reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;

theorem Th7:
  the carrier of L is ClosedSubset of L
proof
  the carrier of L c= the carrier of L;
  then reconsider F=the carrier of L as Subset of L;
A1: p in F & q in F implies p "/\" q in F;
  p in F & q in F implies p "\/" q in F;
  hence thesis by A1,LATTICES:def 24,def 25;
end;
