
theorem Th4:
  for R being Semilattice, D being non empty directed Subset of
  InclPoset Ids R, x being Element of InclPoset Ids R holds sup ({x} "/\" D) =
  union the set of all x /\ d where d is Element of D
proof
  let R be Semilattice, D be non empty directed Subset of InclPoset Ids R, x
  be Element of InclPoset Ids R;
  set I = InclPoset Ids R, A = the set of all x /\ d where d is Element of D;
  set xD = {x "/\" d where d is Element of I: d in D};
  xD c= the carrier of I
  proof
    let a be object;
    assume a in xD;
    then ex d being Element of I st a = x "/\" d & d in D;
    hence thesis;
  end;
  then reconsider xD as Subset of I;
A1: ex_sup_of {x} "/\" D,I by WAYBEL_2:1;
  then ex_sup_of xD,I by YELLOW_4:42;
  then
A2: sup xD is_>=_than xD by YELLOW_0:def 9;
  hereby
    set A = the set of all x /\ w where w is Element of D;
    let a be object;
    ex_sup_of D,I by WAYBEL_0:75;
    then sup ({x} "/\" D) <= x "/\" sup D by A1,YELLOW_4:53;
    then
A3: sup ({x} "/\" D) c= x "/\" sup D by YELLOW_1:3;
    assume a in sup ({x} "/\" D);
    then a in x "/\" sup D by A3;
    then
A4: a in x /\ sup D by YELLOW_2:43;
    then a in sup D by XBOOLE_0:def 4;
    then a in union D by Th3;
    then consider d being set such that
A5: a in d and
A6: d in D by TARSKI:def 4;
    reconsider d as Element of I by A6;
A7: x /\ d in A by A6;
    a in x by A4,XBOOLE_0:def 4;
    then a in x /\ d by A5,XBOOLE_0:def 4;
    hence a in union A by A7,TARSKI:def 4;
  end;
  let a be object;
  assume a in union A;
  then consider Z being set such that
A8: a in Z and
A9: Z in A by TARSKI:def 4;
  consider d being Element of D such that
A10: Z = x /\ d and
  not contradiction by A9;
  reconsider d as Element of I;
A11: xD = {x} "/\" D by YELLOW_4:42;
  then x "/\" d in {x} "/\" D;
  then sup xD >= x "/\" d by A2,A11;
  then
A12: x "/\" d c= sup xD by YELLOW_1:3;
  x /\ d = x "/\" d by YELLOW_2:43;
  then a in sup xD by A12,A8,A10;
  hence thesis by YELLOW_4:42;
end;
