
theorem Th28:
  for S being compact Hausdorff TopLattice, x being Element of S
  st for b being Element of S holds b"/\" is continuous holds downarrow x is
  closed
proof
  let S be compact Hausdorff TopLattice, b be Element of S;
  assume for a being Element of S holds a"/\" is continuous;
  then
A1: b"/\" is continuous;
  set g1 = (rng (b"/\"))|`(b"/\");
A2: g1 = b"/\" by RELAT_1:94;
A3: dom (b"/\") = the carrier of S by FUNCT_2:def 1;
A4: {b} "/\" [#]S = {b "/\" y where y is Element of S: y in [#] S} by
YELLOW_4:42;
A5: rng (b"/\") = {b} "/\" [#]S
  proof
    thus rng (b"/\") c= {b} "/\" [#]S
    proof
      let q be object;
      assume q in rng (b"/\");
      then consider x being object such that
A6:   x in dom (b"/\") and
A7:   (b"/\").x = q by FUNCT_1:def 3;
      reconsider x1 = x as Element of S by A6;
      q = b "/\" x1 by A7,WAYBEL_1:def 18;
      hence thesis by A4;
    end;
    let q be object;
    assume q in {b} "/\" [#]S;
    then consider y being Element of S such that
A8: q = b "/\" y and
    y in [#]S by A4;
    q = (b"/\").y by A8,WAYBEL_1:def 18;
    hence thesis by A3,FUNCT_1:def 3;
  end;
  then rng g1 = {b} "/\" [#]S by RELAT_1:94
    .= [#](S | (rng (b"/\"))) by A5,PRE_TOPC:def 5
    .= the carrier of S|(rng(b"/\"));
  then reconsider g1 as Function of S, S|(rng (b"/\")) by A2,A3,FUNCT_2:1;
  rng g1 = {b} "/\" [#]S by A5,RELAT_1:94
    .= [#](S | ({b}"/\"[#]S)) by PRE_TOPC:def 5;
  then S | ({b} "/\" [#]S) is compact by A1,A2,A5,COMPTS_1:14,PRE_TOPC:27;
  then {b} "/\" [#]S is compact by COMPTS_1:3;
  hence thesis by Th5;
end;
