for b
1 being
RealLinearSpace holds
( ex b
2, b
3 being
VECTOR of b
1 st
for b
4, b
5 being
Real holds
(
(b4 * b2) + (b5 * b3) = 0. b
1 implies ( b
4 = 0 & b
5 = 0 ) ) implies ( not for b
2, b
3 being
Element of
(OASpace b1) 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
(OASpace b1) holds
( b
2,b
3 // b
4,b
4 & ( b
2,b
3 // b
3,b
2 implies b
2 = b
3 ) & ( b
2 <> b
3 & b
2,b
3 // b
6,b
7 & b
2,b
3 // b
8,b
9 implies b
6,b
7 // b
8,b
9 ) & ( b
2,b
3 // b
4,b
5 implies b
3,b
2 // b
5,b
4 ) & ( b
2,b
3 // b
3,b
4 implies b
2,b
3 // b
2,b
4 ) & not ( b
2,b
3 // b
2,b
4 & not b
2,b
3 // b
3,b
4 & not b
2,b
4 // b
4,b
3 ) ) ) & ex b
2, b
3, b
4, b
5 being
Element of
(OASpace b1) st
( not b
2,b
3 // b
4,b
5 & not b
2,b
3 // b
5,b
4 ) & ( for b
2, b
3, b
4 being
Element of
(OASpace b1) holds
ex b
5 being
Element of
(OASpace b1) st
( b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & b
3 <> b
5 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of
(OASpace b1) holds
not ( b
2 <> b
4 & b
4,b
2 // b
2,b
5 & ( for b
6 being
Element of
(OASpace b1) holds
not ( b
3,b
2 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) ) ) ) ) )