reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;

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;
