:: DIRORT semantic presentation
theorem Th1: :: DIRORT:1
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
Element of
(CESpace b1,b2,b3) holds
( b
4,b
4 '//' b
6,b
8 & b
4,b
6 '//' b
8,b
8 & not ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
7,b
5 & not b
4 = b
6 & not b
5 = b
7 ) & not ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
5,b
8 & not b
4,b
6 '//' b
7,b
8 & not b
4,b
6 '//' b
8,b
7 ) & ( b
4,b
6 '//' b
5,b
7 implies b
6,b
4 '//' b
7,b
5 ) & ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
7,b
8 implies b
4,b
6 '//' b
5,b
8 ) & not ( b
4,b
5 '//' b
6,b
7 & not b
6,b
7 '//' b
4,b
5 & not b
6,b
7 '//' b
5,b
4 ) ) ) & ( for b
4, b
5, b
6 being
Element of
(CESpace b1,b2,b3) holds
ex b
7 being
Element of
(CESpace b1,b2,b3) st
( b
6 <> b
7 & b
6,b
7 '//' b
4,b
5 ) ) & ( for b
4, b
5, b
6 being
Element of
(CESpace b1,b2,b3) holds
ex b
7 being
Element of
(CESpace b1,b2,b3) st
( b
6 <> b
7 & b
4,b
5 '//' b
6,b
7 ) ) ) )
theorem Th2: :: DIRORT:2
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
Element of
(CMSpace b1,b2,b3) holds
( b
4,b
4 '//' b
6,b
8 & b
4,b
6 '//' b
8,b
8 & not ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
7,b
5 & not b
4 = b
6 & not b
5 = b
7 ) & not ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
5,b
8 & not b
4,b
6 '//' b
7,b
8 & not b
4,b
6 '//' b
8,b
7 ) & ( b
4,b
6 '//' b
5,b
7 implies b
6,b
4 '//' b
7,b
5 ) & ( b
4,b
6 '//' b
5,b
7 & b
4,b
6 '//' b
7,b
8 implies b
4,b
6 '//' b
5,b
8 ) & not ( b
4,b
5 '//' b
6,b
7 & not b
6,b
7 '//' b
4,b
5 & not b
6,b
7 '//' b
5,b
4 ) ) ) & ( for b
4, b
5, b
6 being
Element of
(CMSpace b1,b2,b3) holds
ex b
7 being
Element of
(CMSpace b1,b2,b3) st
( b
6 <> b
7 & b
6,b
7 '//' b
4,b
5 ) ) & ( for b
4, b
5, b
6 being
Element of
(CMSpace b1,b2,b3) holds
ex b
7 being
Element of
(CMSpace b1,b2,b3) st
( b
6 <> b
7 & b
4,b
5 '//' b
6,b
7 ) ) ) )
definition
let c
1 be non
empty AffinStruct ;
attr a
1 is
Oriented_Orthogonality_Space-like means :
Def1:
:: DIRORT:def 1
( ( for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
( b
1,b
1 '//' b
3,b
5 & b
1,b
3 '//' b
5,b
5 & not ( b
1,b
3 '//' b
2,b
4 & b
1,b
3 '//' b
4,b
2 & not b
1 = b
3 & not b
2 = b
4 ) & not ( b
1,b
3 '//' b
2,b
4 & b
1,b
3 '//' b
2,b
5 & not b
1,b
3 '//' b
4,b
5 & not b
1,b
3 '//' b
5,b
4 ) & ( b
1,b
3 '//' b
2,b
4 implies b
3,b
1 '//' b
4,b
2 ) & ( b
1,b
3 '//' b
2,b
4 & b
1,b
3 '//' b
4,b
5 implies b
1,b
3 '//' b
2,b
5 ) & not ( b
1,b
2 '//' b
3,b
4 & not b
3,b
4 '//' b
1,b
2 & not b
3,b
4 '//' b
2,b
1 ) ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
ex b
4 being
Element of a
1 st
( b
3 <> b
4 & b
3,b
4 '//' 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
3 <> b
4 & b
1,b
2 '//' b
3,b
4 ) ) );
end;
:: deftheorem Def1 defines Oriented_Orthogonality_Space-like DIRORT:def 1 :
for b
1 being non
empty AffinStruct holds
( b
1 is
Oriented_Orthogonality_Space-like iff ( ( for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
2 '//' b
4,b
6 & b
2,b
4 '//' b
6,b
6 & not ( b
2,b
4 '//' b
3,b
5 & b
2,b
4 '//' b
5,b
3 & not b
2 = b
4 & not b
3 = b
5 ) & not ( b
2,b
4 '//' b
3,b
5 & b
2,b
4 '//' b
3,b
6 & not b
2,b
4 '//' b
5,b
6 & not b
2,b
4 '//' b
6,b
5 ) & ( b
2,b
4 '//' b
3,b
5 implies b
4,b
2 '//' b
5,b
3 ) & ( b
2,b
4 '//' b
3,b
5 & b
2,b
4 '//' b
5,b
6 implies b
2,b
4 '//' b
3,b
6 ) & not ( b
2,b
3 '//' b
4,b
5 & not b
4,b
5 '//' b
2,b
3 & not b
4,b
5 '//' b
3,b
2 ) ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st
( b
4 <> b
5 & b
4,b
5 '//' 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
4 <> b
5 & b
2,b
3 '//' b
4,b
5 ) ) ) );
theorem Th3: :: DIRORT:3
canceled;
theorem Th4: :: DIRORT:4
theorem Th5: :: DIRORT:5
theorem Th6: :: DIRORT:6
