theorem
  the_transitive-closure_of (X \/ Y) =
  the_transitive-closure_of X \/ the_transitive-closure_of Y
proof
   X
 c= the_transitive-closure_of X & Y c= the_transitive-closure_of Y by Th52;
then
A1: X \/ Y c= the_transitive-closure_of X \/ the_transitive-closure_of Y by
XBOOLE_1:13;
 the_transitive-closure_of (X \/ Y) c= the_transitive-closure_of (
  the_transitive-closure_of X \/ the_transitive-closure_of Y) by A1,Th58;
  hence the_transitive-closure_of (X \/ Y) c=
  the_transitive-closure_of X \/ the_transitive-closure_of Y by Th49,Th55;
   the_transitive-closure_of X c= the_transitive-closure_of (X \/ Y) &
  the_transitive-closure_of Y c= the_transitive-closure_of (X \/ Y) by Th58,
XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
