reserve i, j, k, l, m, n for Nat,
  a, b, c, t, u for object,
  X, Y, Z for set,
  D, D1, D2, Fml for non empty set;
reserve p, q, r, s for FinSequence;
 reserve R, R1, R2 for Rule;
 reserve A, A1, A2 for non empty set;
 reserve B, B1, B2 for set;
 reserve P, P1, P2 for Formula-sequence;
 reserve S, S1, S2 for Formula-finset;
 reserve C for Extension of B;
 reserve E for Extension of R;
 reserve P for non empty ProofSystem;
 reserve B, B1, B2 for Subset of P;
 reserve F for finite Subset of P;

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;
