:: ANALORT semantic presentation
Lemma1:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1
for b6, b7, b8, b9 being Real holds
( b2 = (b6 * b3) + (b7 * b4) & b5 = (b8 * b3) + (b9 * b4) implies ( b2 + b5 = ((b6 + b8) * b3) + ((b7 + b9) * b4) & b2 - b5 = ((b6 - b8) * b3) + ((b7 - b9) * b4) ) )
Lemma2:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1
for b5, b6, b7 being Real holds
( b2 = (b5 * b3) + (b6 * b4) implies b7 * b2 = ((b7 * b5) * b3) + ((b7 * b6) * b4) )
Lemma3:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Real holds
( Gen b2,b3 & (b4 * b2) + (b5 * b3) = (b6 * b2) + (b7 * b3) implies ( b4 = b6 & b5 = b7 ) )
Lemma4:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds
( Gen b2,b3 implies ( b2 <> 0. b1 & b3 <> 0. b1 ) )
Lemma5:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
( Gen b2,b3 implies b4 = ((pr1 b2,b3,b4) * b2) + ((pr2 b2,b3,b4) * b3) )
Lemma6:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1
for b5, b6 being Real holds
( Gen b2,b3 & b4 = (b5 * b2) + (b6 * b3) implies ( b5 = pr1 b2,b3,b4 & b6 = pr2 b2,b3,b4 ) )
Lemma7:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1
for b6 being Real holds
( Gen b2,b3 implies ( pr1 b2,b3,(b4 + b5) = (pr1 b2,b3,b4) + (pr1 b2,b3,b5) & pr2 b2,b3,(b4 + b5) = (pr2 b2,b3,b4) + (pr2 b2,b3,b5) & pr1 b2,b3,(b4 - b5) = (pr1 b2,b3,b4) - (pr1 b2,b3,b5) & pr2 b2,b3,(b4 - b5) = (pr2 b2,b3,b4) - (pr2 b2,b3,b5) & pr1 b2,b3,(b6 * b4) = b6 * (pr1 b2,b3,b4) & pr2 b2,b3,(b6 * b4) = b6 * (pr2 b2,b3,b4) ) )
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4 be
VECTOR of c
1;
func Ortm c
2,c
3,c
4 -> VECTOR of a
1 equals :: ANALORT:def 1
((pr1 a2,a3,a4) * a2) + ((- (pr2 a2,a3,a4)) * a3);
correctness
coherence
((pr1 c2,c3,c4) * c2) + ((- (pr2 c2,c3,c4)) * c3) is VECTOR of c1;
;
end;
:: deftheorem Def1 defines Ortm ANALORT:def 1 :
theorem Th1: :: ANALORT:1
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies
Ortm b
2,b
3,
(b4 + b5) = (Ortm b2,b3,b4) + (Ortm b2,b3,b5) )
theorem Th2: :: ANALORT:2
theorem Th3: :: ANALORT:3
theorem Th4: :: ANALORT:4
theorem Th5: :: ANALORT:5
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies
Ortm b
2,b
3,
(b4 - b5) = (Ortm b2,b3,b4) - (Ortm b2,b3,b5) )
theorem Th6: :: ANALORT:6
theorem Th7: :: ANALORT:7
theorem Th8: :: ANALORT:8
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4 be
VECTOR of c
1;
func Orte c
2,c
3,c
4 -> VECTOR of a
1 equals :: ANALORT:def 2
((pr2 a2,a3,a4) * a2) + ((- (pr1 a2,a3,a4)) * a3);
correctness
coherence
((pr2 c2,c3,c4) * c2) + ((- (pr1 c2,c3,c4)) * c3) is VECTOR of c1;
;
end;
:: deftheorem Def2 defines Orte ANALORT:def 2 :
theorem Th9: :: ANALORT:9
theorem Th10: :: ANALORT:10
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies
Orte b
2,b
3,
(b4 + b5) = (Orte b2,b3,b4) + (Orte b2,b3,b5) )
theorem Th11: :: ANALORT:11
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies
Orte b
2,b
3,
(b4 - b5) = (Orte b2,b3,b4) - (Orte b2,b3,b5) )
theorem Th12: :: ANALORT:12
theorem Th13: :: ANALORT:13
theorem Th14: :: ANALORT:14
theorem Th15: :: ANALORT:15
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7 be
VECTOR of c
1;
pred c
4,c
5,c
6,c
7 are_COrte_wrt c
2,c
3 means :
Def3:
:: ANALORT:def 3
Orte a
2,a
3,a
4,
Orte a
2,a
3,a
5 // a
6,a
7;
pred c
4,c
5,c
6,c
7 are_COrtm_wrt c
2,c
3 means :
Def4:
:: ANALORT:def 4
Ortm a
2,a
3,a
4,
Ortm a
2,a
3,a
5 // a
