
theorem
  for N being meet-continuous Lawson complete TopLattice st InclPoset
sigma N is continuous holds N is Hausdorff iff for X being Subset of [:N,N qua
  TopSpace:] st X = the InternalRel of N holds X is closed
proof
  let N be meet-continuous Lawson complete TopLattice such that
A1: InclPoset sigma N is continuous;
A2: [:the carrier of N,the carrier of N:] = the carrier of [:N,N qua
  TopSpace:] by BORSUK_1:def 2;
  hereby
    reconsider D = id the carrier of N as Subset of [:N,N qua TopSpace:] by
BORSUK_1:def 2;
    set INF = inf_op N, f = <:pr1(the carrier of N,the carrier of N),INF:>;
    assume N is Hausdorff;
    then
A3: D is closed by YELLOW12:46;
A4: the carrier of [:N,N:] = [:the carrier of N,the carrier of N:] by
YELLOW_3:def 2;
    then reconsider
    f as Function of [:N,N qua TopSpace:], [:N,N qua TopSpace:] by A2,
FUNCT_3:58;
    let X be Subset of [:N,N qua TopSpace:] such that
A5: X = the InternalRel of N;
A6: for x, y being Element of N holds f. [x,y] = [x, x "/\" y]
    proof
      let x, y be Element of N;
A7:   dom (pr1(the carrier of N,the carrier of N)) = [:the carrier of N,
      the carrier of N:] by FUNCT_2:def 1;
A8:   [x,y] in [:the carrier of N,the carrier of N:] by ZFMISC_1:87;
      dom INF = [:the carrier of N,the carrier of N:] by A4,FUNCT_2:def 1;
      hence
      f. [x,y] = [pr1(the carrier of N,the carrier of N).(x,y),INF.(x,y)]
      by A8,A7,FUNCT_3:49
        .= [x, INF.(x,y)] by FUNCT_3:def 4
        .= [x, x "/\" y] by WAYBEL_2:def 4;
    end;
A9: X = f"D
    proof
      hereby
        let a be object;
        assume
A10:    a in X;
        then consider s, t being object such that
A11:    s in the carrier of N and
A12:    t in the carrier of N and
A13:    a = [s,t] by A5,ZFMISC_1:def 2;
        reconsider s, t as Element of N by A11,A12;
        s <= t by A5,A10,A13,ORDERS_2:def 5;
        then s "/\" t = s by YELLOW_0:25;
        then f. [s,t] = [s,s] by A6;
        then
A14:    f.a in D by A13,RELAT_1:def 10;
        dom f = the carrier of [:N,N qua TopSpace:] by FUNCT_2:def 1;
        then a in dom f by A2,A11,A12,A13,ZFMISC_1:87;
        hence a in f"D by A14,FUNCT_1:def 7;
      end;
      let a be object;
      assume
A15:  a in f"D;
      then
A16:  f.a in D by FUNCT_1:def 7;
      consider s, t being object such that
A17:  s in the carrier of N and
A18:  t in the carrier of N and
A19:  a = [s,t] by A2,A15,ZFMISC_1:def 2;
      reconsider s, t as Element of N by A17,A18;
      f.a = [s, s "/\" t] by A6,A19;
      then s = s "/\" t by A16,RELAT_1:def 10;
      then s <= t by YELLOW_0:25;
      hence thesis by A5,A19,ORDERS_2:def 5;
    end;
    reconsider INF as Function of [:N,N qua TopSpace:], N by A2,A4;
    N is topological_semilattice by A1,Th22;
    then
A20: INF is continuous;
    pr1(the carrier of N,the carrier of N) is continuous Function of [:N,
    N qua TopSpace:],N by YELLOW12:39;
    then f is continuous by A20,YELLOW12:41;
    hence X is closed by A9,A3;
  end;
  assume
A21: for X being Subset of [:N,N qua TopSpace:] st X = the InternalRel
  of N holds X is closed;
A22: (the InternalRel of N) /\ the InternalRel of N~ c= id the carrier of N
  by YELLOW12:23;
  id the carrier of N c= (the InternalRel of N) /\ the InternalRel of N~
  by YELLOW12:22;
  then
A23: id the carrier of N = (the InternalRel of N) /\ the InternalRel of N~
  by A22;
  for A being Subset of [:N,N qua TopSpace:] st A = id the carrier of N
  holds A is closed
  proof
    reconsider f = <:pr2(the carrier of N,the carrier of N), pr1(the carrier
of N,the carrier of N):> as continuous Function of [:N,N qua TopSpace:], [:N,N
    qua TopSpace:] by YELLOW12:42;
    reconsider X = the InternalRel of N, Y = the InternalRel of N~ as Subset
    of [:N,N qua TopSpace:] by BORSUK_1:def 2;
    let A be Subset of [:N,N qua TopSpace:] such that
A24: A = id the carrier of N;
    reconsider X, Y as Subset of [:N,N qua TopSpace:];
A25: X is closed by A21;
A26: dom f = [:the carrier of N,the carrier of N:] by YELLOW12:4;
A27: f.:X = Y
    proof
      thus f.:X c= Y
      proof
        let y be object;
        assume y in f.:X;
        then consider x being object such that
        x in dom f and
A28:    x in X and
A29:    y = f.x by FUNCT_1:def 6;
        consider x1, x2 being object such that
A30:    x1 in the carrier of N and
A31:    x2 in the carrier of N and
A32:    x = [x1,x2] by A28,ZFMISC_1:def 2;
        f.x = [x2,x1] by A30,A31,A32,Lm3;
        hence thesis by A28,A29,A32,RELAT_1:def 7;
      end;
      let y be object;
      assume
A33:  y in Y;
      then consider y1, y2 being object such that
A34:  y1 in the carrier of N and
A35:  y2 in the carrier of N and
A36:  y = [y1,y2] by ZFMISC_1:def 2;
A37:  [y2,y1] in X by A33,A36,RELAT_1:def 7;
      f. [y2,y1] = y by A34,A35,A36,Lm3;
      hence thesis by A26,A37,FUNCT_1:def 6;
    end;
    f is being_homeomorphism by YELLOW12:43;
    then Y is closed by A25,A27,TOPS_2:58;
    hence thesis by A23,A24,A25;
  end;
  hence thesis by YELLOW12:46;
end;