theorem Th7: :: DIRORT:7
canceled;
theorem Th8: :: DIRORT:8
:: deftheorem Def2 defines _|_ DIRORT:def 2 :
definition
let c
1 be
Oriented_Orthogonality_Space;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred c
2,c
3 // c
4,c
5 means :
Def3:
:: DIRORT:def 3
ex b
1, b
2 being
Element of a
1 st
( b
1 <> b
2 & b
1,b
2 '//' a
2,a
3 & b
1,b
2 '//' a
4,a
5 );
end;
:: deftheorem Def3 defines // DIRORT:def 3 :
for b
1 being
Oriented_Orthogonality_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 iff ex b
6, b
7 being
Element of b
1 st
( b
6 <> b
7 & b
6,b
7 '//' b
2,b
3 & b
6,b
7 '//' b
4,b
5 ) );
definition
let c
1 be
Oriented_Orthogonality_Space;
attr a
1 is
bach_transitive means :
Def4:
:: DIRORT:def 4
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
not ( b
1,b
2 '//' b
4,b
5 & b
7,b
8 '//' b
4,b
5 & b
7,b
8 '//' b
3,b
6 & not b
7 = b
8 & not b
4 = b
5 & not b
1,b
2 '//' b
3,b
6 );
end;
:: deftheorem Def4 defines bach_transitive DIRORT:def 4 :
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
bach_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
not ( b
2,b
3 '//' b
5,b
6 & b
8,b
9 '//' b
5,b
6 & b
8,b
9 '//' b
4,b
7 & not b
8 = b
9 & not b
5 = b
6 & not b
2,b
3 '//' b
4,b
7 ) );
definition
let c
1 be
Oriented_Orthogonality_Space;
attr a
1 is
right_transitive means :
Def5:
:: DIRORT:def 5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
not ( b
1,b
2 '//' b
4,b
5 & b
4,b
5 '//' b
7,b
8 & b
3,b
6 '//' b
7,b
8 & not b
7 = b
8 & not b
4 = b
5 & not b
1,b
2 '//' b
3,b
6 );
end;
:: deftheorem Def5 defines right_transitive DIRORT:def 5 :
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
right_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
not ( b
2,b
3 '//' b
5,b
6 & b
5,b
6 '//' b
8,b
9 & b
4,b
7 '//' b
8,b
9 & not b
8 = b
9 & not b
5 = b
6 & not b
2,b
3 '//' b
4,b
7 ) );
definition
let c
1 be
Oriented_Orthogonality_Space;
attr a
1 is
left_transitive means :
Def6:
:: DIRORT:def 6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
not ( b
1,b
2 '//' b
4,b
5 & b
4,b
5 '//' b
7,b
8 & b
1,b
2 '//' b
3,b
6 & not b
1 = b
2 & not b
4 = b
5 & not b
3,b
6 '//' b
7,b
8 );
end;
:: deftheorem Def6 defines left_transitive DIRORT:def 6 :
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
left_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
not ( b
2,b
3 '//' b
5,b
6 & b
5,b
6 '//' b
8,b
9 & b
2,b
3 '//' b
4,b
7 & not b
2 = b
3 & not b
5 = b
6 & not b
4,b
7 '//' b
8,b
9 ) );
:: deftheorem Def7 defines Euclidean_like DIRORT:def 7 :
:: deftheorem Def8 defines Minkowskian_like DIRORT:def 8 :
theorem Th9: :: DIRORT:9
theorem Th10: :: DIRORT:10
theorem Th11: :: DIRORT:11
theorem Th12: :: DIRORT:12
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
left_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
6,b
7 & b
2,b
3 '//' b
4,b
5 & b
2 <> b
3 implies b
6,b
7 '//' b
4,b
5 ) )
theorem Th13: :: DIRORT:13
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
bach_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 '//' b
5,b
6 & b
5,b
6 // b
4,b
7 & b
5 <> b
6 implies b
2,b
3 '//' b
4,b
7 ) )
theorem Th14: :: DIRORT:14
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
bach_transitive implies for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 & b
4,b
5 // b
6,b
7 & b
4 <> b
5 implies b
2,b
3 // b
6,b
7 ) )
theorem Th15: :: DIRORT:15
theorem Th16: :: DIRORT:16
theorem Th17: :: DIRORT:17
theorem Th18: :: DIRORT:18
theorem Th19: :: DIRORT:19
for b
1 being
Oriented_Orthogonality_Space holds
( b
1 is
bach_transitive implies ( b
1 is
right_transitive iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 '//' b
6,b
7 & b
4,b
5 '//' b
6,b
7 & b
6 <> b
7 implies b
2,b
3 // b
4,b
5 ) ) )
theorem Th20: :: DIRORT:20