
theorem
  for X being non empty TopSpace st X is regular &
  InclPoset the topology of X is continuous holds X is locally-compact
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
A1: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  assume that
A2: T is regular and
A3: L is continuous;
  let x be Point of T, X be Subset of T;
  assume that
A4: x in X and
A5: X is open;
  reconsider a = X as Element of L by A1,A5;
  a = sup waybelow a by A3,Def5
    .= union waybelow a by YELLOW_1:22;
  then consider Y being set such that
A6: x in Y and
A7: Y in waybelow a by A4,TARSKI:def 4;
  Y in the carrier of L by A7;
  then reconsider Y as Subset of T by A1;
  consider y being Element of L such that
A8: Y = y and
A9: y << a by A7;
  Y is open by A1,A7;
  then consider W being open Subset of T such that
A10: x in W and
A11: Cl W c= Y by A2,A6,URYSOHN1:21;
  take Z = Cl W;
  W c= Z by PRE_TOPC:18;
  hence x in Int Z by A10,TOPS_1:22;
  y <= a by A9,Th1;
  then Y c= X by A8,YELLOW_1:3;
  hence Z c= X by A11;
  let F be Subset-Family of T such that
A12: F is Cover of Z and
A13: F is open;
  reconsider ZZ = {Z`} as Subset-Family of T;
  reconsider ZZ as Subset-Family of T;
  reconsider FZ = F \/ ZZ as Subset-Family of T;
  for x being Subset of T st x in ZZ holds x is open by TARSKI:def 1;
  then ZZ is open;
  then FZ is open by A13,TOPS_2:13;
  then reconsider FF = FZ as Subset of L by YELLOW_1:25;
A14: sup FF = union FZ by YELLOW_1:22
    .= union F \/ union ZZ by ZFMISC_1:78
    .= union F \/ Z` by ZFMISC_1:25;
  Z c= union F by A12,SETFAM_1:def 11;
  then Z \/ Z` c= sup FF by A14,XBOOLE_1:9;
  then [#]T c= sup FF by PRE_TOPC:2;
  then X c= sup FF;
  then a <= sup FF by YELLOW_1:3;
  then consider A being finite Subset of L such that
A15: A c= FF and
A16: y <= sup A by A9,Th18;
A17: sup A = union A by YELLOW_1:22;
  reconsider G = A \ ZZ as Subset-Family of T by A1,XBOOLE_1:1;
  take G;
  thus G c= F by A15,XBOOLE_1:43;
  thus Z c= union G
  proof
    let z be object;
    assume
A18: z in Z;
    then
A19: z in Y by A11;
A20: Y c= union A by A8,A16,A17,YELLOW_1:3;
A21: not z in Z` by A18,XBOOLE_0:def 5;
    consider B being set such that
A22: z in B and
A23: B in A by A19,A20,TARSKI:def 4;
    not B in ZZ by A21,A22,TARSKI:def 1;
    then B in G by A23,XBOOLE_0:def 5;
    hence thesis by A22,TARSKI:def 4;
  end;
  thus thesis;
end;
