:: BHSP_7 semantic presentation
Lemma1:
for b1, b2, b3, b4 being Real holds
not ( abs (b1 - b2) < b4 / 2 & abs (b2 - b3) < b4 / 2 & not abs (b1 - b3) < b4 )
Lemma2:
for b1 being real number holds
not ( b1 > 0 & ( for b2 being Nat holds not 1 / (b2 + 1) < b1 ) )
Lemma3:
for b1 being real number
for b2 being Nat holds
not ( b1 > 0 & ( for b3 being Nat holds
not ( 1 / (b3 + 1) < b1 & b3 >= b2 ) ) )
theorem Th1: :: BHSP_7:1
theorem Th2: :: BHSP_7:2
for b
1 being
RealUnitarySpace holds
( the
add of b
1 is
commutative & the
add of b
1 is
associative & the
add of b
1 has_a_unity implies for b
2 being
finite OrthogonalFamily of b
1 holds
( not b
2 is
empty implies for b
3 being
Function of the
carrier of b
1,the
carrier of b
1 holds
( b
2 c= dom b
3 & ( for b
4 being
Point of b
1 holds
( b
4 in b
2 implies b
3 . b
4 = b
4 ) ) implies for b
4 being
Function of the
carrier of b
1,
REAL holds
( b
2 c= dom b
4 & ( for b
5 being
Point of b
1 holds
( b
5 in b
2 implies b
4 . b
5 = b
5 .|. b
5 ) ) implies
(setopfunc b2,the carrier of b1,the carrier of b1,b3,the add of b1) .|. (setopfunc b2,the carrier of b1,the carrier of b1,b3,the add of b1) = setopfunc b
2,the
carrier of b
1,
REAL ,b
4,
addreal ) ) ) )
theorem Th3: :: BHSP_7:3
theorem Th4: :: BHSP_7:4
theorem Th5: :: BHSP_7:5
theorem Th6: :: BHSP_7:6
theorem Th7: :: BHSP_7:7
Lemma11:
for b1, b2 being real number holds
( ( for b3 being Real holds
not ( 0 < b3 & not abs (b1 - b2) < b3 ) ) implies b1 = b2 )
theorem Th8: :: BHSP_7:8