6,a
7;
end;
:: deftheorem Def3 defines are_COrte_wrt ANALORT:def 3 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 iff
Orte b
2,b
3,b
4,
Orte b
2,b
3,b
5 // b
6,b
7 );
:: deftheorem Def4 defines are_COrtm_wrt ANALORT:def 4 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 iff
Ortm b
2,b
3,b
4,
Ortm b
2,b
3,b
5 // b
6,b
7 );
theorem Th16: :: ANALORT:16
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5 // b
6,b
7 implies
Orte b
2,b
3,b
4,
Orte b
2,b
3,b
5 // Orte b
2,b
3,b
6,
Orte b
2,b
3,b
7 )
theorem Th17: :: ANALORT:17
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5 // b
6,b
7 implies
Ortm b
2,b
3,b
4,
Ortm b
2,b
3,b
5 // Ortm b
2,b
3,b
6,
Ortm b
2,b
3,b
7 )
theorem Th18: :: ANALORT:18
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 implies b
6,b
7,b
5,b
4 are_COrte_wrt b
2,b
3 )
theorem Th19: :: ANALORT:19
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 implies b
6,b
7,b
4,b
5 are_COrtm_wrt b
2,b
3 )
theorem Th20: :: ANALORT:20
theorem Th21: :: ANALORT:21
theorem Th22: :: ANALORT:22
theorem Th23: :: ANALORT:23
theorem Th24: :: ANALORT:24
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies b
4,b
5,
Orte b
2,b
3,b
4,
Orte b
2,b
3,b
5 are_Ort_wrt b
2,b
3 )
theorem Th25: :: ANALORT:25
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds b
2,b
3,
Orte b
4,b
5,b
2,
Orte b
4,b
5,b
3 are_COrte_wrt b
4,b
5
theorem Th26: :: ANALORT:26
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds b
2,b
3,
Ortm b
4,b
5,b
2,
Ortm b
4,b
5,b
3 are_COrtm_wrt b
4,b
5
theorem Th27: :: ANALORT:27
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies ( b
4,b
5 // b
6,b
7 iff ex b
8, b
9 being
VECTOR of b
1 st
( b
8 <> b
9 & b
8,b
9,b
4,b
5 are_COrte_wrt b
2,b
3 & b
8,b
9,b
6,b
7 are_COrte_wrt b
2,b
3 ) ) )
theorem Th28: :: ANALORT:28
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies ( b
4,b
5 // b
6,b
7 iff ex b
8, b
9 being
VECTOR of b
1 st
( b
8 <> b
9 & b
8,b
9,b
4,b
5 are_COrtm_wrt b
2,b
3 & b
8,b
9,b
6,b
7 are_COrtm_wrt b
2,b
3 ) ) )
theorem Th29: :: ANALORT:29
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies ( b
4,b
5,b
6,b
7 are_Ort_wrt b
2,b
3 iff ( b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 or b
4,b
5,b
7,b
6 are_COrte_wrt b
2,b
3 ) ) )
theorem Th30: :: ANALORT:30
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
4,b
5,b
7,b
6 are_COrte_wrt b
2,b
3 & not b
4 = b
5 & not b
6 = b
7 )
theorem Th31: :: ANALORT:31
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
4,b
5,b
7,b
6 are_COrtm_wrt b
2,b
3 & not b
4 = b
5 & not b
6 = b
7 )
theorem Th32: :: ANALORT:32
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
4,b
5,b
6,b
8 are_COrte_wrt b
2,b
3 & not b
4,b
5,b
7,b
8 are_COrte_wrt b
2,b
3 & not b
4,b
5,b
8,b
7 are_COrte_wrt b
2,b
3 )
theorem Th33: :: ANALORT:33
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
4,b
5,b
6,b
8 are_COrtm_wrt b
2,b
3 & not b
4,b
5,b
7,b
8 are_COrtm_wrt b
2,b
3 & not b
4,b
5,b
8,b
7 are_COrtm_wrt b
2,b
3 )
theorem Th34: :: ANALORT:34
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
2,b
3,b
4,b
5 are_COrte_wrt b
6,b
7 implies b
3,b
2,b
5,b
4 are_COrte_wrt b
6,b
7 )
theorem Th35: :: ANALORT:35
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
2,b
3,b
4,b
5 are_COrtm_wrt b
6,b
7 implies b
3,b
2,b
5,b
4 are_COrtm_wrt b
6,b
7 )
theorem Th36: :: ANALORT:36
