theorem
  the_transitive-closure_of (X /\ Y) c=
  the_transitive-closure_of X /\ the_transitive-closure_of Y
proof
   the_transitive-closure_of (X /\ Y) c= the_transitive-closure_of X &
  the_transitive-closure_of (X /\ Y) c= the_transitive-closure_of Y by Th58,
XBOOLE_1:17;
  hence thesis by XBOOLE_1:19;
end;
