS1:def 6;
assume
A2: B
= (
RN_Base n);
A3: { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
c= B
proof
let y be
object;
assume y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
then ex x be
Element of (
REAL n) st y
= x & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i));
hence y
in B by
A2;
end;
B
c= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) }
proof
let y be
object;
assume y
in B;
then ex i2 be
Element of
NAT st y
= (
Base_FinSeq (n,i2)) & 1
<= i2 & i2
<= n by
A2;
hence y
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) };
end;
then
A4: B
= { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) } by
A3,
XBOOLE_0:def 10;
A5: { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 }
c= B
proof
let x2 be
object;
assume x2
in { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 };
then ex x be
Element of (
REAL n) st x
= x2 & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 ;
hence x2
in B by
A4;
end;
then
reconsider B0 = { x where x be
Element of (
REAL n) : ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 } as
Subset of (
REAL-US n) by
XBOOLE_1: 1;
defpred
R[
object,
object] means $1
in B0 implies ex i be
Element of
NAT st $2
= i & 1
<= i & i
<= n & $1
= (
Base_FinSeq (n,i));
A6: for x be
object st x
in B0 holds ex y be
object st y
in (
Seg n) &
R[x, y]
proof
let x be
object;
assume x
in B0;
then
consider x2 be
Element of (
REAL n) such that
A7: x
= x2 and
A8: ex i be
Element of
NAT st 1
<= i & i
<= n & x2
= (
Base_FinSeq (n,i)) &
|(x1, x2)|
<>
0 ;
consider i0 be
Element of
NAT such that
A9: 1
<= i0 and
A10: i0
<= n and
A11: x2
= (
Base_FinSeq (n,i0)) and
|(x1, x2)|
<>
0 by
A8;
i0
in (
Seg n) by
A9,
A10,
FINSEQ_1: 1;
hence ex y be
object st y
in (
Seg n) &
R[x, y] by
A7,
A9,
A10,
A11;
end;
consider f be
Function of B0, (
Seg n) such that
A12: for x be
object st x
in B0 holds
R[x, (f
. x)] from
FUNCT_2:sch 1(
A6);
defpred
Q[
object,
object] means ($1
in B0 implies (for i be
Element of
NAT st 1
<= i & i
<= n & $1
= (
Base_FinSeq (n,i)) holds $2
=
|(x1, (
Base_FinSeq (n,i)))|)) & ( not $1
in B0 implies $2
=
0 );
A13: for x be
object st x
in the
carrier of (
REAL-US n) holds ex y be
object st y
in
REAL &
Q[x, y]
proof
let x be
object;
assume x
in the
carrier of (
REAL-US n);
per cases ;
suppose
A14: x
in B0;
then
consider x2 be
Element of (
REAL n) such that
A15: x2
= x and ex i be
Element of
NAT st 1
<= i & i
<= n & x2
= (
Base_FinSeq (n,i)) &
|(x1, x2)|
<>
0 ;
reconsider y0 =
|(x1, x2)| as
Element of
REAL ;
for i2 be
Element of
NAT st 1
<= i2 & i2
<= n & x
= (
Base_FinSeq (n,i2)) holds y0
=
|(x1, (
Base_FinSeq (n,i2)))| by
A15;
hence ex y be
object st y
in
REAL &
Q[x, y] by
A14;
end;
suppose not x
in B0;
hence ex y be
object st y
in
REAL &
Q[x, y] by
Lm3;
end;
end;
consider l2 be
Function of the
carrier of (
REAL-US n),
REAL such that
A16: for x be
object st x
in the
carrier of (
REAL-US n) holds
Q[x, (l2
. x)] from
FUNCT_2:sch 1(
A13);
A17: l2 is
Element of (
Funcs (the
carrier of (
REAL-US n),
REAL )) by
FUNCT_2: 8;
for v be
Element of (
REAL-US n) st not v
in B0 holds (l2
. v)
=
0 by
A16;
then
reconsider l3 = l2 as
Linear_Combination of (
REAL-US n) by
A2,
A5,
A17,
RLVECT_2:def 3;
(
Carrier l3)
c= B0
proof
let x be
object;
assume x
in (
Carrier l3);
then x
in { v where v be
Element of (
REAL-US n) : (l3
. v)
<>
0 } by
RLVECT_2:def 4;
then ex v be
Element of (
REAL-US n) st x
= v & (l3
. v)
<>
0 ;
hence x
in B0 by
A16;
end;
then
reconsider l0 = l3 as
Linear_Combination of B0 by
RLVECT_2:def 6;
A18: (
Carrier l0)
c= B0 by
RLVECT_2:def 6;
then (
Carrier l0)
c= B by
A5;
then
reconsider l2 = l0 as
Linear_Combination of B by
RLVECT_2:def 6;
A19: B0
c= (
Carrier l0)
proof
let x be
object;
assume
A20: x
in B0;
then
consider x6 be
Element of (
REAL n) such that
A21: x
= x6 and
A22: ex i be
Element of
NAT st 1
<= i & i
<= n & x6
= (
Base_FinSeq (n,i)) &
|(x1, x6)|
<>
0 ;
reconsider x66 = x6 as
Element of (
REAL-US n) by
REAL_NS1:def 6;
consider i8 be
Element of
NAT such that 1
<= i8 and i8
<= n and
A23: x6
= (
Base_FinSeq (n,i8)) and
|(x1, x6)|
<>
0 by
A22;
(l0
. x66)
=
|(x1, (
Base_FinSeq (n,i8)))| by
A16,
A20,
A21,
A22,
A23;
then x
in { v where v be
Element of (
REAL-US n) : (l0
. v)
<>
