:: GOBRD12 semantic presentation
Lemma1:
for b1 being non constant standard special_circular_sequence holds (L~ b1) ` = the carrier of ((TOP-REAL 2) | ((L~ b1) ` ))
Lemma2:
the carrier of (TOP-REAL 2) = REAL 2
by EUCLID:25;
theorem Th1: :: GOBRD12:1
canceled;
theorem Th2: :: GOBRD12:2
theorem Th3: :: GOBRD12:3
theorem Th4: :: GOBRD12:4
Lemma6:
for b1 being non constant standard special_circular_sequence holds
( Cl (Down (LeftComp b1),((L~ b1) ` )) is connected & Down (LeftComp b1),((L~ b1) ` ) = LeftComp b1 & Down (LeftComp b1),((L~ b1) ` ) is_a_component_of (TOP-REAL 2) | ((L~ b1) ` ) )
Lemma7:
for b1 being non constant standard special_circular_sequence holds
( Cl (Down (RightComp b1),((L~ b1) ` )) is connected & Down (RightComp b1),((L~ b1) ` ) = RightComp b1 & Down (RightComp b1),((L~ b1) ` ) is_a_component_of (TOP-REAL 2) | ((L~ b1) ` ) )
theorem Th5: :: GOBRD12:5
theorem Th6: :: GOBRD12:6
Lemma10:
for b1 being non constant standard special_circular_sequence
for b2, b3, b4, b5 being Nat holds
( b2 <= len (GoB b1) & b3 <= width (GoB b1) & b4 <= len (GoB b1) & b5 <= width (GoB b1) & ( b4 = b2 + 1 or b2 = b4 + 1 ) & b3 = b5 & Int (cell (GoB b1),b2,b3) c= (LeftComp b1) \/ (RightComp b1) implies Int (cell (GoB b1),b4,b5) c= (LeftComp b1) \/ (RightComp b1) )
Lemma11:
for b1 being non constant standard special_circular_sequence
for b2, b3, b4, b5 being Nat holds
( b2 <= len (GoB b1) & b3 <= width (GoB b1) & b4 <= len (GoB b1) & b5 <= width (GoB b1) & ( b5 = b3 + 1 or b3 = b5 + 1 ) & b2 = b4 & Int (cell (GoB b1),b2,b3) c= (LeftComp b1) \/ (RightComp b1) implies Int (cell (GoB b1),b4,b5) c= (LeftComp b1) \/ (RightComp b1) )
theorem Th7: :: GOBRD12:7
for b
1 being non
constant standard special_circular_sequencefor b
2, b
3, b
4, b
5 being
Nat holds
( b
2 <= len (GoB b1) & b
3 <= width (GoB b1) & b
4 <= len (GoB b1) & b
5 <= width (GoB b1) & b
2,b
3,b
4,b
5 are_adjacent2 implies (
Int (cell (GoB b1),b2,b3) c= (LeftComp b1) \/ (RightComp b1) iff
Int (cell (GoB b1),b4,b5) c= (LeftComp b1) \/ (RightComp b1) ) )
theorem Th8: :: GOBRD12:8
for b
1 being non
constant standard special_circular_sequencefor b
2, b
3 being
FinSequence of
NAT holds
(
len b
2 = len b
3 & ex b
4 being
Nat st
( b
4 in dom b
2 &
Int (cell (GoB b1),(b2 /. b4),(b3 /. b4)) c= (LeftComp b1) \/ (RightComp b1) ) & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 < len b
2 implies b
2 /. b
4,b
3 /. b
4,b
2 /. (b4 + 1),b
3 /. (b4 + 1) are_adjacent2 ) ) & ( for b
4, b
5, b
6 being
Nat holds
( b
4 in dom b
2 & b
5 = b
2 . b
4 & b
6 = b
3 . b
4 implies ( b
5 <= len (GoB b1) & b
6 <= width (GoB b1) ) ) ) implies for b
4 being
Nat holds
( b
4 in dom b
2 implies
Int (cell (GoB b1),(b2 /. b4),(b3 /. b4)) c= (LeftComp b1) \/ (RightComp b1) ) )
theorem Th9: :: GOBRD12:9
theorem Th10: :: GOBRD12:10
theorem Th11: :: GOBRD12:11