
theorem Th57:
  for L being complete non empty Poset holds [IdsMap L,SupMap L]
  is Galois & SupMap L is sups-preserving
proof
  let L be complete non empty Poset;
  set g = IdsMap L, d = SupMap L;
  now
    let I be Element of InclPoset(Ids L), x be Element of L;
    reconsider I9 = I as Ideal of L by YELLOW_2:41;
    hereby
      assume I <= g.x;
      then I c= g.x by YELLOW_1:3;
      then I9 c= downarrow x by YELLOW_2:def 4;
      then x is_>=_than I9 by YELLOW_2:1;
      then sup I9 <= x by YELLOW_0:32;
      hence d.I <= x by YELLOW_2:def 3;
    end;
    assume d.I <= x;
    then
A1: sup I9 <= x by YELLOW_2:def 3;
    sup I9 is_>=_than I9 by YELLOW_0:32;
    then x is_>=_than I9 by A1,YELLOW_0:4;
    then I9 c= downarrow x by YELLOW_2:1;
    then I c= g.x by YELLOW_2:def 4;
    hence I <= g.x by YELLOW_1:3;
  end;
  hence [IdsMap L,SupMap L] is Galois;
  then SupMap L is lower_adjoint;
  hence thesis;
end;
