reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: ZFMISC_1:82
  bool A (\/) bool B = bool (A (\/) B) implies
   for i be set st i in I holds  A.i,B.i are_c=-comparable
proof
  assume
A1: bool A (\/) bool B = bool (A (\/) B);
  let i be set such that
A2: i in I;
  bool (A.i \/ B.i) = (bool A (\/) bool B).i by A1,A2,Lm2
    .= (bool A).i \/ (bool B).i by A2,PBOOLE:def 4
    .= (bool A).i \/ (bool (B.i)) by A2,Def1
    .= (bool (A.i)) \/ (bool (B.i)) by A2,Def1;
  hence thesis by ZFMISC_1:70;
end;
