:: TOPREAL8 semantic presentation
theorem Th1: :: TOPREAL8:1
for b
1, b
2, b
3 being
set holds
( b
1 c= {b2,b3} & b
2 in b
1 & not b
3 in b
1 implies b
1 = {b2} )
theorem Th2: :: TOPREAL8:2
theorem Th3: :: TOPREAL8:3
theorem Th4: :: TOPREAL8:4
theorem Th5: :: TOPREAL8:5
theorem Th6: :: TOPREAL8:6
theorem Th7: :: TOPREAL8:7
theorem Th8: :: TOPREAL8:8
theorem Th9: :: TOPREAL8:9
theorem Th10: :: TOPREAL8:10
theorem Th11: :: TOPREAL8:11
theorem Th12: :: TOPREAL8:12
Lemma12:
for b1 being FinSequence
for b2, b3 being Nat holds
( 1 <= b2 & b2 <= b3 + 1 & b3 <= len b1 implies ( (len (b2,b3 -cut b1)) + b2 = b3 + 1 & ( for b4 being Nat holds
( b4 < len (b2,b3 -cut b1) implies (b2,b3 -cut b1) . (b4 + 1) = b1 . (b2 + b4) ) ) ) )
theorem Th13: :: TOPREAL8:13
theorem Th14: :: TOPREAL8:14
theorem Th15: :: TOPREAL8:15
theorem Th16: :: TOPREAL8:16
theorem Th17: :: TOPREAL8:17
theorem Th18: :: TOPREAL8:18
theorem Th19: :: TOPREAL8:19
theorem Th20: :: TOPREAL8:20
theorem Th21: :: TOPREAL8:21
theorem Th22: :: TOPREAL8:22
theorem Th23: :: TOPREAL8:23
theorem Th24: :: TOPREAL8:24
theorem Th25: :: TOPREAL8:25
theorem Th26: :: TOPREAL8:26
theorem Th27: :: TOPREAL8:27
theorem Th28: :: TOPREAL8:28
theorem Th29: :: TOPREAL8:29
theorem Th30: :: TOPREAL8:30
theorem Th31: :: TOPREAL8:31
theorem Th32: :: TOPREAL8:32
Lemma33:
for b1 being non empty one-to-one unfolded s.n.c. FinSequence of (TOP-REAL 2)
for b2 being one-to-one non trivial unfolded s.n.c. FinSequence of (TOP-REAL 2)
for b3, b4 being Nat holds
( b3 < len b1 & 1 < b3 implies for b5 being Point of (TOP-REAL 2) holds
not ( b5 in (LSeg (b1 ^' b2),b3) /\ (LSeg (b1 ^' b2),b4) & not b5 <> b1 /. 1 ) )
Lemma34:
for b1 being non empty one-to-one unfolded s.n.c. FinSequence of (TOP-REAL 2)
for b2 being one-to-one non trivial unfolded s.n.c. FinSequence of (TOP-REAL 2)
for b3, b4 being Nat holds
( b4 > len b1 & b4 + 1 < len (b1 ^' b2) implies for b5 being Point of (TOP-REAL 2) holds
not ( b5 in (LSeg (b1 ^' b2),b3) /\ (LSeg (b1 ^' b2),b4) & not b5 <> b2 /. (len b2) ) )
theorem Th33: :: TOPREAL8:33
theorem Th34: :: TOPREAL8:34
theorem Th35: :: TOPREAL8:35
theorem Th36: :: TOPREAL8:36