0 } by
A21,
A22,
A23;
hence x
in (
Carrier l0) by
RLVECT_2:def 4;
end;
A24: (
dom (
Sgm (
rng f)) qua
Function)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
A25: (
rng ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function))
c= (
rng (f qua
Function
" ) qua
Function) by
RELAT_1: 26;
A26: (
dom (
Sgm (
rng f)) qua
Function)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
for x1,x2 be
object st x1
in (
dom f) & x2
in (
dom f) & (f
. x1)
= (f
. x2) holds x1
= x2
proof
let x1,x2 be
object;
assume that
A27: x1
in (
dom f) and
A28: x2
in (
dom f) and
A29: (f
. x1)
= (f
. x2);
A30: ex i2 be
Element of
NAT st (f
. x2)
= i2 & 1
<= i2 & i2
<= n & x2
= (
Base_FinSeq (n,i2)) by
A12,
A28;
ex i1 be
Element of
NAT st (f
. x1)
= i1 & 1
<= i1 & i1
<= n & x1
= (
Base_FinSeq (n,i1)) by
A12,
A27;
hence x1
= x2 by
A29,
A30;
end;
then
A31: f is
one-to-one by
FUNCT_1:def 4;
A32: (
Seg n)
=
{} implies B0
=
{}
proof
set y = the
Element of B0;
assume
A33: (
Seg n)
=
{} ;
assume B0
<>
{} ;
then y
in B0;
then ex x be
Element of (
REAL n) st x
= y & ex i be
Element of
NAT st 1
<= i & i
<= n & x
= (
Base_FinSeq (n,i)) &
|(x1, x)|
<>
0 ;
hence contradiction by
A33;
end;
A34: for i3 be
Element of
NAT st i3
in (
dom (
ProjFinSeq x1)) & not i3
in (
rng (
Sgm (
rng f))) holds ((
ProjFinSeq x1)
. i3)
= (
0* n)
proof
let i3 be
Element of
NAT ;
assume that
A35: i3
in (
dom (
ProjFinSeq x1)) and
A36: not i3
in (
rng (
Sgm (
rng f)));
A37: i3
in (
Seg (
len (
ProjFinSeq x1))) by
A35,
FINSEQ_1:def 3;
then
A38: 1
<= i3 by
FINSEQ_1: 1;
(
len (
ProjFinSeq x1))
= n by
Def12;
then
A39: i3
<= n by
A37,
FINSEQ_1: 1;
A40: not i3
in (
rng f) by
A36,
FINSEQ_1:def 13;
A41:
now
assume
|(x1, (
Base_FinSeq (n,i3)))|
<>
0 ;
then
A42: (
Base_FinSeq (n,i3))
in B0 by
A38,
A39;
then
consider i5 be
Element of
NAT such that
A43: (f
. (
Base_FinSeq (n,i3)))
= i5 and 1
<= i5 and i5
<= n and
A44: (
Base_FinSeq (n,i3))
= (
Base_FinSeq (n,i5)) by
A12;
A45: (
Base_FinSeq (n,i3))
in (
dom f) by
A32,
A42,
FUNCT_2:def 1;
i3
= i5 by
A38,
A39,
A44,
Th24;
hence contradiction by
A40,
A43,
A45,
FUNCT_1:def 3;
end;
((
ProjFinSeq x1)
. i3)
= (
|(x1, (
Base_FinSeq (n,i3)))|
* (
Base_FinSeq (n,i3))) by
A38,
A39,
Def12;
hence ((
ProjFinSeq x1)
. i3)
= (
0* n) by
A41,
EUCLID_4: 3;
end;
A46: (
rng ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function))
c= (
REAL n);
A47: (
rng (
Sgm (
rng f)))
= (
rng f) by
FINSEQ_1:def 13;
A48: (
dom f)
= B0 by
A32,
FUNCT_2:def 1;
then (
rng (f qua
Function
" ))
= B0 by
A31,
FUNCT_1: 33;
then
A49: (
rng ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function))
c= the
carrier of (
REAL-US n) by
A25,
XBOOLE_1: 1;
(
dom ((f qua
Function
" ) qua
Function
* (
Sgm (
rng f)) qua
Function))
= ((
Sgm (
rng f)) qua
Function
" (
dom (f qua
Function
" ))) by
RELAT_1: 147
.= ((
Sgm (
rng f)) qua
Function
" (
rng f)) by
A31,
FUNCT_1: 33
.= (
dom (
Sgm (
rng f))) by
A47,
Th1;
then ((f qua
Function
" ) qua
Function
* (
Sgm (
rng f)) qua
Function) is
FinSequence by
A26,
FINSEQ_1:def 2;
then
reconsider F0 = ((f qua
Function
" )
* (
Sgm (
rng f)) qua
Function) as
FinSequence of the
carrier of (
REAL-US n) by
A49,
FINSEQ_1:def 4;
(
dom F0)
= ((
Sgm (
rng f)) qua
Function
" (
dom (f qua
Function
" ))) by
RELAT_1: 147
.= ((
Sgm (
rng f)) qua
Function
" (
rng f)) by
A31,
FUNCT_1: 33
.= (
dom (
Sgm (
rng f)) qua
Function) by
A47,
Th1;
then
A50: (
dom F0)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
A51: (
dom (
ProjFinSeq x1))
= (
Seg (
len (
ProjFinSeq x1))) by
FINSEQ_1:def 3
.= (
Seg n) by
Def12;
then (
dom ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function))
= ((
Sgm (
rng f)) qua
Function
" (
Seg n)) by
RELAT_1: 147
.= (
dom (
Sgm (
rng f))) by
A47,
Th1;
then ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function) is
FinSequence by
A24,
FINSEQ_1:def 2;
then
reconsider F2 = ((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function) as
FinSequence of the
carrier of (
REAL-US n) by
A1,
A46,
FINSEQ_1:def 4;
(
dom F2)
= ((
Sgm (
rng f)) qua
Function
" (
Seg n)) by
A51,
RELAT_1: 147
.= (
dom (
Sgm (
rng f))) by
A47,
Th1;
then
A52: (
dom F2)
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
then
A53: (
Seg (
len F2))
= (
Seg (
len (
