reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];

theorem
  a \/ (b /\ c) = (a \/ b) /\ (a \/ c)
proof
  thus a \/ (b /\ c) = a \/ c /\ a \/ c /\ b by Th6
    .= a \/ (c /\ a \/ c /\ b) by Th3
    .= a \/ c /\ (a \/ b) by Th5
    .= (a \/ b) /\ a \/ (a \/ b) /\ c by Th7
    .= (a \/ b) /\ (a \/ c) by Th5;
end;
