:: JORDAN1D semantic presentation
1 = (2 * 0) + 1
;
then Lemma1:
1 div 2 = 0
by NAT_1:def 1;
2 = (2 * 1) + 0
;
then Lemma2:
2 div 2 = 1
by NAT_1:def 1;
Lemma3:
for b1, b2, b3, b4, b5 being set holds
( b1 in ((b2 \/ b3) \/ b4) \/ b5 iff not ( not b1 in b2 & not b1 in b3 & not b1 in b4 & not b1 in b5 ) )
Lemma4:
for b1, b2, b3, b4 being set holds union {b1,b2,b3,b4} = ((b1 \/ b2) \/ b3) \/ b4
theorem Th1: :: JORDAN1D:1
for b
1, b
2 being
set holds
( ( for b
3 being
set holds
not ( b
3 in b
1 & ( for b
4 being
set holds
not ( b
4 c= b
2 & b
3 c= union b
4 ) ) ) ) implies
union b
1 c= union b
2 )
theorem Th2: :: JORDAN1D:2
theorem Th3: :: JORDAN1D:3
theorem Th4: :: JORDAN1D:4
theorem Th5: :: JORDAN1D:5
for b
1, b
2 being
Nat holds
( b
1 > 0 & b
2 mod b
1 <> 0 implies
- (b2 div b1) = ((- b2) div b1) + 1 )
theorem Th6: :: JORDAN1D:6
for b
1, b
2 being
Nat holds
( b
1 > 0 & b
2 mod b
1 = 0 implies
- (b2 div b1) = (- b2) div b
1 )
Lemma14:
for b1, b2 being Nat
for b3 being real number holds
( 2 <= b1 implies (b3 / (2 |^ b2)) * (b1 - 2) = (b3 / (2 |^ (b2 + 1))) * (((2 * b1) -' 2) - 2) )
Lemma15:
for b1 being Nat holds
( 2 <= b1 implies 1 <= (2 * b1) -' 2 )
Lemma16:
for b1 being Nat holds
( 1 <= b1 implies 1 <= (2 * b1) -' 1 )
Lemma17:
for b1, b2 being Nat holds
not ( b1 < (2 |^ b2) + 3 & not (2 * b1) -' 2 < (2 |^ (b2 + 1)) + 3 )
E18:
now
let c
1 be
Nat;
assume
2
<= c
1
;
hence (((2 * c1) -' 2) + 1) - 2 =
(((2 * c1) - 2) + 1) - 2
by Lemma11
.=
(2 * c1) - 3
;
end;
theorem Th7: :: JORDAN1D:7
for b
1, b
2, b
3, b
4 being
Natfor b
5 being non
empty Subset of
(TOP-REAL 2) holds
( 2
<= b
1 & b
1 < len (Gauge b5,b2) & 1
<= b
3 & b
3 <= len (Gauge b5,b2) & 1
<= b
4 & b
4 <= len (Gauge b5,(b2 + 1)) implies
((Gauge b5,b2) * b1,b3) `1 = ((Gauge b5,(b2 + 1)) * ((2 * b1) -' 2),b4) `1 )
theorem Th8: :: JORDAN1D:8
for b
1, b
2, b
3, b
4 being
Natfor b
5 being non
empty Subset of
(TOP-REAL 2) holds
( 2
<= b
1 & b
1 < len (Gauge b5,b2) & 1
<= b
3 & b
3 <= len (Gauge b5,b2) & 1
<= b
4 & b
4 <= len (Gauge b5,(b2 + 1)) implies
((Gauge b5,b2) * b3,b1) `2 = ((Gauge b5,(b2 + 1)) * b4,((2 * b1) -' 2)) `2 )
Lemma21:
for b1, b2 being Nat
for b3 being non empty Subset of (TOP-REAL 2) holds
not ( b1 + 1 < len (Gauge b3,b2) & not (2 * b1) -' 1 < len (Gauge b3,(b2 + 1)) )
theorem Th9: :: JORDAN1D:9
for b
1, b
2, b
3 being
Natfor b
4 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
( 2
<= b
1 & b
1 + 1
< len (Gauge b4,b2) & 2
<= b
3 & b
3 + 1
< len (Gauge b4,b2) implies
cell (Gauge b4,b2),b
1,b
3 = (((cell (Gauge b4,(b2 + 1)),((2 * b1) -' 2),((2 * b3) -' 2)) \/ (cell (Gauge b4,(b2 + 1)),((2 * b1) -' 1),((2 * b3) -' 2))) \/ (cell (Gauge b4,(b2 + 1)),((2 * b1) -' 2),((2 * b3) -' 1))) \/ (cell (Gauge b4,(b2 + 1)),((2 * b1) -' 1),((2 * b3) -' 1)) )
theorem Th10: :: JORDAN1D:10
for b
1, b
2, b
3 being
Natfor b
4 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
5 being
Nat holds
( 2
<= b
1 & b
1 + 1
< len (Gauge b4,b2) & 2
<= b
3 & b
3 + 1
< len (Gauge b4,b2) implies
cell (Gauge b4,b2),b
1,b
3 = union { (cell (Gauge b4,(b2 + b5)),b6,b7) where B is Nat, B is Nat : ( (((2 |^ b5) * b1) - (2 |^ (b5 + 1))) + 2 <= b6 & b6 <= (((2 |^ b5) * b1) - (2 |^ b5)) + 1 & (((2 |^ b5) * b3) - (2 |^ (b5 + 1))) + 2 <= b7 & b7 <= (((2 |^ b5) * b3) - (2 |^ b5)) + 1 ) } )
theorem Th11: :: JORDAN1D:11
theorem Th12: :: JORDAN1D:12
theorem Th13: :: JORDAN1D:13
theorem Th14: :: JORDAN1D:14
theorem Th15: :: JORDAN1D:15
theorem Th16: :: JORDAN1D:16
theorem Th17: :: JORDAN1D:17
theorem Th18: :: JORDAN1D:18
theorem Th19: :: JORDAN1D:19
theorem Th20: :: JORDAN1D:20
theorem Th21: :: JORDAN1D:21
theorem Th22: :: JORDAN1D:22
theorem Th23: :: JORDAN1D:23
theorem Th24: :: JORDAN1D:24
theorem Th25: :: JORDAN1D:25
theorem Th26: :: JORDAN1D:26
theorem Th27: :: JORDAN1D:27
theorem Th28: :: JORDAN1D:28
theorem Th29: :: JORDAN1D:29
theorem Th30: :: JORDAN1D:30
theorem Th31: :: JORDAN1D:31
theorem Th32: :: JORDAN1D:32
theorem Th33: :: JORDAN1D:33
theorem Th34: :: JORDAN1D:34
theorem Th35: :: JORDAN1D:35
theorem Th36: :: JORDAN1D:36