:: FINTOPO4 semantic presentation
:: deftheorem Def1 defines are_separated FINTOPO4:def 1 :
theorem Th1: :: FINTOPO4:1
theorem Th2: :: FINTOPO4:2
theorem Th3: :: FINTOPO4:3
theorem Th4: :: FINTOPO4:4
theorem Th5: :: FINTOPO4:5
theorem Th6: :: FINTOPO4:6
theorem Th7: :: FINTOPO4:7
theorem Th8: :: FINTOPO4:8
theorem Th9: :: FINTOPO4:9
theorem Th10: :: FINTOPO4:10
theorem Th11: :: FINTOPO4:11
theorem Th12: :: FINTOPO4:12
theorem Th13: :: FINTOPO4:13
:: deftheorem Def2 defines is_continuous FINTOPO4:def 2 :
theorem Th14: :: FINTOPO4:14
theorem Th15: :: FINTOPO4:15
theorem Th16: :: FINTOPO4:16
theorem Th17: :: FINTOPO4:17
definition
let c
1 be
Nat;
func Nbdl1 c
1 -> Function of
Seg a
1,
bool (Seg a1) means :
Def3:
:: FINTOPO4:def 3
(
dom a
2 = Seg a
1 & ( for b
1 being
Nat holds
( b
1 in Seg a
1 implies a
2 . b
1 = {b1,(max (b1 -' 1),1),(min (b1 + 1),a1)} ) ) );
existence
ex b1 being Function of Seg c1, bool (Seg c1) st
( dom b1 = Seg c1 & ( for b2 being Nat holds
( b2 in Seg c1 implies b1 . b2 = {b2,(max (b2 -' 1),1),(min (b2 + 1),c1)} ) ) )
uniqueness
for b1, b2 being Function of Seg c1, bool (Seg c1) holds
( dom b1 = Seg c1 & ( for b3 being Nat holds
( b3 in Seg c1 implies b1 . b3 = {b3,(max (b3 -' 1),1),(min (b3 + 1),c1)} ) ) & dom b2 = Seg c1 & ( for b3 being Nat holds
( b3 in Seg c1 implies b2 . b3 = {b3,(max (b3 -' 1),1),(min (b3 + 1),c1)} ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines Nbdl1 FINTOPO4:def 3 :
:: deftheorem Def4 defines FTSL1 FINTOPO4:def 4 :
theorem Th18: :: FINTOPO4:18
theorem Th19: :: FINTOPO4:19
definition
let c
1 be
Nat;
func Nbdc1 c
1 -> Function of
Seg a
1,
bool (Seg a1) means :
Def5:
:: FINTOPO4:def 5
(
dom a
2 = Seg a
1 & ( for b
1 being
Nat holds
( b
1 in Seg a
1 implies ( ( 1
< b
1 & b
1 < a
1 implies a
2 . b
1 = {b1,(b1 -' 1),(b1 + 1)} ) & ( b
1 = 1 & b
1 < a
1 implies a
2 . b
1 = {b1,a1,(b1 + 1)} ) & ( 1
< b
1 & b
1 = a
1 implies a
2 . b
1 = {b1,(b1 -' 1),1} ) & ( b
1 = 1 & b
1 = a
1 implies a
2 . b
1 = {b1} ) ) ) ) );
existence
ex b1 being Function of Seg c1, bool (Seg c1) st
( dom b1 = Seg c1 & ( for b2 being Nat holds
( b2 in Seg c1 implies ( ( 1 < b2 & b2 < c1 implies b1 . b2 = {b2,(b2 -' 1),(b2 + 1)} ) & ( b2 = 1 & b2 < c1 implies b1 . b2 = {b2,c1,(b2 + 1)} ) & ( 1 < b2 & b2 = c1 implies b1 . b2 = {b2,(b2 -' 1),1} ) & ( b2 = 1 & b2 = c1 implies b1 . b2 = {b2} ) ) ) ) )
uniqueness
for b1, b2 being Function of Seg c1, bool (Seg c1) holds
( dom b1 = Seg c1 & ( for b3 being Nat holds
( b3 in Seg c1 implies ( ( 1 < b3 & b3 < c1 implies b1 . b3 = {b3,(b3 -' 1),(b3 + 1)} ) & ( b3 = 1 & b3 < c1 implies b1 . b3 = {b3,c1,(b3 + 1)} ) & ( 1 < b3 & b3 = c1 implies b1 . b3 = {b3,(b3 -' 1),1} ) & ( b3 = 1 & b3 = c1 implies b1 . b3 = {b3} ) ) ) ) & dom b2 = Seg c1 & ( for b3 being Nat holds
( b3 in Seg c1 implies ( ( 1 < b3 & b3 < c1 implies b2 . b3 = {b3,(b3 -' 1),(b3 + 1)} ) & ( b3 = 1 & b3 < c1 implies b2 . b3 = {b3,c1,(b3 + 1)} ) & ( 1 < b3 & b3 = c1 implies b2 . b3 = {b3,(b3 -' 1),1} ) & ( b3 = 1 & b3 = c1 implies b2 . b3 = {b3} ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Nbdc1 FINTOPO4:def 5 :
for b
1 being
Natfor b
2 being
Function of
Seg b
1,
bool (Seg b1) holds
( b
2 = Nbdc1 b
1 iff (
dom b
2 = Seg b
1 & ( for b
3 being
Nat holds
( b
3 in Seg b
1 implies ( ( 1
< b
3 & b
3 < b
1 implies b
2 . b
3 = {b3,(b3 -' 1),(b3 + 1)} ) & ( b
3 = 1 & b
3 < b
1 implies b
2 . b
3 = {b3,b1,(b3 + 1)} ) & ( 1
< b
3 & b
3 = b
1 implies b
2 . b
3 = {b3,(b3 -' 1),1} ) & ( b
3 = 1 & b
3 = b
1 implies b
2 . b
3 = {b3} ) ) ) ) ) );
:: deftheorem Def6 defines FTSC1 FINTOPO4:def 6 :
theorem Th20: :: FINTOPO4:20
theorem Th21: :: FINTOPO4:21