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;
