theorem
  (A*) ^^ (A*) = A*
proof
  <%>E in A* by Th48;
  then
A1: A* c= (A*) ^^ (A*) by Th16;
  (A*) ^^ (A*) c= A* by Th46;
  hence thesis by A1,XBOOLE_0:def 10;
end;
