reserve X,Y,Z,Z1,Z2,D for set,x,y for object;

theorem
  X <> {} & Y <> {} implies meet (X \/ Y) = meet X /\ meet Y
proof
  assume that
A1: X <> {} and
A2: Y <> {};
A3: meet X /\ meet Y c= meet (X \/ Y)
  proof
    let x be object;
    assume x in meet X /\ meet Y;
    then
A4: x in meet X & x in meet Y by XBOOLE_0:def 4;
    now
      let Z;
      assume Z in X \/ Y;
      then Z in X or Z in Y by XBOOLE_0:def 3;
      hence x in Z by A4,Def1;
    end;
    hence thesis by A1,Def1;
  end;
  meet (X \/ Y) c= meet X & meet (X \/ Y) c= meet Y by A1,A2,Th6,XBOOLE_1:7;
  then meet (X \/ Y) c= meet X /\ meet Y by XBOOLE_1:19;
  hence thesis by A3;
end;
