:: ANALMETR 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 being VECTOR of b1 holds (0 * b2) + (0 * b3) = 0. b1
Lemma3:
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) )
:: deftheorem Def1 defines Gen ANALMETR:def 1 :
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 iff ( ( for b
4 being
VECTOR of b
1 holds
ex b
5, b
6 being
Real st b
4 = (b5 * b2) + (b6 * b3) ) & ( for b
4, b
5 being
Real holds
(
(b4 * b2) + (b5 * b3) = 0. b
1 implies ( b
4 = 0 & b
5 = 0 ) ) ) ) );
:: deftheorem Def2 defines are_Ort_wrt ANALMETR:def 2 :
Lemma6:
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 ) )
theorem Th1: :: ANALMETR:1
canceled;
theorem Th2: :: ANALMETR:2
canceled;
theorem Th3: :: ANALMETR:3
canceled;
theorem Th4: :: ANALMETR:4
canceled;
theorem Th5: :: ANALMETR:5
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
(
Gen b
4,b
5 implies ( b
2,b
3 are_Ort_wrt b
4,b
5 iff for b
6, b
7, b
8, b
9 being
Real holds
( b
2 = (b6 * b4) + (b7 * b5) & b
3 = (b8 * b4) + (b9 * b5) implies
(b6 * b8) + (b7 * b9) = 0 ) ) )
Lemma8:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds
( Gen b2,b3 implies ( b2 <> 0. b1 & b3 <> 0. b1 ) )
theorem Th6: :: ANALMETR:6
theorem Th7: :: ANALMETR:7
theorem Th8: :: ANALMETR:8
theorem Th9: :: ANALMETR:9
theorem Th10: :: ANALMETR:10
theorem Th11: :: ANALMETR:11
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1for b
6, b
7 being
Real holds
( b
2,b
3 are_Ort_wrt b
4,b
5 implies ( b
6 * b
2,b
3 are_Ort_wrt b
4,b
5 & b
2,b
7 * b
3 are_Ort_wrt b
4,b
5 ) )
theorem Th12: :: ANALMETR:12
theorem Th13: :: ANALMETR:13
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5 are_Ort_wrt b
2,b
3 & b
4,b
6 are_Ort_wrt b
2,b
3 & b
4 <> 0. b
1 & ( for b
7, b
8 being
Real holds
not ( b
7 * b
5 = b
8 * b
6 & not ( not b
7 <> 0 & not b
8 <> 0 ) ) ) )
theorem Th14: :: ANALMETR:14
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5 are_Ort_wrt b
2,b
3 & b
4,b
6 are_Ort_wrt b
2,b
3 implies ( b
4,b
5 + b
6 are_Ort_wrt b
2,b
3 & b
4,b
5 - b
6 are_Ort_wrt b
2,b
3 ) )
theorem Th15: :: ANALMETR:15
theorem Th16: :: ANALMETR:16
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5 - b
6 are_Ort_wrt b
2,b
3 & b
5,b
6 - b
4 are_Ort_wrt b
2,b
3 implies b
6,b
4 - b
5 are_Ort_wrt b
2,b
3 )
theorem Th17: :: ANALMETR:17
theorem Th18: :: ANALMETR:18
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
( not ( ( b
2,b
3 // b
4,b
5 or b
2,b
3 // b
5,b
4 ) & ( for b
6, b
7 being
Real holds
not ( b
6 * (b3 - b2) = b
7 * (b5 - b4) & not ( not b
6 <> 0 & not b
7 <> 0 ) ) ) ) & not ( ex b
6, b
7 being
Real st
( b
6 * (b3 - b2) = b
7 * (b5 - b4) & not ( not b
6 <> 0 & not b
7 <> 0 ) ) & not b
2,b
3 // b
4,b
5 & not b
2,b
3 // b
5,b
4 ) )
theorem Th19: :: ANALMETR:19
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
2,c
3,c
4,c
5 are_Ort_wrt c
6,c
7 means :
Def3:
:: ANALMETR:def 3
a
3 - a
2,a
5 - a
4 are_Ort_wrt a
6,a
7;
end;
:: deftheorem Def3 defines are_Ort_wrt ANALMETR: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
