theorem
  for P, B st P is consistent non paraconsistent & B is consistent
      ex B1 st B c= B1 & B1 is maximally-consistent
proof
  let P, B;
  assume A1: P is consistent non paraconsistent;
  assume A2: B is consistent;
  consider S being finite Subset of P such that A3: S is inconsistent by A1;
  A5: for B1 holds B1 is consistent iff B1 is S-omitting by A3;
  then consider B1 such that
    A10: B c= B1 and
    A11: B1 is S-maximally-omitting by A2, Th61;
  take B1;
  thus B c= B1 by A10;
  thus thesis by A5, A11;
end;
