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 = bool (A \/ B) implies A,B are_c=-comparable
proof
  assume
A1: bool A \/ bool B = bool (A \/ B);
  A \/ B in bool (A \/ B) by Def1;
  then A \/ B in bool A or A \/ B in bool B by A1,XBOOLE_0:def 3;
  then
A2: A \/ B c= A or A \/ B c= B by Def1;
  A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
  hence A c= B or B c= A by A2;
end;