Sgm (
rng f)))) by
FINSEQ_1:def 3;
A54: (
len F2)
= (
len (
Sgm (
rng f))) by
A52,
FINSEQ_1:def 3;
(
dom (f qua
Function
" ))
= (
rng f) by
A31,
FUNCT_1: 33;
then (
rng F0)
= (
rng (f qua
Function
" )) by
A47,
RELAT_1: 28
.= (
dom f) by
A31,
FUNCT_1: 33;
then
A55: (
rng F0)
= (
Carrier l0) by
A48,
A18,
A19,
XBOOLE_0:def 10;
A56: for i be
Nat st i
in (
dom F2) holds (F2
. i)
= ((l0
. (F0
/. i))
* (F0
/. i))
proof
let i be
Nat;
A57: (
Sgm (
rng f)) is
one-to-one by
FINSEQ_3: 92;
assume i
in (
dom F2);
then
A58: i
in (
Seg (
len F2)) by
FINSEQ_1:def 3;
then
A59: i
in (
dom (
Sgm (
rng f))) by
A54,
FINSEQ_1:def 3;
then ((
Sgm (
rng f))
. i)
in (
rng (
Sgm (
rng f))) by
FUNCT_1:def 3;
then
reconsider i2 = ((
Sgm (
rng f))
. i) as
Element of
NAT ;
reconsider r = (
Base_FinSeq (n,i2)) as
Element of (
REAL-US n) by
REAL_NS1:def 6;
i2
in (
rng (
Sgm (
rng f))) by
A59,
FUNCT_1:def 3;
then
consider x2 be
object such that
A60: x2
in (
dom f) and
A61: (f
. x2)
= i2 by
A47,
FUNCT_1:def 3;
(
dom f)
= B0 by
A32,
FUNCT_2:def 1;
then
reconsider r2 = x2 as
Element of (
REAL-US n) by
A60;
A62: ex i22 be
Element of
NAT st (f
. r2)
= i22 & 1
<= i22 & i22
<= n & r2
= (
Base_FinSeq (n,i22)) by
A12,
A60;
then
consider i4 be
Element of
NAT such that
A63: (f
. r)
= i4 and 1
<= i4 and i4
<= n and
A64: r
= (
Base_FinSeq (n,i4)) by
A61;
A65: (
dom f)
= B0 by
A32,
FUNCT_2:def 1;
(F0
. i)
= ((f qua
Function
" )
. ((
Sgm (
rng f)) qua
Function
. i)) by
A59,
FUNCT_1: 13
.= (
Base_FinSeq (n,i2)) by
A31,
A60,
A61,
A62,
FUNCT_1: 32;
then (
Base_FinSeq (n,i2))
in (
rng F0) by
A50,
A53,
A58,
FUNCT_1:def 3;
then (
Base_FinSeq (n,i2))
in { v where v be
Element of (
REAL-US n) : (l0
. v)
<>
0 } by
A55,
RLVECT_2:def 4;
then
A66: ex v0 be
Element of (
REAL-US n) st (
Base_FinSeq (n,i2))
= v0 & (l0
. v0)
<>
0 ;
then (
Base_FinSeq (n,i2))
in B0 by
A16;
then
A67: ((f qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
= (
Base_FinSeq (n,i2)) by
A31,
A65,
FUNCT_1: 34;
then
A68: (((f qua
Function
" )
* (
Sgm (
rng f)))
. i)
= (
Base_FinSeq (n,i2)) by
A59,
A61,
A62,
FUNCT_1: 13;
A69: i2
in (
rng f) by
A47,
A59,
FUNCT_1:def 3;
then
A70: 1
<= i2 by
FINSEQ_1: 1;
A71: i2
<= n by
A69,
FINSEQ_1: 1;
then i4
= i2 by
A70,
A64,
Th24;
then
A72: (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
= i by
A59,
A63,
A57,
FUNCT_1: 32;
A73: (f
. (
Base_FinSeq (n,i2)))
in (
rng (
Sgm (
rng f))) by
A47,
A60,
A61,
A62,
FUNCT_1:def 3;
then
A74: ((f qua
Function
" )
. ((
Sgm (
rng f))
. (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))))
= (
Base_FinSeq (n,i2)) by
A57,
A67,
FUNCT_1: 35;
(
dom ((
Sgm (
rng f)) qua
Function
" ))
= (
rng (
Sgm (
rng f))) by
A57,
FUNCT_1: 33;
then (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
in (
rng ((
Sgm (
rng f)) qua
Function
" )) by
A73,
FUNCT_1:def 3;
then
A75: (((
Sgm (
rng f)) qua
Function
" )
. (f
. (
Base_FinSeq (n,i2))))
in (
dom (
Sgm (
rng f))) by
A57,
FUNCT_1: 33;
(l0
. (F0
/. i))
= (l0
. (((f qua
Function
" )
* (
Sgm (
rng f)))
. i)) by
A50,
A54,
A58,
PARTFUN1:def 6
.= (l0
. (
Base_FinSeq (n,i2))) by
A72,
A75,
A74,
FUNCT_1: 13
.=
|(x1, (
Base_FinSeq (n,i2)))| by
A16,
A70,
A71,
A66;
then ((l0
. (F0
/. i))
* (F0
/. i))
= (
|(x1, (
Base_FinSeq (n,i2)))|
* (
Base_FinSeq (n,i2))) by
A50,
A54,
A58,
A68,
PARTFUN1:def 6
.= ((
ProjFinSeq x1)
. ((
Sgm (
rng f))
. i)) by
A70,
A71,
Def12
.= (((
ProjFinSeq x1) qua
Function
* (
Sgm (
rng f)) qua
Function)
. i) by
A59,
FUNCT_1: 13;
hence (F2
. i)
= ((l0
. (F0
/. i))
* (F0
/. i));
end;
A76: (
Sgm (
rng f)) qua
Function is
one-to-one by
FINSEQ_3: 92;
reconsider F3 = F2 as
FinSequence of (
REAL n) by
REAL_NS1:def 6;
A77: x0
= (
Sum (
ProjFinSeq x1)) by
Th30
.= (
Sum F3) by
A47,
A51,
A34,
Th23,
FINSEQ_3: 92
.= (
Sum F2) by
Th33;
(
len F2)
= (
len F0) by
A50,
A53,
FINSEQ_1:def 3;
then x1
= (
Sum (l0
(#) F0)) by
A77,
A56,
RLVECT_2:def 7;
then x1
= (
Sum l2) by
A31,
A76,
A55,
RLVECT_2:def 8;
hence ex l be
Linear_Combination of B st x0
= (
Sum l);
end;
theorem ::
EUCLID_7:45
Th44: for B be
Subset of (
RealVectSpace (
Seg n)) st B
= (
RN_Base n) holds B is
Basis of (
RealVectSpace (
Seg n))
proof
let B be
Subset of (
RealVectSpace (
Seg n));
set V = (
RealVectSpace (
Seg n));
assume
A1: B
= (
RN_Base n);