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
4,b
5,b
7,b
8 are_COrte_wrt b
2,b
3 implies b
4,b
5,b
6,b
8 are_COrte_wrt b
2,b
3 )
theorem Th37: :: ANALORT:37
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
4,b
5,b
7,b
8 are_COrtm_wrt b
2,b
3 implies b
4,b
5,b
6,b
8 are_COrtm_wrt b
2,b
3 )
theorem Th38: :: ANALORT:38
theorem Th39: :: ANALORT:39
theorem Th40: :: ANALORT:40
theorem Th41: :: ANALORT:41
theorem Th42: :: ANALORT:42
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
8,b
9,b
6,b
7 are_COrte_wrt b
2,b
3 & b
8,b
9,b
10,b
11 are_COrte_wrt b
2,b
3 & not b
8 = b
9 & not b
6 = b
7 & not b
4,b
5,b
10,b
11 are_COrte_wrt b
2,b
3 )
theorem Th43: :: ANALORT:43
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
8,b
9,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
8,b
9,b
10,b
11 are_COrtm_wrt b
2,b
3 & not b
8 = b
9 & not b
6 = b
7 & not b
4,b
5,b
10,b
11 are_COrtm_wrt b
2,b
3 )
theorem Th44: :: ANALORT:44
canceled;
theorem Th45: :: ANALORT:45
canceled;
theorem Th46: :: ANALORT:46
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_COrte_wrt b
2,b
3 & b
10,b
11,b
8,b
9 are_COrte_wrt b
2,b
3 & not b
4,b
5,b
10,b
11 are_COrte_wrt b
2,b
3 & not b
6 = b
7 & not b
8 = b
9 )
theorem Th47: :: ANALORT:47
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_COrtm_wrt b
2,b
3 & b
10,b
11,b
8,b
9 are_COrtm_wrt b
2,b
3 & not b
4,b
5,b
10,b
11 are_COrtm_wrt b
2,b
3 & not b
6 = b
7 & not b
8 = b
9 )
theorem Th48: :: ANALORT:48
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrte_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_COrte_wrt b
2,b
3 & b
4,b
5,b
10,b
11 are_COrte_wrt b
2,b
3 & not b
10,b
11,b
8,b
9 are_COrte_wrt b
2,b
3 & not b
6 = b
7 & not b
4 = b
5 )
theorem Th49: :: ANALORT:49
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_COrtm_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_COrtm_wrt b
2,b
3 & b
4,b
5,b
10,b
11 are_COrtm_wrt b
2,b
3 & not b
10,b
11,b
8,b
9 are_COrtm_wrt b
2,b
3 & not b
6 = b
7 & not b
4 = b
5 )
theorem Th50: :: ANALORT:50
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
not ( not b
4,b
7,b
5,b
6 are_COrte_wrt b
2,b
3 & not b
4,b
7,b
6,b
5 are_COrte_wrt b
2,b
3 & b
6,b
8,b
6,b
5 are_COrte_wrt b
2,b
3 & ( for b
9 being
VECTOR of b
1 holds
not ( ( b
4,b
7,b
4,b
9 are_COrte_wrt b
2,b
3 or b
4,b
7,b
9,b
4 are_COrte_wrt b
2,b
3 ) & ( b
6,b
8,b
6,b
9 are_COrte_wrt b
2,b
3 or b
6,b
8,b
9,b
6 are_COrte_wrt b
2,b
3 ) ) ) ) )
theorem Th51: :: ANALORT:51
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & ( for b
4, b
5, b
6 being
VECTOR of b
1 holds
not ( b
4,b
5,b
4,b
6 are_COrte_wrt b
2,b
3 & ( for b
7, b
8 being
VECTOR of b
1 holds
not ( b
7,b
8,b
4,b
5 are_COrte_wrt b
2,b
3 & not ( not b
7,b
8,b
4,b
6 are_COrte_wrt b
2,b
3 & not b
7,b
8,b
6,b
4 are_COrte_wrt b
2,b
3 ) & not b
7 = b
8 ) ) ) ) )
theorem Th52: :: ANALORT:52
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
not ( not b
4,b
7,b
5,b
6 are_COrtm_wrt b
2,b
3 & not b
4,b
7,b
6,b
5 are_COrtm_wrt b
2,b
3 & b
6,b
8,b
6,b
5 are_COrtm_wrt b
2,b
3 & ( for b
9 being
VECTOR of b
1 holds
not ( ( b
4,b
7,b
4,b
9 are_COrtm_wrt b
2,b
3 or b
4,b
7,b
9,b
4 are_COrtm_wrt b
2,b
3 ) & ( b
6,b
8,b
6,b
9 are_COrtm_wrt b
2,b
3 or b
6,b
8,b
9,b
6 are_COrtm_wrt b
2,b
3 ) ) ) ) )
theorem Th53: :: ANALORT:53
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & ( for b
4, b
5, b
6 being
VECTOR of b
