
theorem Th7:
  for R being /\-complete Semilattice, Z be net of R, D be Subset of R st
  D = the set of all "/\"({Z.k where k is Element of Z: k >= j},R)
  where j is Element of Z holds D is non empty directed
proof
  let R be /\-complete Semilattice, Z be net of R, D be Subset of R;
  assume
A1: D = the set of all "/\"({Z.k where k is Element of Z: k >= j},R)
  where j is Element of Z;
  set j = the Element of Z;
  "/\"({Z.k where k is Element of Z: k >= j},R) in D by A1;
  hence D is non empty;
  let x,y be Element of R;
  assume x in D;
  then consider jx being Element of Z such that
A2: x = "/\"({Z.k where k is Element of Z: k >= jx},R) by A1;
  assume y in D;
  then consider jy being Element of Z such that
A3: y = "/\"({Z.k where k is Element of Z: k >= jy},R) by A1;
  reconsider jx, jy as Element of Z;
  consider j being Element of Z such that
A4: j >= jx and
A5: j >= jy by YELLOW_6:def 3;
  consider j9 being Element of Z such that
A6: j9 >= j and j9 >= j by YELLOW_6:def 3;
  deffunc F(Element of Z) = Z.$1;
  defpred Px[Element of Z] means $1 >= jx;
  defpred Py[Element of Z] means $1 >= jy;
  defpred P[Element of Z] means $1 >= j;
  set Ex = {F(k) where k is Element of Z: Px[k]},
  Ey = {F(k) where k is Element of Z: Py[k]},
  E = {F(k) where k is Element of Z: P[k]};
A7: Z.j in Ex by A4;
A8: Z.j in Ey by A5;
A9: Z.j9 in E by A6;
A10: Ex is Subset of R from DOMAIN_1:sch 8;
A11: Ey is Subset of R from DOMAIN_1:sch 8;
A12: E is Subset of R from DOMAIN_1:sch 8;
  take z = "/\"({Z.k where k is Element of Z: k >= j},R);
  reconsider Ex9= Ex as non empty Subset of R by A7,A10;
  reconsider Ey9 = Ey as non empty Subset of R by A8,A11;
  reconsider E9 = E as non empty Subset of R by A9,A12;
A13: ex_inf_of E9,R by WAYBEL_0:76;
A14: ex_inf_of Ex9,R by WAYBEL_0:76;
A15: ex_inf_of Ey9,R by WAYBEL_0:76;
  thus z in D by A1;
  E9 c= Ex9
  proof
    let e be object;
    assume e in E9;
    then consider k being Element of Z such that
A16: e = Z.k and
A17: k >= j;
    reconsider k as Element of Z;
    k >= jx by A4,A17,YELLOW_0:def 2;
    hence thesis by A16;
  end;
  hence x <= z by A2,A13,A14,YELLOW_0:35;
  E9 c= Ey9
  proof
    let e be object;
    assume e in E9;
    then consider k being Element of Z such that
A18: e = Z.k and
A19: k >= j;
    reconsider k as Element of Z;
    k >= jy by A5,A19,YELLOW_0:def 2;
    hence thesis by A18;
  end;
  hence y <= z by A3,A13,A15,YELLOW_0:35;
end;
