:: JGRAPH_8 semantic presentation
Lemma1:
I[01] = TopSpaceMetr (Closed-Interval-MSpace 0,1)
by TOPMETR:27, TOPMETR:def 8;
Lemma2:
for b1 being set
for b2 being FinSequence holds
( 1 <= len b2 implies ( (b2 ^ <*b1*>) . 1 = b2 . 1 & (<*b1*> ^ b2) . ((len b2) + 1) = b2 . (len b2) ) )
Lemma3:
for b1 being FinSequence of REAL holds
( ( for b2 being Nat holds
not ( 1 <= b2 & b2 < len b1 & not b1 /. b2 < b1 /. (b2 + 1) ) ) implies b1 is increasing )
theorem Th1: :: JGRAPH_8:1
theorem Th2: :: JGRAPH_8:2
theorem Th3: :: JGRAPH_8:3
theorem Th4: :: JGRAPH_8:4
theorem Th5: :: JGRAPH_8:5
theorem Th6: :: JGRAPH_8:6
for b
1, b
2, b
3, b
4 being
real number for b
5, b
6, b
7, b
8 being
Point of
(Trectangle b1,b2,b3,b4)for b
9 being
Path of b
5,b
6for b
10 being
Path of b
8,b
7for b
11, b
12, b
13, b
14 being
Point of
(TOP-REAL 2) holds
not ( b
11 `1 = b
1 & b
12 `1 = b
2 & b
13 `2 = b
3 & b
14 `2 = b
4 & b
5 = b
11 & b
6 = b
12 & b
7 = b
13 & b
8 = b
14 & ( for b
15, b
16 being
Point of
I[01] holds
not b
9 . b
15 = b
10 . b
16 ) )