2,b
3,b
4,b
5 are_Ort_wrt b
6,b
7 iff b
3 - b
2,b
5 - b
4 are_Ort_wrt b
6,b
7 );
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func Orthogonality c
1,c
2,c
3 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def4:
:: ANALMETR:def 4
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_Ort_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_Ort_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_Ort_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_Ort_wrt c2,c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Orthogonality ANALMETR:def 4 :
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 = Orthogonality 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_Ort_wrt b
2,b
3 ) ) );
theorem Th20: :: ANALMETR:20
canceled;
theorem Th21: :: ANALMETR:21
canceled;
theorem Th22: :: ANALMETR:22
theorem Th23: :: ANALMETR:23
theorem Th24: :: ANALMETR:24
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1for b
6, b
7, b
8, b
9 being
Element of
(Lambda (OASpace b1)) holds
( b
6 = b
2 & b
7 = b
3 & b
8 = b
4 & b
9 = b
5 implies ( b
6,b
7 // b
8,b
9 iff ex b
10, b
11 being
Real st
( b
10 * (b3 - b2) = b
11 * (b5 - b4) & not ( not b
10 <> 0 & not b
11 <> 0 ) ) ) )
:: deftheorem Def5 ANALMETR:def 5 :
canceled;
:: deftheorem Def6 defines _|_ ANALMETR:def 6 :
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func AMSpace c
1,c
2,c
3 -> strict ParOrtStr equals :: ANALMETR:def 7
ParOrtStr(# the
carrier of a
1,
(lambda (DirPar a1)),
(Orthogonality a1,a2,a3) #);
correctness
coherence
ParOrtStr(# the carrier of c1,(lambda (DirPar c1)),(Orthogonality c1,c2,c3) #) is strict ParOrtStr ;
;
end;
:: deftheorem Def7 defines AMSpace ANALMETR:def 7 :
theorem Th25: :: ANALMETR:25
canceled;
theorem Th26: :: ANALMETR:26
canceled;
theorem Th27: :: ANALMETR:27
canceled;
theorem Th28: :: ANALMETR:28
:: deftheorem Def8 defines Af ANALMETR:def 8 :
theorem Th29: :: ANALMETR:29
canceled;
theorem Th30: :: ANALMETR:30
theorem Th31: :: ANALMETR:31
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
(AMSpace b1,b6,b7) holds
( b
8 = b
2 & b
9 = b
3 & b
10 = b
4 & b
11 = b
5 implies ( b
8,b
10 _|_ b
9,b
11 iff b
2,b
4,b
3,b
5 are_Ort_wrt b
6,b
7 ) )
theorem Th32: :: ANALMETR:32
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
(AMSpace b1,b2,b3) holds
( b
8 = b
4 & b
9 = b
5 & b
10 = b
6 & b
11 = b
7 implies ( b
8,b
9 // b
10,b
11 iff ex b
12, b
13 being
Real st
( b
12 * (b5 - b4) = b
13 * (b7 - b6) & not ( not b
12 <> 0 & not b
13 <> 0 ) ) ) )
theorem Th33: :: ANALMETR:33
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7 being
Element of
(AMSpace b1,b2,b3) holds
( b
4,b
5 _|_ b
6,b
7 implies b
6,b
7 _|_ b
4,b
5 )
theorem Th34: :: ANALMETR:34
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7 being
Element of
(AMSpace b1,b2,b3) holds
( b
4,b
5 _|_ b
6,b
7 implies b
4,b
5 _|_ b
7,b
6 )
theorem Th35: :: ANALMETR:35
theorem Th36: :: ANALMETR:36
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of