then
reconsider B1 = B as
R-orthonormal
Subset of V;
A2: the
carrier of (
Lin B)
= the set of all (
Sum l) where l be
Linear_Combination of B by
RLVECT_3:def 2;
A3:
now
assume not the
carrier of V
c= the
carrier of (
Lin B);
then
consider x be
object such that
A4: x
in the
carrier of V and
A5: not x
in the
carrier of (
Lin B);
reconsider x0 = x as
Element of V by
A4;
ex l be
Linear_Combination of B st x0
= (
Sum l) by
A1,
Th42;
hence contradiction by
A2,
A5;
end;
the
carrier of (
Lin B)
c= the
carrier of V
proof
let x be
object;
assume x
in the
carrier of (
Lin B);
then ex l be
Linear_Combination of B st x
= (
Sum l) by
A2;
hence x
in the
carrier of V;
end;
then the
carrier of (
Lin B)
= the
carrier of V by
A3,
XBOOLE_0:def 10;
then (
Lin B)
= V by
Th8;
then B1 is
Basis of (
RealVectSpace (
Seg n)) by
RLVECT_3:def 3;
hence thesis;
end;
registration
let n;
cluster (
RealVectSpace (
Seg n)) ->
finite-dimensional;
coherence
proof
reconsider B = (
RN_Base n) as
Subset of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
B is
Basis of (
RealVectSpace (
Seg n)) by
Th44;
hence thesis by
RLVECT_5:def 1;
end;
end
theorem ::
EUCLID_7:46
(
dim (
RealVectSpace (
Seg n)))
= n
proof
reconsider B = (
RN_Base n) as
Subset of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
A1: for I be
Basis of (
RealVectSpace (
Seg n)) holds n
= (
card I)
proof
let I be
Basis of (
RealVectSpace (
Seg n));
B is
Basis of (
RealVectSpace (
Seg n)) by
Th44;
then (
card B)
= (
card I) by
RLVECT_5: 25;
hence n
= (
card I) by
Lm5;
end;
n
in
NAT by
ORDINAL1:def 12;
hence thesis by
A1,
RLVECT_5:def 2;
end;
theorem ::
EUCLID_7:47
Th46: for B be
Subset of (
RealVectSpace (
Seg n)) st B is
Basis of (
RealVectSpace (
Seg n)) holds (
card B)
= n
proof
let B be
Subset of (
RealVectSpace (
Seg n));
assume
A1: B is
Basis of (
RealVectSpace (
Seg n));
reconsider Br = (
RN_Base n) as
Subset of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
Br is
Basis of (
RealVectSpace (
Seg n)) by
Th44;
then (
card Br)
= (
card B) by
A1,
RLVECT_5: 25;
hence (
card B)
= n by
Lm5;
end;
theorem ::
EUCLID_7:48
Th47:
{} is
Basis of (
RealVectSpace (
Seg
0 ))
proof
consider A be
finite
Subset of (
RealVectSpace (
Seg
0 )) such that
A1: A is
Basis of (
RealVectSpace (
Seg
0 )) by
RLVECT_5:def 1;
(
card A)
=
0 by
A1,
Th46;
then A
=
{} ;
hence thesis by
A1;
end;
theorem ::
EUCLID_7:49
Th48: for n be
Element of
NAT holds (
RN_Base n) is
Basis of (
REAL-US n)
proof
let n be
Element of
NAT ;
reconsider B = (
RN_Base n) as
Subset of (
REAL-US n) by
REAL_NS1:def 6;
set V = (
REAL-US n);
A1: the
carrier of (
Lin B)
= the set of all (
Sum l) where l be
Linear_Combination of B by
RUSUB_3:def 1;
A2:
now
assume not the
carrier of V
c= the
carrier of (
Lin B);
then
consider x be
object such that
A3: x
in the
carrier of V and
A4: not x
in the
carrier of (
Lin B);
reconsider x0 = x as
Element of V by
A3;
ex l be
Linear_Combination of B st x0
= (
Sum l) by
Th43;
hence contradiction by
A1,
A4;
end;
the
carrier of (
Lin B)
c= the
carrier of V
proof
let x be
object;
assume x
in the
carrier of (
Lin B);
then ex l be
Linear_Combination of B st x
= (
Sum l) by
A1;
hence x
in the
carrier of V;
end;
then the
carrier of (
Lin B)
= the
carrier of V by
A2,
XBOOLE_0:def 10;
then (
Lin B)
= V by
RUSUB_1: 26;
hence thesis by
RUSUB_3:def 2;
end;
theorem ::
EUCLID_7:50
Th49: for D be
Orthogonal_Basis of n holds D is
Basis of (
RealVectSpace (
Seg n))
proof
let D be
Orthogonal_Basis of n;
set V = (
RealVectSpace (
Seg n));
reconsider B = D as
R-orthonormal
Subset of V by
FINSEQ_2: 93;
per cases ;
suppose n
=
0 ;
hence thesis by
Th17,
Th47;
end;
suppose
A1: n
<>
0 ;
reconsider D0 = D as
R-orthonormal
Subset of (
REAL n);
consider I be
Basis of V such that
A2: B
c= I by
RLVECT_5: 2;
(
card I)
= n by
Th46;
then
A3: I is
finite;
A4: for D2 be
R-orthonormal
Subset of (
REAL n) st D0
c= D2 holds D2
= D0 by
Def6;
A5:
now
assume that B
<> I and
A6: not I
c= the
carrier of (
Lin B);
consider x0 be
object such that
A7: x0
in I and
A8: not x0
in the
carrier of (
Lin B) by
A6;
reconsider z0 = x0 as
Element of (
REAL n) by
A7,
FINSEQ_2: 93;
not x0
in (
Lin B) by
A8;
then
A9: not x0
in B by
RLVECT_3: 15;
consider p be
