
theorem Th31:
  for N being topological_semilattice with_infima TopPoset, C
  being Subset of N st subrelstr C is meet-inheriting holds subrelstr Cl C is
  meet-inheriting
proof
  let N be topological_semilattice with_infima TopPoset, C be Subset of N
  such that
A1: subrelstr C is meet-inheriting;
  set A = Cl C, S = subrelstr A;
  let x, y be Element of N such that
A2: x in the carrier of S and
A3: y in the carrier of S and
  ex_inf_of {x,y},N;
A4: the carrier of S = A by YELLOW_0:def 15;
  for V being a_neighborhood of x "/\" y holds V meets C
  proof
    set NN = [:N,N qua TopSpace:];
    let V be a_neighborhood of x "/\" y;
A5: the carrier of NN = [:the carrier of N,the carrier of N:] by BORSUK_1:def 2
;
    then reconsider xy = [x,y] as Point of NN by YELLOW_3:def 2;
    the carrier of [:N,N:] = [:the carrier of N,the carrier of N:] by
YELLOW_3:def 2;
    then reconsider f = inf_op N as Function of NN, N by A5;
A6: f.xy = f.(x,y) .= x "/\" y by WAYBEL_2:def 4;
    f is continuous by YELLOW13:def 5;
    then consider H being a_neighborhood of xy such that
A7: f.:H c= V by A6,BORSUK_1:def 1;
    consider A being Subset-Family of NN such that
A8: Int H = union A and
A9: for e being set st e in A ex X1, Y1 being Subset of N st e = [:X1
    ,Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
    xy in union A by A8,CONNSP_2:def 1;
    then consider K being set such that
A10: xy in K and
A11: K in A by TARSKI:def 4;
    consider Ix, Iy being Subset of N such that
A12: K = [:Ix,Iy:] and
A13: Ix is open and
A14: Iy is open by A9,A11;
A15: x in Ix by A10,A12,ZFMISC_1:87;
A16: y in Iy by A10,A12,ZFMISC_1:87;
A17: Ix = Int Ix by A13,TOPS_1:23;
    Iy = Int Iy by A14,TOPS_1:23;
    then reconsider Iy as a_neighborhood of y by A16,CONNSP_2:def 1;
    Iy meets C by A3,A4,CONNSP_2:27;
    then consider iy being object such that
A18: iy in Iy and
A19: iy in C by XBOOLE_0:3;
    reconsider Ix as a_neighborhood of x by A15,A17,CONNSP_2:def 1;
    Ix meets C by A2,A4,CONNSP_2:27;
    then consider ix being object such that
A20: ix in Ix and
A21: ix in C by XBOOLE_0:3;
    reconsider ix, iy as Element of N by A20,A18;
    now
      [ix,iy] in K by A12,A20,A18,ZFMISC_1:87;
      then
A22:  [ix,iy] in Int H by A8,A11,TARSKI:def 4;
      take a = ix "/\" iy;
A23:  dom f = the carrier of [:N,N:] by FUNCT_2:def 1;
A24:  Int H c= H by TOPS_1:16;
      f.(ix,iy) = ix "/\" iy by WAYBEL_2:def 4;
      then ix "/\" iy in f.:H by A24,A23,A22,FUNCT_1:def 6;
      hence a in V by A7;
A25:  ex_inf_of {ix,iy},N by YELLOW_0:21;
      the carrier of subrelstr C = C by YELLOW_0:def 15;
      then inf{ix,iy} in C by A25,A1,A21,A19;
      hence a in C by YELLOW_0:40;
    end;
    hence thesis by XBOOLE_0:3;
  end;
  then x "/\" y in Cl C by CONNSP_2:27;
  hence thesis by A4,YELLOW_0:40;
end;