(AMSpace b1,b2,b3) holds
not ( b
4,b
5 _|_ b
6,b
7 & b
4,b
5 // b
8,b
9 & not b
4 = b
5 & not b
6,b
7 _|_ b
8,b
9 )
theorem Th37: :: ANALMETR:37
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 being
Element of
(AMSpace b1,b2,b3) holds
ex b
7 being
Element of
(AMSpace b1,b2,b3) st
( b
4,b
5 _|_ b
6,b
7 & b
6 <> b
7 ) )
theorem Th38: :: ANALMETR:38
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of
(AMSpace b1,b2,b3) holds
not (
Gen b
2,b
3 & b
4,b
5 _|_ b
6,b
7 & b
4,b
5 _|_ b
8,b
9 & not b
4 = b
5 & not b
6,b
7 // b
8,b
9 )
theorem Th39: :: ANALMETR:39
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7, b
8 being
Element of
(AMSpace b1,b2,b3) holds
(
Gen b
2,b
3 & b
4,b
5 _|_ b
6,b
7 & b
4,b
5 _|_ b
6,b
8 implies b
4,b
5 _|_ b
7,b
8 )
theorem Th40: :: ANALMETR:40
theorem Th41: :: ANALMETR:41
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5, b
6, b
7 being
Element of
(AMSpace b1,b2,b3) holds
(
Gen b
2,b
3 & b
4,b
5 _|_ b
6,b
7 & b
6,b
5 _|_ b
7,b
4 implies b
7,b
5 _|_ b
4,b
6 )
theorem Th42: :: ANALMETR:42
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4, b
5 being
Element of
(AMSpace b1,b2,b3) holds
(
Gen b
2,b
3 & b
4 <> b
5 implies for b
6 being
Element of
(AMSpace b1,b2,b3) holds
ex b
7 being
Element of
(AMSpace b1,b2,b3) st
( b
4,b
5 // b
4,b
7 & b
4,b
5 _|_ b
7,b
6 ) )
consider c1 being RealLinearSpace such that
Lemma39:
ex b1, b2 being VECTOR of c1 st Gen b1,b2
by Th7;
consider c2, c3 being VECTOR of c1 such that
Lemma40:
Gen c2,c3
by Lemma39;
E41:
now
set c
4 =
AffinStruct(# the
carrier of
(AMSpace c1,c2,c3),the
CONGR of
(AMSpace c1,c2,c3) #);
AffinStruct(# the
carrier of
(AMSpace c1,c2,c3),the
CONGR of
(AMSpace c1,c2,c3) #)
= Af (AMSpace c1,c2,c3)
;
then E42:
AffinStruct(# the
carrier of
(AMSpace c1,c2,c3),the
CONGR of
(AMSpace c1,c2,c3) #)
= Lambda (OASpace c1)
by Th30;
for b
1, b
2 being
Real holds
(
(b1 * c2) + (b2 * c3) = 0. c
1 implies ( b
1 = 0 & b
2 = 0 ) )
by Def1, Lemma40;
then
OASpace c
1 is
OAffinSpace
by ANALOAF:38;
hence
(
AffinStruct(# the
carrier of
(AMSpace c1,c2,c3),the
CONGR of
(AMSpace c1,c2,c3) #) is
AffinSpace & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of
(AMSpace c1,c2,c3) holds
( ( b
1,b
2 _|_ b
1,b
2 implies b
1 = b
2 ) & b
1,b
2 _|_ b
3,b
3 & ( b
1,b
2 _|_ b
3,b
4 implies ( b
1,b
2 _|_ b
4,b
3 & b
3,b
4 _|_ b
1,b
2 ) ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 // b
7,b
8 & not b
5,b
6 _|_ b
7,b
8 & not b
1 = b
2 ) & ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 _|_ b
5,b
8 implies b
1,b
2 _|_ b
6,b
8 ) ) ) & ( for b
1, b
2, b
3 being
Element of
(AMSpace c1,c2,c3) holds
not ( b
1 <> b
2 & ( for b
4 being
Element of
(AMSpace c1,c2,c3) holds
not ( b
1,b
2 // b
1,b
4 & b
1,b
2 _|_ b
4,b
3 ) ) ) ) & ( for b
1, b
2, b
3 being
Element of
(AMSpace c1,c2,c3) holds
ex b
4 being
Element of
(AMSpace c1,c2,c3) st
( b
1,b
2 _|_ b
3,b
4 & b
3 <> b
4 ) ) )
by E42, Lemma40, Th33, Th34, Th35, Th36, Th37, Th39, Th40, Th42, DIRAF:48;