FinSequence such that
A10: (
rng p)
= B and
A11: p is
one-to-one by
A2,
A3,
FINSEQ_4: 58;
A12: 1
<= ((
len p)
+ 1) by
NAT_1: 12;
reconsider p0 = p as
FinSequence of (
REAL n) by
A10,
FINSEQ_1:def 4;
defpred
P[
Nat] means 1
<= $1 & $1
<= ((
len p)
+ 1) implies ex q be
FinSequence of (
REAL n) st (
len q)
= $1 & ($1
<= (
len p) implies for d be
Real holds not (q
. $1)
= (d
* (p0
/. $1))) & (q
. 1)
= z0 & for i be
Nat, a,b be
Element of (
REAL n) st 1
<= i & i
< $1 & a
= (q
/. i) & b
= (p0
/. i) holds (q
/. (i
+ 1))
<> (
0* n) & (q
. (i
+ 1))
= ((q
/. i)
- (
|(a, b)|
* (p0
/. i)));
A13: I is
linearly-independent by
RLVECT_3:def 3;
A14: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat;
assume
A15:
P[k];
per cases ;
suppose
A16: 1
<= (k
+ 1) & (k
+ 1)
<= ((
len p)
+ 1);
k
< (k
+ 1) by
XREAL_1: 29;
then
A17: k
< ((
len p)
+ 1) by
A16,
XXREAL_0: 2;
A18: k
<= (
len p) by
A16,
XREAL_1: 6;
per cases ;
suppose 1
<= k;
then
consider q0 be
FinSequence of (
REAL n) such that
A19: (
len q0)
= k and
A20: 1
<= k and k
<= ((
len p)
+ 1) and
A21: k
<= (
len p) implies for d be
Real holds not (q0
. k)
= (d
* (p0
/. k)) and
A22: (q0
. 1)
= z0 and
A23: for i be
Nat, a,b be
Element of (
REAL n) st 1
<= i & i
< k & a
= (q0
/. i) & b
= (p0
/. i) holds (q0
/. (i
+ 1))
<> (
0* n) & (q0
. (i
+ 1))
= ((q0
/. i)
- (
|(a, b)|
* (p0
/. i))) by
A15,
A17;
reconsider a2 = (q0
/. k), b2 = (p0
/. k) as
Element of (
REAL n);
reconsider q3 =
<*((q0
/. k)
- (
|(a2, b2)|
* (p0
/. k)))*> as
FinSequence of (
REAL n);
reconsider q1 = (q0
^ q3) as
FinSequence of (
REAL n);
1
in (
Seg (
len q0)) by
A19,
A20,
FINSEQ_1: 1;
then
A24: 1
in (
dom q0) by
FINSEQ_1:def 3;
then
A25: (q1
. 1)
= z0 by
A22,
FINSEQ_1:def 7;
A26: 1
<= (k
+ 1) by
NAT_1: 12;
A27: (k
+ 1)
<= (
len p) implies (p0
/. (k
+ 1))
in B
proof
assume
A28: (k
+ 1)
<= (
len p);
then (k
+ 1)
in (
dom p) by
A26,
FINSEQ_3: 25;
then (p
. (k
+ 1))
in (
rng p) by
FUNCT_1:def 3;
hence (p0
/. (k
+ 1))
in B by
A10,
A28,
FINSEQ_4: 15,
NAT_1: 12;
end;
A29: (
len q1)
= ((
len q0)
+ (
len q3)) by
FINSEQ_1: 22
.= (k
+ 1) by
A19,
FINSEQ_1: 40;
A30: for i be
Nat, a,b be
Element of (
REAL n) st 1
<= i & i
< (k
+ 1) & a
= (q1
/. i) & b
= (p0
/. i) holds (q1
/. (i
+ 1))
<> (
0* n) & (q1
. (i
+ 1))
= ((q1
/. i)
- (
|(a, b)|
* (p0
/. i)))
proof
let i be
Nat, a,b be
Element of (
REAL n);
assume that
A31: 1
<= i and
A32: i
< (k
+ 1) and
A33: a
= (q1
/. i) and
A34: b
= (p0
/. i);
A35: i
<= k by
A32,
NAT_1: 13;
A36: (i
+ 1)
<= (k
+ 1) by
A32,
NAT_1: 13;
A37: 1
<= (i
+ 1) by
NAT_1: 12;
per cases by
A35,
XXREAL_0: 1;
suppose
A38: i
< k;
then
A39: (i
+ 1)
<= k by
NAT_1: 13;
then
A40: (q1
. (i
+ 1))
= (q0
. (i
+ 1)) by
A19,
FINSEQ_1: 64,
NAT_1: 12;
A41: (q1
/. (i
+ 1))
= (q1
. (i
+ 1)) by
A29,
A37,
A36,
FINSEQ_4: 15;
A42: (q0
/. (i
+ 1))
= (q0
. (i
+ 1)) by
A19,
A39,
FINSEQ_4: 15,
NAT_1: 12;
(q1
/. i)
= (q1
. i) by
A29,
A31,
A32,
FINSEQ_4: 15
.= (q0
. i) by
A19,
A31,
A35,
FINSEQ_1: 64
.= (q0
/. i) by
A19,
A31,
A35,
FINSEQ_4: 15;
hence (q1
/. (i
+ 1))
<> (
0* n) & (q1
. (i
+ 1))
= ((q1
/. i)
- (
|(a, b)|
* (p0
/. i))) by
A23,
A31,
A33,
A34,
A38,
A40,
A41,
A42;
end;
suppose
A43: i
= k;
then
A44: (q1
/. (i
+ 1))
= (q1
. (k
+ 1)) by
A29,
A26,
FINSEQ_4: 15
.= (q3
. ((k
+ 1)
- (
len q0))) by
A19,
A29,
FINSEQ_1: 24,
XREAL_1: 29
.= ((q0
/. k)
- (
|(a2, b2)|
* (p0
/. k))) by
A19,
FINSEQ_1: 40;
A45:
now
assume (q1
/. (i
+ 1))
= (
0* n);
then (((q0
/. k)
- (
|(a2, b2)|
* (p0
/. k)))
+ (
|(a2, b2)|
* (p0
/. k)))
= (
|(a2, b2)|
* (p0
/. k)) by
A44,
EUCLID_4: 1;
then
A46: (q0
/. k)
= (
|(a2, b2)|
* (p0
/. k)) by
RVSUM_1: 43;
(q0
/. k)
= (q0
. k) by
A19,
A20,
FINSEQ_4: 15;
hence contradiction by
A16,
A21,
A46,
XREAL_1: 6;
end;
k
< (k
+ 1) by
XREAL_1: 29;
then
A47: (q1
/. k)
= (q1
. k) by
A20,
A29,
FINSEQ_4: 15;
A48: (q0
/. k)
= (q0
. k) by
A19,
A20,
FINSEQ_4: 15;
(q1
. k)
= (q0
. k) by
A19,
A20,
FINSEQ_1: 64;
hence (q1
/. (i
+ 1))
<> (
0* n) & (q1
. (i
+ 1))
= ((q1
/. i)
- (
|(a, b)|
* (p0
/. i))) by
A29,
A33,
A34,
A37,
A43,
A48,
A47,
A44,
A45,
FINSEQ_4: 15;
end;
end;
A49: for s1 be
Element of (
RealVectSpace (
Seg n)), a01 be
Real st s1
in B holds (a01
* s1)
in the
carrier of (
Lin B)
proof
let s1 be
Element of (
RealVectSpace (
Seg n)), a01 be
Real;
assume
