
theorem
  for T being non empty TopSpace st T is locally-compact
  holds InclPoset the topology of T is continuous
proof
  let T be non empty TopSpace such that
A1: T is locally-compact;
  set L = InclPoset the topology of T;
A2: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  thus for x being Element of L holds waybelow x is non empty directed;
  thus L is up-complete;
  let x be Element of L;
  x in the topology of T by A2;
  then reconsider X = x as Subset of T;
  thus x c= sup waybelow x
  proof
    let a be object;
    assume
A3: a in x;
    X is open by A2;
    then consider Y being Subset of T such that
A4: a in Int Y and
A5: Y c= X and
A6: Y is compact by A1,A3;
    reconsider iY = Int Y as Subset of T;
    reconsider y = iY as Element of L by A2,PRE_TOPC:def 2;
    y << x by A5,A6,Th38,TOPS_1:16;
    then y in waybelow x;
    then y c= union waybelow x by ZFMISC_1:74;
    then y c= sup waybelow x by YELLOW_1:22;
    hence thesis by A4;
  end;
  let a be object;
  assume a in sup waybelow x;
  then a in union waybelow x by YELLOW_1:22;
  then consider Y being set such that
A7: a in Y and
A8: Y in waybelow x by TARSKI:def 4;
  consider y being Element of L such that
A9: Y = y and
A10: y << x by A8;
  y <= x by A10,Th1;
  then Y c= x by A9,YELLOW_1:3;
  hence thesis by A7;
end;
