:: EUCLMETR semantic presentation
:: deftheorem Def1 defines Euclidean EUCLMETR:def 1 :
:: deftheorem Def2 defines Pappian EUCLMETR:def 2 :
:: deftheorem Def3 defines Desarguesian EUCLMETR:def 3 :
:: deftheorem Def4 defines Fanoian EUCLMETR:def 4 :
:: deftheorem Def5 defines Moufangian EUCLMETR:def 5 :
:: deftheorem Def6 defines translation EUCLMETR:def 6 :
definition
let c
1 be
OrtAfSp;
attr a
1 is
Homogeneous means :
Def7:
:: EUCLMETR:def 7
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
( b
1,b
2 _|_ b
1,b
3 & b
1,b
4 _|_ b
1,b
5 & b
1,b
6 _|_ b
1,b
7 & b
2,b
4 _|_ b
3,b
5 & b
2,b
6 _|_ b
3,b
7 & not b
1,b
6 // b
1,b
2 & not b
1,b
2 // b
1,b
4 implies b
4,b
6 _|_ b
5,b
7 );
end;
:: deftheorem Def7 defines Homogeneous EUCLMETR:def 7 :
for b
1 being
OrtAfSp holds
( b
1 is
Homogeneous iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3 _|_ b
2,b
4 & b
2,b
5 _|_ b
2,b
6 & b
2,b
7 _|_ b
2,b
8 & b
3,b
5 _|_ b
4,b
6 & b
3,b
7 _|_ b
4,b
8 & not b
2,b
7 // b
2,b
3 & not b
2,b
3 // b
2,b
5 implies b
5,b
7 _|_ b
6,b
8 ) );
theorem Th1: :: EUCLMETR:1
for b
1 being
OrtAfSpfor b
2, b
3, b
4 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 implies ( b
2 <> b
3 & b
3 <> b
4 & b
2 <> b
4 ) )
theorem Th2: :: EUCLMETR:2
theorem Th3: :: EUCLMETR:3
theorem Th4: :: EUCLMETR:4
for b
1 being
OrtAfSpfor b
2, b
3, b
4 being
Element of b
1 holds
(
LIN b
2,b
3,b
4 implies (
LIN b
2,b
4,b
3 &
LIN b
3,b
2,b
4 &
LIN b
3,b
4,b
2 &
LIN b
4,b
2,b
3 &
LIN b
4,b
3,b
2 ) )
theorem Th5: :: EUCLMETR:5
for b
1 being
OrtAfPlfor b
2, b
3, b
4 being
Element of b
1 holds
not ( not
LIN b
2,b
3,b
4 & ( for b
5 being
Element of b
1 holds
not ( b
5,b
2 _|_ b
3,b
4 & b
5,b
3 _|_ b
2,b
4 ) ) )
theorem Th6: :: EUCLMETR:6
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & b
5,b
2 _|_ b
3,b
4 & b
5,b
3 _|_ b
2,b
4 & b
6,b
2 _|_ b
3,b
4 & b
6,b
3 _|_ b
2,b
4 implies b
5 = b
6 )
theorem Th7: :: EUCLMETR:7
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 & b
3,b
4 _|_ b
2,b
5 &
LIN b
2,b
3,b
4 & not b
2 = b
4 & not b
2 = b
3 & not b
3 = b
4 )
theorem Th8: :: EUCLMETR:8
theorem Th9: :: EUCLMETR:9
Lemma16:
for b1 being OrtAfPl holds
( PAP_holds_in b1 iff Af b1 satisfies_PAP' )
theorem Th10: :: EUCLMETR:10
Lemma18:
for b1 being OrtAfPl holds
( DES_holds_in b1 iff Af b1 satisfies_DES' )
theorem Th11: :: EUCLMETR:11
theorem Th12: :: EUCLMETR:12
Lemma20:
for b1 being OrtAfPl holds
( des_holds_in b1 iff Af b1 satisfies_des' )
theorem Th13: :: EUCLMETR:13
theorem Th14: :: EUCLMETR:14
theorem Th15: :: EUCLMETR:15
theorem Th16: :: EUCLMETR:16
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not
LIN b
2,b
3,b
5 & b
2 <> b
4 & b
2,b
3 _|_ b
2,b
4 & b
2,b
5 _|_ b
2,b
6 & b
2,b
5 _|_ b
2,b
7 & b
3,b
5 _|_ b
4,b
6 & b
3,b
5 _|_ b
4,b
7 implies b
6 = b
7 )
theorem Th17: :: EUCLMETR:17
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not
LIN b
2,b
3,b
5 & b
2 <> b
4 & ( for b
6 being
Element of b
1 holds
not ( b
2,b
5 _|_ b
2,b
6 & b
3,b
5 _|_ b
4,b
6 ) ) )
Lemma24:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
( (b2 - b4) - (b3 - b4) = b2 - b3 & (b2 + b4) - (b3 + b4) = b2 - b3 )
Lemma25:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds
( Gen b2,b3 implies for b4, b5, b6 being Real holds
( PProJ b2,b3,((b4 * b2) + (b5 * b3)),(((b6 * b5) * b2) + ((- (b6 * b4)) * b3)) = 0 & (b4 * b2) + (b5 * b3),((b6 * b5) * b2) + ((- (b6 * b4)) * b3) are_Ort_wrt b2,b3 ) )
theorem Th18: :: EUCLMETR:18
theorem Th19: :: EUCLMETR:19
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 &
0. b
1 <> b
4 &
0. b
1 <> b
5 & b
4,b
5 are_Ort_wrt b
2,b
3 & ( for b
6 being
Real holds
not for b
7, b
8 being
Real holds
(
(b7 * b2) + (b8 * b3),
((b6 * b8) * b2) + ((- (b6 * b7)) * b3) are_Ort_wrt b
2,b
3 &
((b7 * b2) + (b8 * b3)) - b
4,
(((b6 * b8) * b2) + ((- (b6 * b7)) * b3)) - b
5 are_Ort_wrt b
2,b
3 ) ) )
Lemma28:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Real holds ((b4 * b2) + (b5 * b3)) - ((b6 * b2) + (b7 * b3)) = ((b4 - b6) * b2) + ((b5 - b7) * b3)
Lemma29:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds
( Gen b2,b3 implies for b4, b5, b6, b7 being Real holds
( (b4 * b2) + (b6 * b3) = (b5 * b2) + (b7 * b3) implies ( b4 = b5 & b6 = b7 ) ) )
theorem Th20: :: EUCLMETR:20
canceled;
theorem Th21: :: EUCLMETR:21
theorem Th22: :: EUCLMETR:22
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3 _|_ b
2,b
4 & b
2,b
5 _|_ b
2,b
6 & b
2,b
7 _|_ b
2,b
8 & b
3,b
5 _|_ b
4,b
6 & b
3,b
7 _|_ b
4,b
8 & not b
2,b
7 // b
2,b
3 & not b
2,b
3 // b
2,b
5 & b
2 = b
4 implies b
5,b
7 _|_ b
6,b
8 )
theorem Th23: :: EUCLMETR:23
theorem Th24: :: EUCLMETR:24