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;

theorem
  F c= <.F \/ H.) & H c= <.F \/ H.)
proof
A1: F \/ H c= <.F \/ H.) by FILTER_0:def 4;
  F c= F \/ H & H c= F \/ H by XBOOLE_1:7;
  hence thesis by A1;
end;