1 holds
not ( b
4,b
5,b
4,b
6 are_COrtm_wrt b
2,b
3 & ( for b
7, b
8 being
VECTOR of b
1 holds
not ( b
7,b
8,b
4,b
5 are_COrtm_wrt b
2,b
3 & not ( not b
7,b
8,b
4,b
6 are_COrtm_wrt b
2,b
3 & not b
7,b
8,b
6,b
4 are_COrtm_wrt b
2,b
3 ) & not b
7 = b
8 ) ) ) ) )
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func CORTE c
1,c
2,c
3 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def5:
:: ANALORT:def 5
for b
1, b
2 being
set holds
(
[b1,b2] in a
4 iff ex b
3, b
4, b
5, b
6 being
VECTOR of a
1 st
( b
1 = [b3,b4] & b
2 = [b5,b6] & b
3,b
4,b
5,b
6 are_COrte_wrt a
2,a
3 ) );
existence
ex b1 being Relation of [:the carrier of c1,the carrier of c1:] st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4, b5, b6, b7 being VECTOR of c1 st
( b2 = [b4,b5] & b3 = [b6,b7] & b4,b5,b6,b7 are_COrte_wrt c2,c3 ) )
uniqueness
for b1, b2 being Relation of [:the carrier of c1,the carrier of c1:] holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_COrte_wrt c2,c3 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_COrte_wrt c2,c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines CORTE ANALORT:def 5 :
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4 being
Relation of
[:the carrier of b1,the carrier of b1:] holds
( b
4 = CORTE b
1,b
2,b
3 iff for b
5, b
6 being
set holds
(
[b5,b6] in b
4 iff ex b
7, b
8, b
9, b
10 being
VECTOR of b
1 st
( b
5 = [b7,b8] & b
6 = [b9,b10] & b
7,b
8,b
9,b
10 are_COrte_wrt b
2,b
3 ) ) );
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func CORTM c
1,c
2,c
3 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def6:
:: ANALORT:def 6
for b
1, b
2 being
set holds
(
[b1,b2] in a
4 iff ex b
3, b
4, b
5, b
6 being
VECTOR of a
1 st
( b
1 = [b3,b4] & b
2 = [b5,b6] & b
3,b
4,b
5,b
6 are_COrtm_wrt a
2,a
3 ) );
existence
ex b1 being Relation of [:the carrier of c1,the carrier of c1:] st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4, b5, b6, b7 being VECTOR of c1 st
( b2 = [b4,b5] & b3 = [b6,b7] & b4,b5,b6,b7 are_COrtm_wrt c2,c3 ) )
uniqueness
for b1, b2 being Relation of [:the carrier of c1,the carrier of c1:] holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_COrtm_wrt c2,c3 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_COrtm_wrt c2,c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines CORTM ANALORT:def 6 :
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4 being
Relation of
[:the carrier of b1,the carrier of b1:] holds
( b
4 = CORTM b
1,b
2,b
3 iff for b
5, b
6 being
set holds
(
[b5,b6] in b
4 iff ex b
7, b
8, b
9, b
10 being
VECTOR of b
1 st
( b
5 = [b7,b8] & b
6 = [b9,b10] & b
7,b
8,b
9,b
10 are_COrtm_wrt b
2,b
3 ) ) );
:: deftheorem Def7 defines CESpace ANALORT:def 7 :
:: deftheorem Def8 defines CMSpace ANALORT:def 8 :
theorem Th54: :: ANALORT:54
theorem Th55: :: ANALORT:55
theorem Th56: :: ANALORT:56
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1for b
8, b
9, b
10, b
11 being
Element of
(CESpace b1,b6,b7) holds
( b
2 = b
8 & b
3 = b
9 & b
4 = b
10 & b
5 = b
11 implies ( b
8,b
9 // b
10,b
11 iff b
2,b
3,b
4,b
5 are_COrte_wrt b
6,b
7 ) )
theorem Th57: :: ANALORT:57
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1for b
8, b
9, b
10, b
11 being
Element of
(CMSpace b1,b6,b7) holds
( b
2 = b
8 & b
3 = b
9 & b
4 = b
10 & b
5 = b
11 implies ( b
8,b
9 // b
10,b
11 iff b
2,b
3,b
4,b
5 are_COrtm_wrt b
6,b
7 ) )