end;
definition
let c
4 be non
empty ParOrtStr ;
attr a
1 is
OrtAfSp-like means :
Def9:
:: ANALMETR:def 9
(
AffinStruct(# the
carrier of a
1,the
CONGR of a
1 #) is
AffinSpace & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( ( b
1,b
2 _|_ b
1,b
2 implies b
1 = b
2 ) & b
1,b
2 _|_ b
3,b
3 & ( b
1,b
2 _|_ b
3,b
4 implies ( b
1,b
2 _|_ b
4,b
3 & b
3,b
4 _|_ b
1,b
2 ) ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 // b
7,b
8 & not b
5,b
6 _|_ b
7,b
8 & not b
1 = b
2 ) & ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 _|_ b
5,b
8 implies b
1,b
2 _|_ b
6,b
8 ) ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
not ( b
1 <> b
2 & ( for b
4 being
Element of a
1 holds
not ( b
1,b
2 // b
1,b
4 & b
1,b
2 _|_ b
4,b
3 ) ) ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
ex b
4 being
Element of a
1 st
( b
1,b
2 _|_ b
3,b
4 & b
3 <> b
4 ) ) );
end;
:: deftheorem Def9 defines OrtAfSp-like ANALMETR:def 9 :
for b
1 being non
empty ParOrtStr holds
( b
1 is
OrtAfSp-like iff (
AffinStruct(# the
carrier of b
1,the
CONGR of b
1 #) is
AffinSpace & ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( ( b
2,b
3 _|_ b
2,b
3 implies b
2 = b
3 ) & b
2,b
3 _|_ b
4,b
4 & ( b
2,b
3 _|_ b
4,b
5 implies ( b
2,b
3 _|_ b
5,b
4 & b
4,b
5 _|_ b
2,b
3 ) ) & not ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 // b
8,b
9 & not b
6,b
7 _|_ b
8,b
9 & not b
2 = b
3 ) & ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 _|_ b
6,b
9 implies b
2,b
3 _|_ b
7,b
9 ) ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
not ( b
2 <> b
3 & ( for b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
5 & b
2,b
3 _|_ b
5,b
4 ) ) ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st
( b
2,b
3 _|_ b
4,b
5 & b
4 <> b
5 ) ) ) );
theorem Th43: :: ANALMETR:43
canceled;
theorem Th44: :: ANALMETR:44
consider c4 being RealLinearSpace such that
Lemma43:
ex b1, b2 being VECTOR of c4 st Gen b1,b2
by Th7;
consider c5, c6 being VECTOR of c4 such that
Lemma44:
Gen c5,c6
by Lemma43;
E45:
now
set c
7 =
AffinStruct(# the
carrier of
(AMSpace c4,c5,c6),the
CONGR of
(AMSpace c4,c5,c6) #);
AffinStruct(# the
carrier of
(AMSpace c4,c5,c6),the
CONGR of
(AMSpace c4,c5,c6) #)
= Af (AMSpace c4,c5,c6)
;
then E46:
AffinStruct(# the
carrier of
(AMSpace c4,c5,c6),the
CONGR of
(AMSpace c4,c5,c6) #)
= Lambda (OASpace c4)
by Th30;
( ( for b
1, b
2 being
Real holds
(
(b1 * c5) + (b2 * c6) = 0. c
4 implies ( b
1 = 0 & b
2 = 0 ) ) ) & ( for b
1 being
VECTOR of c
4 holds
ex b
2, b
3 being
Real st b
1 = (b2 * c5) + (b3 * c6) ) )
by Def1, Lemma44;
then
OASpace c
4 is
OAffinPlane
by ANALOAF:51;
hence
(
AffinStruct(# the
carrier of
(AMSpace c4,c5,c6),the
CONGR of
(AMSpace c4,c5,c6) #) is
AffinPlane & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of
(AMSpace c4,c5,c6) holds
( ( b
1,b
2 _|_ b
1,b
2 implies b
1 = b
2 ) & b
1,b
2 _|_ b
3,b
3 & ( b
1,b
2 _|_ b
3,b
4 implies ( b
1,b
2 _|_ b
4,b
3 & b
3,b
4 _|_ b
1,b
