theorem
  bool A \/ bool B c= bool (A \/ B)
proof
  let x;
   reconsider xx=x as set by TARSKI:1;
  assume x in bool A \/ bool B;
  then x in bool A or x in bool B by XBOOLE_0:def 3;
  then
A1: xx c= A or xx c= B by Def1;
  A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
  then xx c= A \/ B by A1;
  hence thesis by Def1;
end;
