
theorem Th1:
  for X,Y,Z being set st Y c= the_universe_of X & Z c= the_universe_of X
  holds [:Y,Z:] c= the_universe_of X
proof
  let X,Y,Z be set;
  assume Y c= the_universe_of X; then
  A1: Y c= Tarski-Class the_transitive-closure_of X by YELLOW_6:def 1;
  assume Z c= the_universe_of X; then
   Z c= Tarski-Class the_transitive-closure_of X by YELLOW_6:def 1;
  then
[:Y,Z:] c= Tarski-Class the_transitive-closure_of X by A1,CLASSES1:28;
  hence [:Y,Z:] c= the_universe_of X by YELLOW_6:def 1;
end;