A50: s1
in B;
{s1}
c= B by
A50,
TARSKI:def 1;
then (
Lin
{s1}) is
Subspace of (
Lin B) by
RLVECT_3: 20;
then
A51: the
carrier of (
Lin
{s1})
c= the
carrier of (
Lin B) by
RLSUB_1:def 2;
(a01
* s1)
in (
Lin
{s1}) by
RLVECT_4: 8;
then (a01
* s1)
in the
carrier of (
Lin
{s1});
hence (a01
* s1)
in the
carrier of (
Lin B) by
A51;
end;
A52: for s1 be
Element of (
REAL n), a01 be
Real st s1
in B holds (a01
* s1)
in the
carrier of (
Lin B)
proof
let s1 be
Element of (
REAL n), a01 be
Real;
reconsider s10 = s1 as
Element of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
reconsider aa = a01 as
Element of
REAL by
XREAL_0:def 1;
A53: (aa
* s1)
= (aa
* s10);
assume s1
in B;
hence (a01
* s1)
in the
carrier of (
Lin B) by
A49,
A53;
end;
defpred
G[
Nat] means ($1
+ 1)
<= (k
+ 1) implies not (q1
. ($1
+ 1))
in the
carrier of (
Lin B);
A54: for j be
Nat st
G[j] holds
G[(j
+ 1)]
proof
let j be
Nat;
assume
A55:
G[j];
((j
+ 1)
+ 1)
<= (k
+ 1) implies not (q1
. ((j
+ 1)
+ 1))
in the
carrier of (
Lin B)
proof
A56: B
c= the
carrier of (
Lin B)
proof
let z be
object;
assume z
in B;
then z
in (
Lin B) by
RLVECT_3: 15;
hence z
in the
carrier of (
Lin B);
end;
assume ((j
+ 1)
+ 1)
<= (k
+ 1);
then
A57: (j
+ 1)
<= k by
XREAL_1: 6;
per cases by
A57,
XXREAL_0: 1;
suppose
A58: (j
+ 1)
= k;
1
<= (j
+ 1) by
NAT_1: 12;
then (j
+ 1)
in (
dom p0) by
A18,
A58,
FINSEQ_3: 25;
then
A59: (p0
. (j
+ 1))
in (
rng p) by
FUNCT_1:def 3;
A60: (p0
/. (j
+ 1))
= (p0
. (j
+ 1)) by
A18,
A58,
FINSEQ_4: 15,
NAT_1: 12;
1
<= ((j
+ 1)
+ 1) by
NAT_1: 12;
then
A61: (q1
/. ((j
+ 1)
+ 1))
= (q1
. ((j
+ 1)
+ 1)) by
A29,
A58,
FINSEQ_4: 15;
A62: (q1
. ((j
+ 1)
+ 1))
= ((q0
/. k)
- (
|(a2, b2)|
* (p0
/. k))) by
A19,
A58,
FINSEQ_1: 42;
now
assume
A63: (q1
. ((j
+ 1)
+ 1))
in the
carrier of (
Lin B);
((q1
/. ((j
+ 1)
+ 1))
+ (
|(a2, b2)|
* (p0
/. (j
+ 1))))
= (q0
/. (j
+ 1)) by
A58,
A62,
A61,
RVSUM_1: 43;
then (q0
/. (j
+ 1))
in the
carrier of (
Lin B) by
A10,
A56,
A61,
A59,
A60,
A63,
Lm4;
then
A64: (q0
/. (j
+ 1))
in (
Lin B);
A65: not (q1
. (j
+ 1))
in (
Lin B) by
A55,
A58,
XREAL_1: 29;
(q1
. (j
+ 1))
= (q0
. (j
+ 1)) by
A19,
A20,
A58,
FINSEQ_1: 64;
hence contradiction by
A19,
A20,
A58,
A64,
A65,
FINSEQ_4: 15;
end;
hence not (q1
. ((j
+ 1)
+ 1))
in the
carrier of (
Lin B);
end;
suppose
A66: (j
+ 1)
< k;
reconsider a11 = (q0
/. (j
+ 1)), b11 = (p0
/. (j
+ 1)) as
Element of (
REAL n);
A67: ((j
+ 1)
+ 1)
<= k by
A66,
NAT_1: 13;
then
A68: (q0
/. ((j
+ 1)
+ 1))
= (q0
. ((j
+ 1)
+ 1)) by
A19,
FINSEQ_4: 15,
NAT_1: 12;
A69: (j
+ 1)
<= (
len p0) by
A18,
A66,
XXREAL_0: 2;
then
A70: (p0
/. (j
+ 1))
= (p0
. (j
+ 1)) by
FINSEQ_4: 15,
NAT_1: 12;
A71: 1
<= (j
+ 1) by
NAT_1: 12;
then (j
+ 1)
in (
dom p0) by
A69,
FINSEQ_3: 25;
then
A72: (p0
. (j
+ 1))
in (
rng p) by
FUNCT_1:def 3;
A73: (q1
. ((j
+ 1)
+ 1))
= (q0
. ((j
+ 1)
+ 1)) by
A19,
A67,
FINSEQ_1: 64,
NAT_1: 12;
A74: (q0
. ((j
+ 1)
+ 1))
= ((q0
/. (j
+ 1))
- (
|(a11, b11)|
* (p0
/. (j
+ 1)))) by
A23,
A66,
A71;
now
assume
A75: (q1
. ((j
+ 1)
+ 1))
in the
carrier of (
Lin B);
((q0
/. ((j
+ 1)
+ 1))
+ (
|(a11, b11)|
* (p0
/. (j
+ 1))))
= (q0
/. (j
+ 1)) by
A74,
A68,
RVSUM_1: 43;
then (q0
/. (j
+ 1))
in the
carrier of (
Lin B) by
A10,
A56,
A68,
A72,
A70,
A73,
A75,
Lm4;
then
A76: (q0
/. (j
+ 1))
in (
Lin B);
k
< (k
+ 1) by
XREAL_1: 29;
then
A77: not (q1
. (j
+ 1))
in (
Lin B) by
A55,
A66,
XXREAL_0: 2;
(q1
. (j
+ 1))
= (q0
. (j
+ 1)) by
A19,
A66,
FINSEQ_1: 64,
NAT_1: 12;
hence contradiction by
A19,
A66,
A76,
A77,
FINSEQ_4: 15,
NAT_1: 12;
end;
hence not (q1
. ((j
+ 1)
+ 1))
in the
carrier of (
Lin B);
end;
end;
hence
G[(j
+ 1)];
end;
A78:
G[
0 ] by
A8,
A22,
A24,
FINSEQ_1:def 7;
for j be
Nat holds
G[j] from
NAT_1:sch 2(
A78,
A54);
then (k
+ 1)
<= (
len p) implies for d be
Real holds not (q1
. (k
+ 1))
= (d
* (p0
/. (k
+ 1))) by
A52,
A27;
hence
P[(k
+ 1)] by
A29,
A25,
A30;
end;
suppose
A79: 1
> k;
reconsider q1 =
<*z0*> as
FinSequence of (
REAL n);
A80: (
len q1)
= 1 by
FINSEQ_1: 40;
A81: (q1
. 1)
= z0 by
FINSEQ_1: 40;
A82: 1
<= (
len p) implies for d be
Real holds not (q1
. 1)
= (d
* (p0
/. 1))
proof
assume
A83: 1
<= (
len p);
thus for d be
Real holds not (q1
. 1)
= (d
* (p0
/. 1))
proof
let d be
Real;
A84: (q1
. 1)
= z0 by
FINSEQ_1: 40;
1
in (
dom p0) by
