:: 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 n be
Nat;
func Nbdl1 c1 -> Relation of
Seg a1 means :
Def3:
:: FINTOPO4:def 3
for
i being
Element of
NAT st
i in Seg n holds
it .: {i} = {i,(max (i -' 1),1),(min (i + 1),n)};
existence
ex b1 being Relation of Seg n st
for i being Element of NAT st i in Seg n holds
b1 .: {i} = {i,(max (i -' 1),1),(min (i + 1),n)}
uniqueness
for b1, b2 being Relation of Seg n st ( for i being Element of NAT st i in Seg n holds
b1 .: {i} = {i,(max (i -' 1),1),(min (i + 1),n)} ) & ( for i being Element of NAT st i in Seg n holds
b2 .: {i} = {i,(max (i -' 1),1),(min (i + 1),n)} ) holds
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 n be
Nat;
func Nbdc1 c1 -> Relation of
Seg a1 means :
Def5:
:: FINTOPO4:def 5
for
i being
Element of
NAT st
i in Seg n holds
( ( 1
< i &
i < n implies
it .: {i} = {i,(i -' 1),(i + 1)} ) & (
i = 1 &
i < n implies
it .: {i} = {i,n,(i + 1)} ) & ( 1
< i &
i = n implies
it .: {i} = {i,(i -' 1),1} ) & (
i = 1 &
i = n implies
it .: {i} = {i} ) );
existence
ex b1 being Relation of Seg n st
for i being Element of NAT st i in Seg n holds
( ( 1 < i & i < n implies b1 .: {i} = {i,(i -' 1),(i + 1)} ) & ( i = 1 & i < n implies b1 .: {i} = {i,n,(i + 1)} ) & ( 1 < i & i = n implies b1 .: {i} = {i,(i -' 1),1} ) & ( i = 1 & i = n implies b1 .: {i} = {i} ) )
uniqueness
for b1, b2 being Relation of Seg n st ( for i being Element of NAT st i in Seg n holds
( ( 1 < i & i < n implies b1 .: {i} = {i,(i -' 1),(i + 1)} ) & ( i = 1 & i < n implies b1 .: {i} = {i,n,(i + 1)} ) & ( 1 < i & i = n implies b1 .: {i} = {i,(i -' 1),1} ) & ( i = 1 & i = n implies b1 .: {i} = {i} ) ) ) & ( for i being Element of NAT st i in Seg n holds
( ( 1 < i & i < n implies b2 .: {i} = {i,(i -' 1),(i + 1)} ) & ( i = 1 & i < n implies b2 .: {i} = {i,n,(i + 1)} ) & ( 1 < i & i = n implies b2 .: {i} = {i,(i -' 1),1} ) & ( i = 1 & i = n implies b2 .: {i} = {i} ) ) ) holds
b1 = b2
end;
:: deftheorem Def5 defines Nbdc1 FINTOPO4:def 5 :
:: deftheorem Def6 defines FTSC1 FINTOPO4:def 6 :
theorem Th20: :: FINTOPO4:20
theorem Th21: :: FINTOPO4:21