let T be LinearTopSpace; :: thesis: for L being non empty reflexive transitive RelStr

for f being Function of ([#] L), the carrier of T

for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let L be non empty reflexive transitive RelStr ; :: thesis: for f being Function of ([#] L), the carrier of T

for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let f be Function of ([#] L), the carrier of T; :: thesis: for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let x be Point of T; :: thesis: for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let V be local_base of T; :: thesis: ( [#] L is directed implies ( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) )

assume A1: [#] L is directed ; :: thesis: ( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

set B = { (x + U) where U is Subset of T : ( U in V & U is a_neighborhood of 0. T ) } ;

reconsider B = { (x + U) where U is Subset of T : ( U in V & U is a_neighborhood of 0. T ) } as Subset-Family of T by Lm2;

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) ; :: thesis: verum

for f being Function of ([#] L), the carrier of T

for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let L be non empty reflexive transitive RelStr ; :: thesis: for f being Function of ([#] L), the carrier of T

for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let f be Function of ([#] L), the carrier of T; :: thesis: for x being Point of T

for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let x be Point of T; :: thesis: for V being local_base of T st [#] L is directed holds

( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

let V be local_base of T; :: thesis: ( [#] L is directed implies ( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) )

assume A1: [#] L is directed ; :: thesis: ( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v )

set B = { (x + U) where U is Subset of T : ( U in V & U is a_neighborhood of 0. T ) } ;

reconsider B = { (x + U) where U is Subset of T : ( U in V & U is a_neighborhood of 0. T ) } as Subset-Family of T by Lm2;

now :: thesis: ( ( x in lim_f f implies for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v ) & ( ( for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) implies x in lim_f f ) )

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ; :: thesis: x in lim_f f

for b being Element of B ex i being Element of L st

for j being Element of L st i <= j holds

f . j in b

end;

hence
( x in lim_f f iff for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v ) & ( ( for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) implies x in lim_f f ) )

hereby :: thesis: ( ( for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) implies x in lim_f f )

assume A4:
for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) implies x in lim_f f )

assume A2:
x in lim_f f
; :: thesis: for v being Subset of T st v in V /\ (NeighborhoodSystem (0. T)) holds

ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v

let v be Subset of T; :: thesis: ( v in V /\ (NeighborhoodSystem (0. T)) implies ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v )

assume v in V /\ (NeighborhoodSystem (0. T)) ; :: thesis: ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v

then ( v in V & v in NeighborhoodSystem (0. T) ) by XBOOLE_0:def 4;

then ( v in V & v is a_neighborhood of 0. T ) by YELLOW19:2;

then x + v in B ;

then reconsider b = x + v as Element of B ;

consider i0 being Element of L such that

A3: for j being Element of L st i0 <= j holds

f . j in b by A1, A2, Lm3;

take i0 = i0; :: thesis: for j being Element of L st i0 <= j holds

f . j in x + v

thus for j being Element of L st i0 <= j holds

f . j in x + v by A3; :: thesis: verum

end;ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v

let v be Subset of T; :: thesis: ( v in V /\ (NeighborhoodSystem (0. T)) implies ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v )

assume v in V /\ (NeighborhoodSystem (0. T)) ; :: thesis: ex i0 being Element of L st

for j being Element of L st i0 <= j holds

f . j in x + v

then ( v in V & v in NeighborhoodSystem (0. T) ) by XBOOLE_0:def 4;

then ( v in V & v is a_neighborhood of 0. T ) by YELLOW19:2;

then x + v in B ;

then reconsider b = x + v as Element of B ;

consider i0 being Element of L such that

A3: for j being Element of L st i0 <= j holds

f . j in b by A1, A2, Lm3;

take i0 = i0; :: thesis: for j being Element of L st i0 <= j holds

f . j in x + v

thus for j being Element of L st i0 <= j holds

f . j in x + v by A3; :: thesis: verum

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ; :: thesis: x in lim_f f

for b being Element of B ex i being Element of L st

for j being Element of L st i <= j holds

f . j in b

proof

hence
x in lim_f f
by A1, Lm3; :: thesis: verum
let b be Element of B; :: thesis: ex i being Element of L st

for j being Element of L st i <= j holds

f . j in b

not B is empty by Lm2;

then b in B ;

then consider U1 being Subset of T such that

A5: b = x + U1 and

A6: U1 in V and

A7: U1 is a_neighborhood of 0. T ;

U1 in NeighborhoodSystem (0. T) by A7, YELLOW19:2;

then U1 in V /\ (NeighborhoodSystem (0. T)) by A6, XBOOLE_0:def 4;

then consider i0 being Element of L such that

A8: for j being Element of L st i0 <= j holds

f . j in x + U1 by A4;

take i0 ; :: thesis: for j being Element of L st i0 <= j holds

f . j in b

thus for j being Element of L st i0 <= j holds

f . j in b by A5, A8; :: thesis: verum

end;for j being Element of L st i <= j holds

f . j in b

not B is empty by Lm2;

then b in B ;

then consider U1 being Subset of T such that

A5: b = x + U1 and

A6: U1 in V and

A7: U1 is a_neighborhood of 0. T ;

U1 in NeighborhoodSystem (0. T) by A7, YELLOW19:2;

then U1 in V /\ (NeighborhoodSystem (0. T)) by A6, XBOOLE_0:def 4;

then consider i0 being Element of L such that

A8: for j being Element of L st i0 <= j holds

f . j in x + U1 by A4;

take i0 ; :: thesis: for j being Element of L st i0 <= j holds

f . j in b

thus for j being Element of L st i0 <= j holds

f . j in b by A5, A8; :: thesis: verum

ex i being Element of L st

for j being Element of L st i <= j holds

f . j in x + v ) ; :: thesis: verum