A83,
FINSEQ_3: 25;
then
A85: (p0
. 1)
in B by
A10,
FUNCT_1:def 3;
(p0
/. 1)
= (p0
. 1) by
A83,
FINSEQ_4: 15;
hence not (q1
. 1)
= (d
* (p0
/. 1)) by
A2,
A7,
A9,
A13,
A84,
A85,
Th39;
end;
end;
A86: for i be
Nat, a,b be
Element of (
REAL n) st 1
<= i & i
< 1 & a
= (q1
/. i) & b
= (p0
/. i) holds (q1
/. (i
+ 1))
<> (
0* n) & (q1
. (i
+ 1))
= ((q1
/. i)
- (
|(a, b)|
* (p0
/. i)));
k
=
0 by
A79,
NAT_1: 14;
hence
P[(k
+ 1)] by
A80,
A81,
A82,
A86;
end;
end;
suppose not (1
<= (k
+ 1) & (k
+ 1)
<= ((
len p)
+ 1));
hence
P[(k
+ 1)];
end;
end;
A87:
P[
0 ];
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A87,
A14);
then
consider q be
FinSequence of (
REAL n) such that
A88: (
len q)
= ((
len p)
+ 1) and ((
len p)
+ 1)
<= (
len p) implies for d be
Real holds not (q
. ((
len p)
+ 1))
= (d
* (p0
/. ((
len p)
+ 1))) and (q
. 1)
= z0 and
A89: for i be
Nat, a,b be
Element of (
REAL n) st 1
<= i & i
< ((
len p)
+ 1) & a
= (q
/. i) & b
= (p0
/. i) holds (q
/. (i
+ 1))
<> (
0* n) & (q
. (i
+ 1))
= ((q
/. i)
- (
|(a, b)|
* (p0
/. i))) by
A12;
reconsider u4 = (q
/. (
len q)) as
Element of (
REAL n);
A90: (
len p)
< ((
len p)
+ 1) by
XREAL_1: 29;
set u0 = ((1
/
|.u4.|)
* u4);
reconsider B3 = (B
\/
{u0}) as
Subset of (
REAL n) by
XBOOLE_1: 8;
A91: for i be
Nat, s be
Element of (
REAL n) st 1
<= i & i
<= (
len p) & (p0
/. i)
= s holds
|(s, u4)|
=
0
proof
defpred
Q[
Nat] means for s2 be
Element of (
REAL n), u2 be
Element of (
REAL n), k be
Nat st u2
= (q
/. $1) & 1
<= k & k
< $1 & $1
<= ((
len p)
+ 1) & (p0
/. k)
= s2 holds
|(s2, u2)|
=
0 ;
let i be
Nat, s be
Element of (
REAL n);
assume that
A92: 1
<= i and
A93: i
<= (
len p) and
A94: (p0
/. i)
= s;
A95: for k be
Nat st
Q[k] holds
Q[(k
+ 1)]
proof
let k be
Nat;
assume
A96:
Q[k];
for s2 be
Element of (
REAL n), u2 be
Element of (
REAL n), k2 be
Nat st u2
= (q
/. (k
+ 1)) & 1
<= k2 & k2
< (k
+ 1) & (k
+ 1)
<= ((
len p)
+ 1) & (p0
/. k2)
= s2 holds
|(s2, u2)|
=
0
proof
A97: k
< (k
+ 1) by
XREAL_1: 29;
let s2 be
Element of (
REAL n), u2 be
Element of (
REAL n), k2 be
Nat;
assume that
A98: u2
= (q
/. (k
+ 1)) and
A99: 1
<= k2 and
A100: k2
< (k
+ 1) and
A101: (k
+ 1)
<= ((
len p)
+ 1) and
A102: (p0
/. k2)
= s2;
A103: k2
<= k by
A100,
NAT_1: 13;
per cases ;
suppose k
=
0 ;
hence
|(s2, u2)|
=
0 by
A99,
A100;
end;
suppose
A104: k
<>
0 ;
reconsider a = (q
/. k), b = (p0
/. k) as
Element of (
REAL n);
A105: k
< ((
len p)
+ 1) by
A101,
A97,
XXREAL_0: 2;
then
A106: k
<= (
len p) by
NAT_1: 13;
1
<= (k
+ 1) by
A99,
A100,
XXREAL_0: 2;
then
A107: u2
= (q
. (k
+ 1)) by
A88,
A98,
A101,
FINSEQ_4: 15;
A108: (
0
+ 1)
<= k by
A104,
NAT_1: 13;
then (q
. (k
+ 1))
= ((q
/. k)
- (
|(a, b)|
* (p0
/. k))) by
A89,
A105;
then
A109:
|(s2, u2)|
= (
|(s2, a)|
-
|(s2, (
|(a, b)|
* b))|) by
A107,
EUCLID_4: 26
.= (
|(s2, a)|
- (
|(a, b)|
*
|(s2, b)|)) by
EUCLID_4: 21;
per cases by
A103,
XXREAL_0: 1;
suppose
A110: k2
< k;
then
A111: 1
< k by
A99,
XXREAL_0: 2;
then
A112: (p0
/. k)
= (p0
. k) by
A106,
FINSEQ_4: 15;
k
in (
Seg (
len p0)) by
A106,
A111,
FINSEQ_1: 1;
then
A113: k
in (
dom p0) by
FINSEQ_1:def 3;
then
A114: b
in D by
A10,
A112,
FUNCT_1:def 3;
A115: k
< ((
len p)
+ 1) by
A101,
A97,
XXREAL_0: 2;
A116: k2
< (
len p) by
A106,
A110,
XXREAL_0: 2;
then
A117: (p0
/. k2)
= (p0
. k2) by
A99,
FINSEQ_4: 15;
k2
in (
Seg (
len p0)) by
A99,
A116,
FINSEQ_1: 1;
then
A118: k2
in (
dom p0) by
FINSEQ_1:def 3;
then
A119: s2
in (
rng p) by
A102,
A117,
FUNCT_1:def 3;
s2
<> b by
A11,
A102,
A110,
A117,
A118,
A112,
A113,
FUNCT_1:def 4;
then
|(s2, b)|
=
0 by
A10,
A119,
A114,
Def3;
hence
|(s2, u2)|
=
0 by
A96,
A99,
A102,
A109,
A110,
A115;
end;
suppose
A120: k2
= k;
k
in (
Seg (
len p0)) by
A108,
A106,
FINSEQ_1: 1;
then
A121: k
in (
dom p0) by
FINSEQ_1:def 3;
(p0
/. k)
= (p0
. k) by
A108,
A106,
FINSEQ_4: 15;
then b
in (
rng p0) by
A121,
FUNCT_1:def 3;
then
|.b.|
= 1 by
A10,
Def4;
then (
|.b.|
^2 )
= 1;
hence
|(s2, u2)|
= (
|(b, a)|
- (
|(a, b)|
* 1)) by
A102,
A109,
A120,
EUCLID_2: 4
.=
0 ;
end;
end;
end;
hence
Q[(k
+ 1)];
end;
A122:
Q[
0 ];
A123: for k be
Nat holds
Q[k] from
NAT_1:sch 2(
A122,
A95);
(
len p)
< ((
len p)
+ 1) by
XREAL_1: 29;
then i
< (
len q) by
A88,
A93,
XXREAL_0: 2;
hence
|(s, u4)|
=
0 by
A88,
A92,
A94,
A123;
end;
A124: for i be
Nat, s be
