let T be non empty TopSpace; :: thesis:
let s be sequence of the carrier of T; :: thesis:
let x be Point of T; :: thesis:
let B be basis of (BOOL2F (NeighborhoodSystem x)); :: thesis:

hereby :: thesis:

assume A1: x in lim_f s ; :: thesis:

now :: thesis:

let b be Element of B; :: thesis:
consider i0 being Element of NAT such that
A2: for j being Element of NAT st i0 <= j holds
s . j in b by A1, Th45;
reconsider i1 = i0 as Nat ;
take i1 = i1; :: thesis:

now :: thesis:

let k be Nat; :: thesis:
assume A3: i1 <= k ; :: thesis:
reconsider k0 = k as Element of NAT by ORDINAL1:def 12;
i0 <= k0 by A3;
hence s . k in b by A2; :: thesis:

end;

hence for k being Nat st i1 <= k holds
s . k in b ; :: thesis:

end;

hence for b being Element of B ex i being Nat st
for j being Nat st i <= j holds
s . j in b ; :: thesis:

end;

assume A4: for b being Element of B ex i being Nat st
for j being Nat st i <= j holds
s . j in b ; :: thesis:

now :: thesis:

let b be Element of B; :: thesis:
consider i0 being Nat such that
A5: for j being Nat st i0 <= j holds
s . j in b by A4;
reconsider i1 = i0 as Element of NAT by ORDINAL1:def 12;
take i1 = i1; :: thesis:
thus for j being Element of NAT st i1 <= j holds
s . j in b by A5; :: thesis:

end;

hence x in lim_f s by Th45; :: thesis: