reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th45:
  for L being lower-bounded sup-Semilattice for X being non empty
  Subset of Ids L holds meet X is Ideal of L
proof
  let L be lower-bounded sup-Semilattice;
  let X be non empty Subset of Ids L;
A1: for J being set st J in X holds J is Ideal of L
  proof
    let J be set;
    assume J in X;
    then J in Ids L;
    then ex Y being Ideal of L st Y = J;
    hence thesis;
  end;
  X c= bool the carrier of L
  proof
    let x be object;
    assume x in X;
    then x is Ideal of L by A1;
    hence thesis;
  end;
  then reconsider F = X as Subset-Family of L;
  for J being Subset of L st J in F holds J is lower by A1;
  then reconsider I = meet X as lower Subset of L by Th36;
  for J being set st J in X holds Bottom L in J
  proof
    let J be set;
    assume J in X;
    then reconsider J9= J as Ideal of L by A1;
    set j = the Element of J9;
    Bottom L <= j by YELLOW_0:44;
    hence thesis by WAYBEL_0:def 19;
  end;
  then reconsider I as non empty lower Subset of L by SETFAM_1:def 1;
  for J being Subset of L st J in F holds J is lower directed by A1;
  then I is Ideal of L by Th38;
  hence thesis;
end;