Element of (
REAL n) st 1
<= i & i
<= (
len p) & (p0
/. i)
= s holds
|(s, u0)|
=
0
proof
let i be
Nat, s be
Element of (
REAL n);
assume that
A125: 1
<= i and
A126: i
<= (
len p) and
A127: (p0
/. i)
= s;
A128:
|(s, u0)|
= ((1
/
|.u4.|)
*
|(s, u4)|) by
EUCLID_4: 22;
|(s, u4)|
=
0 by
A91,
A125,
A126,
A127;
hence
|(s, u0)|
=
0 by
A128;
end;
for x,y be
real-valued
FinSequence st x
in B3 & y
in B3 & x
<> y holds
|(x, y)|
=
0
proof
let x,y be
real-valued
FinSequence;
assume that
A129: x
in B3 and
A130: y
in B3 and
A131: x
<> y;
per cases by
A129,
A130,
XBOOLE_0:def 3;
suppose x
in B & y
in B;
hence
|(x, y)|
=
0 by
A131,
Def3;
end;
suppose
A132: x
in B & y
in
{u0};
then
consider x3 be
object such that
A133: x3
in (
dom p0) and
A134: x
= (p0
. x3) by
A10,
FUNCT_1:def 3;
reconsider j = x3 as
Element of
NAT by
A133;
A135: x3
in (
Seg (
len p0)) by
A133,
FINSEQ_1:def 3;
then
A136: j
<= (
len p0) by
FINSEQ_1: 1;
A137: y
= u0 by
A132,
TARSKI:def 1;
A138: 1
<= j by
A135,
FINSEQ_1: 1;
then (p0
. x3)
= (p0
/. j) by
A136,
FINSEQ_4: 15;
hence
|(x, y)|
=
0 by
A124,
A137,
A134,
A138,
A136;
end;
suppose
A139: x
in
{u0} & y
in B;
then
consider y3 be
object such that
A140: y3
in (
dom p0) and
A141: y
= (p0
. y3) by
A10,
FUNCT_1:def 3;
reconsider j = y3 as
Element of
NAT by
A140;
A142: y3
in (
Seg (
len p0)) by
A140,
FINSEQ_1:def 3;
then
A143: j
<= (
len p0) by
FINSEQ_1: 1;
A144: x
= u0 by
A139,
TARSKI:def 1;
A145: 1
<= j by
A142,
FINSEQ_1: 1;
then (p0
. y3)
= (p0
/. j) by
A143,
FINSEQ_4: 15;
hence
|(x, y)|
=
0 by
A124,
A144,
A141,
A145,
A143;
end;
suppose
A146: x
in
{u0} & y
in
{u0};
then y
= u0 by
TARSKI:def 1;
hence
|(x, y)|
=
0 by
A131,
A146,
TARSKI:def 1;
end;
end;
then
A147: B3 is
R-orthogonal;
1
in (
dom p) by
A1,
A10,
FINSEQ_3: 32;
then
A148: 1
<= (
len p) by
FINSEQ_3: 25;
set aq = (q
/. (
len p)), bq = (p0
/. (
len p));
A149: bq
= (p0
/. (
len p));
aq
= (q
/. (
len p));
then
A150:
|.u4.|
<>
0 by
A148,
A88,
A89,
A149,
A90,
EUCLID: 8;
A151:
|.(1
/
|.u4.|).|
= (1
/
|.u4.|) by
ABSVALUE:def 1;
A152: u0
in
{u0} by
TARSKI:def 1;
A153:
|.u0.|
= (
|.(1
/
|.u4.|).|
*
|.u4.|) by
EUCLID: 11
.= 1 by
A150,
A151,
XCMPLX_1: 106;
then B3 is
R-normal by
Th16;
then D0
= B3 by
A4,
A147,
XBOOLE_1: 7;
then u0
in B by
A152,
XBOOLE_0:def 3;
then
consider x3 be
object such that
A154: x3
in (
dom p0) and
A155: u0
= (p0
. x3) by
A10,
FUNCT_1:def 3;
reconsider j = x3 as
Element of
NAT by
A154;
A156: x3
in (
Seg (
len p0)) by
A154,
FINSEQ_1:def 3;
then
A157: j
<= (
len p0) by
FINSEQ_1: 1;
A158: 1
<= j by
A156,
FINSEQ_1: 1;
then (p0
. x3)
= (p0
/. j) by
A157,
FINSEQ_4: 15;
then
|(u0, u0)|
=
0 by
A124,
A155,
A158,
A157;
hence B is
Basis of (
RealVectSpace (
Seg n)) by
A153,
EUCLID_4: 16;
end;
(
Lin B) is
Subspace of (
Lin I) by
A2,
RLVECT_3: 20;
then
A159: the
carrier of (
Lin B)
c= the
carrier of (
Lin I) by
RLSUB_1:def 2;
A160: (
Lin I)
= V by
RLVECT_3:def 3;
now
assume I
c= the
carrier of (
Lin B);
then (
Lin I) is
Subspace of (
Lin B) by
RLVECT_5: 19;
then the
carrier of (
Lin I)
c= the
carrier of (
Lin B) by
RLSUB_1:def 2;
then the
carrier of (
Lin I)
= the
carrier of (
Lin B) by
A159,
XBOOLE_0:def 10;
then (
Lin B)
= (
Lin I) by
RLSUB_1: 30;
hence B is
Basis of (
RealVectSpace (
Seg n)) by
A160,
RLVECT_3:def 3;
end;
hence thesis by
A5;
end;
end;
registration
let n be
Element of
NAT ;
cluster (
REAL-US n) ->
finite-dimensional;
coherence
proof
reconsider B = (
RN_Base n) as
Subset of (
REAL-US n) by
REAL_NS1:def 6;
B is
Basis of (
REAL-US n) by
Th48;
hence thesis by
RUSUB_4:def 1;
end;
end
theorem ::
EUCLID_7:51
for n be
Element of
NAT holds (
dim (
REAL-US n))
= n
proof
let n be
Element of
NAT ;
reconsider B = (
RN_Base n) as
Subset of (
REAL-US n) by
REAL_NS1:def 6;
for I be
Basis of (
REAL-US n) holds n
= (
card I)
proof
let I be
Basis of (
REAL-US n);
B is
Basis of (
REAL-US n) by
Th48;
then (
card B)
= (
card I) by
RUSUB_4: 5;
hence n
= (
card I) by
Lm5;
end;
hence thesis by
RUSUB_4:def 2;
end;
theorem ::
EUCLID_7:52
for B be
Orthogonal_Basis of n holds (
card B)
= n
proof
let B be
Orthogonal_Basis of n;
reconsider B0 = B as
Subset of (
RealVectSpace (
Seg n)) by
FINSEQ_2: 93;
B0 is
Basis of (
RealVectSpace (
Seg n)) by
Th49;
hence (
card B)
= n by
Th46;
end;