reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

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;