2 ) ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 // b
7,b
8 & not b
5,b
6 _|_ b
7,b
8 & not b
1 = b
2 ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 _|_ b
7,b
8 & not b
5,b
6 // b
7,b
8 & not b
1 = b
2 ) ) ) & ( for b
1, b
2, b
3 being
Element of
(AMSpace c4,c5,c6) holds
ex b
4 being
Element of
(AMSpace c4,c5,c6) st
( b
1,b
2 _|_ b
3,b
4 & b
3 <> b
4 ) ) )
by E46, Lemma44, Th33, Th34, Th35, Th36, Th37, Th38, Th40, DIRAF:53;
end;
definition
let c
7 be non
empty ParOrtStr ;
attr a
1 is
OrtAfPl-like means :
Def10:
:: ANALMETR:def 10
(
AffinStruct(# the
carrier of a
1,the
CONGR of a
1 #) is
AffinPlane & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( ( b
1,b
2 _|_ b
1,b
2 implies b
1 = b
2 ) & b
1,b
2 _|_ b
3,b
3 & ( b
1,b
2 _|_ b
3,b
4 implies ( b
1,b
2 _|_ b
4,b
3 & b
3,b
4 _|_ b
1,b
2 ) ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 // b
7,b
8 & not b
5,b
6 _|_ b
7,b
8 & not b
1 = b
2 ) & not ( b
1,b
2 _|_ b
5,b
6 & b
1,b
2 _|_ b
7,b
8 & not b
5,b
6 // b
7,b
8 & not b
1 = b
2 ) ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
ex b
4 being
Element of a
1 st
( b
1,b
2 _|_ b
3,b
4 & b
3 <> b
4 ) ) );
end;
:: deftheorem Def10 defines OrtAfPl-like ANALMETR:def 10 :
for b
1 being non
empty ParOrtStr holds
( b
1 is
OrtAfPl-like iff (
AffinStruct(# the
carrier of b
1,the
CONGR of b
1 #) is
AffinPlane & ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( ( b
2,b
3 _|_ b
2,b
3 implies b
2 = b
3 ) & b
2,b
3 _|_ b
4,b
4 & ( b
2,b
3 _|_ b
4,b
5 implies ( b
2,b
3 _|_ b
5,b
4 & b
4,b
5 _|_ b
2,b
3 ) ) & not ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 // b
8,b
9 & not b
6,b
7 _|_ b
8,b
9 & not b
2 = b
3 ) & not ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 _|_ b
8,b
9 & not b
6,b
7 // b
8,b
9 & not b
2 = b
3 ) ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st
( b
2,b
3 _|_ b
4,b
5 & b
4 <> b
5 ) ) ) );
theorem Th45: :: ANALMETR:45
canceled;
theorem Th46: :: ANALMETR:46
theorem Th47: :: ANALMETR:47
theorem Th48: :: ANALMETR:48
for b
1 being non
empty ParOrtStr for b
2, b
3, b
4, b
5 being
Element of b
1for b
6, b
7, b
8, b
9 being
Element of
(Af b1) holds
( b
2 = b
6 & b
3 = b
7 & b
4 = b
8 & b
5 = b
9 implies ( b
2,b
3 // b
4,b
5 iff b
6,b
7 // b
8,b
9 ) )
Lemma48:
for b1 being OrtAfSp holds
Af b1 is AffinSpace
by Def9;
Lemma49:
for b1 being OrtAfPl holds
Af b1 is AffinPlane
by Def10;
theorem Th49: :: ANALMETR:49
theorem Th50: :: ANALMETR:50
theorem Th51: :: ANALMETR:51
for b
1 being non
empty ParOrtStr holds
( b
1 is
OrtAfPl-like iff ( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 & ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & b
2,b
3 // b
4,b
4 & not ( b
2,b
3 // b
6,b
7 & b
2,b
3 // b
8,b
9 & not b
6,b
7 // b
8,b
9 & not b
2 = b
3 ) & ( b
2,b
3 // b
2,b
4 implies b
3,b
2 // b
3,b
4 ) & ex b
10 being
Element of b
1 st
( b
2,b
3 // b
4,b
10 & b
2,b
4 // b
3,b
10 ) & not for b
10, b
11, b
12 being
Element of b
1 holds b
10,b
11 // b
10,b
12 & ex b
10 being
Element of b
1 st
( b
2,b
3 // b
4,b
10 & b
4 <> b
10 ) & not ( b
2,b
3 // b
3,b
5 & b
3 <> b
2 & ( for b
10 being
Element of b
1 holds
not ( b
4,b
3 // b
3,b
10 & b
4,b
2 // b
5,b
10 ) ) ) & ( b
2,b
3 _|_ b
2,b
3 implies b
2 = b
3 ) & b
2,b
3 _|_ b
4,b
4 & ( b
2,b
3 _|_ b
4,b
5 implies ( b
2,b
3 _|_ b
5,b
4 & b
4,b
5 _|_ b
2,b
3 ) ) & not ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 // b
8,b
9 & not b
6,b
7 _|_ b
8,b
9 & not b
2 = b
3 ) & not ( b
2,b
3 _|_ b
6,b
7 & b
2,b
3 _|_ b
8,b
9 & not b
6,b
7 // b
8,b
9 & not b
2 = b
3 ) & ex b
10 being
Element of b
1 st
( b
2,b
3 _|_ b
4,b
10 & b
4 <> b
10 ) & not ( not b
2,b
3 // b
4,b
5 & ( for b
10 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
10 & b
4,b
5 // b
4,b
10 ) ) ) ) ) ) )
:: deftheorem Def11 defines LIN ANALMETR:def 11 :
definition
let c
7 be non
empty ParOrtStr ;
let c
8, c
9 be
Element of c
7;
func Line c
2,c
3 -> Subset of a
1 means :
Def12:
:: ANALMETR:def 12
for b
1 being
Element of a
1 holds
( b
1 in a
4 iff
LIN a
2,a
3,b
1 );
existence
ex b1 being Subset of c7 st
for b2 being Element of c7 holds
( b2 in b1 iff LIN c8,c9,b2 )
uniqueness
for b1, b2 being Subset of c7 holds
( ( for b3 being Element of c7 holds
( b3 in b1 iff LIN c8,c9,b3 ) ) & ( for b3 being Element of c7 holds
( b3 in b2 iff LIN c8,c9,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines Line ANALMETR:def 12 :
:: deftheorem Def13 defines being_line ANALMETR:def 13 :
theorem Th52: :: ANALMETR:52
canceled;
theorem Th53: :: ANALMETR:53
canceled;
theorem Th54: :: ANALMETR:54
canceled;
theorem Th55: :: ANALMETR:55
for b
1 being
OrtAfSpfor b
2, b
3, b
4 being
Element of b
1for b
5, b
6, b
7 being
Element of
(Af b1) holds
( b
2 = b
5 & b
3 = b
6 & b
4 = b
7 implies (
LIN b
2,b
3,b
4 iff
LIN b
5,b
6,b
7 ) )
theorem Th56: :: ANALMETR:56
theorem Th57: :: ANALMETR:57
theorem Th58: :: ANALMETR:58
:: deftheorem Def14 defines _|_ ANALMETR:def 14 :
:: deftheorem Def15 defines _|_ ANALMETR:def 15 :
:: deftheorem Def16 defines // ANALMETR:def 16 :
theorem Th59: :: ANALMETR:59
canceled;
theorem Th60: :: ANALMETR:60
canceled;
theorem Th61: :: ANALMETR:61
canceled;
theorem Th62: :: ANALMETR:62
theorem Th63: :: ANALMETR:63
theorem Th64: :: ANALMETR:64
for b
1 being
OrtAfSpfor b
2, b
3 being
Subset of b
1for b
4, b
5 being
Subset of
(Af b1) holds
( b
2 = b
4 & b
3 = b
5 implies ( b
2 // b
3 iff b
4 // b
5 ) )
theorem Th65: :: ANALMETR:65
theorem Th66: :: ANALMETR:66
for b
1 being
OrtAfSpfor b
2 being
Subset of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
3,b
4 _|_ b
2 & ( b
3,b
4 // b
5,b
6 or b
5,b
6 // b
3,b
4 ) & b
3 <> b
4 implies b
5,b
6 _|_ b
2 )
theorem Th67: :: ANALMETR:67
theorem Th68: :: ANALMETR:68
canceled;
theorem Th69: :: ANALMETR:69
theorem Th70: :: ANALMETR:70
canceled;
theorem Th71: :: ANALMETR:71
theorem Th72: :: ANALMETR:72
canceled;
theorem Th73: :: ANALMETR:73
theorem Th74: :: ANALMETR:74
canceled;
theorem Th75: :: ANALMETR:75
for b
1 being
OrtAfSpfor b
2 being
Subset of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
3 in b
2 & b
4 in b
2 & b
5,b
6 _|_ b
2 implies ( b
5,b
6 _|_ b
3,b
4 & b
3,b
4 _|_ b
5,b
6 ) )
theorem Th76: :: ANALMETR:76
theorem Th77: :: ANALMETR:77
theorem Th78: :: ANALMETR:78
for b
1 being
OrtAfSpfor b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
4 in b
2 & b
5 in b
2 & b
6 in b
3 & b
7 in b
3 & b
2 _|_ b
3 implies b
4,b
5 _|_ b
6,b
7 )
theorem Th79: :: ANALMETR:79
theorem Th80: :: ANALMETR:80
for b
1 being
OrtAfSpfor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 _|_ b
4,b
4 & b
4,b
4 _|_ b
2,b
3 & b
2,b
3 // b
4,b
4 & b
4,b
4 // b
2,b
3 )
theorem Th81: :: ANALMETR:81
for b
1 being
OrtAfSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies ( b
2,b
3 // b
5,b
4 & b
3,b
2 // b
4,b
5 & b
3,b
2 // b
5,b
4 & b
4,b
5 // b
2,b
3 & b
4,b
5 // b
3,b
2 & b
5,b
4 // b
2,b
3 & b
5,b
4 // b
3,b
2 ) )
theorem Th82: :: ANALMETR:82
for b
1 being
OrtAfSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & not ( not ( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 ) & not ( b
2,b
3 // b
4,b
5 & b
6,b
7 // b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
6,b
7 // b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
2,b
3 // b
6,b
7 ) ) implies b
4,b
5 // b
6,b
7 )
theorem Th83: :: ANALMETR:83
for b
1 being
OrtAfSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 _|_ b
4,b
5 implies ( b
2,b
3 _|_ b
5,b
4 & b
3,b
2 _|_ b
4,b
5 & b
3,b
2 _|_ b
5,b
4 & b
4,b
5 _|_ b
2,b
3 & b
4,b
5 _|_ b
3,b
2 & b
5,b
4 _|_ b
2,b
3 & b
5,b
4 _|_ b
3,b
2 ) )
theorem Th84: :: ANALMETR:84
for b
1 being
OrtAfSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & not ( not ( b
2,b
3 // b
4,b
5 & b
2,b
3 _|_ b
6,b
7 ) & not ( b
2,b
3 // b
6,b
7 & b
2,b
3 _|_ b
4,b
5 ) & not ( b
2,b
3 // b
4,b
5 & b
6,b
7 _|_ b
2,b
3 ) & not ( b
2,b
3 // b
6,b
7 & b
4,b
5 _|_ b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
6,b
7 _|_ b
2,b
3 ) & not ( b
6,b
7 // b
2,b
3 & b
4,b
5 _|_ b
2,b
3 ) & not ( b
4,b
5 // b
2,b
3 & b
2,b
3 _|_ b
6,b
7 ) & not ( b
6,b
7 // b
2,b
3 & b
2,b
3 _|_ b
4,b
5 ) ) implies b
4,b
5 _|_ b
6,b
7 )
theorem Th85: :: ANALMETR:85
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & not ( not ( b
2,b
3 _|_ b
4,b
5 & b
2,b
3 _|_ b
6,b
7 ) & not ( b
2,b
3 _|_ b
4,b
5 & b
6,b
7 _|_ b
2,b
3 ) & not ( b
4,b
5 _|_ b
2,b
3 & b
6,b
7 _|_ b
2,b
3 ) & not ( b
4,b
5 _|_ b
2,b
3 & b
2,b
3 _|_ b
6,b
7 ) ) implies b
4,b
5 // b
6,b
7 )
theorem Th86: :: ANALMETR:86
theorem Th87: :: ANALMETR:87
theorem Th88: :: ANALMETR:88
for b
1 being
OrtAfPlfor b
2, b
3 being
Subset of b
1 holds
not ( b
2 _|_ b
3 & ( for b
4 being
Element of b
1 holds
not ( b
4 in b
2 & b
4 in b
3 ) ) )
theorem Th89: :: ANALMETR:89
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 _|_ b
4,b
5 & ( for b
6 being
Element of b
1 holds
not (
LIN b
2,b
3,b
6 &
LIN b
4,b
5,b
6 ) ) )
theorem Th90: :: ANALMETR:90
theorem Th91: :: ANALMETR:91
theorem Th92: :: ANALMETR:92