
theorem Th14:
  for M being non void finite-degree SubsetFamilyStr for C,A being
Subset of M st A c= C & A is independent ex B being independent Subset of M st
  A c= B & B is_maximal_independent_in C
proof
  let M be non void finite-degree SubsetFamilyStr;
  let C,A0 be Subset of M;
  assume that
A1: A0 c= C and
A2: A0 is independent;
  reconsider AA = A0 as independent Subset of M by A2;
  defpred P[Nat] means for A being finite Subset of M st A0 c= A & A c= C & A
  is independent holds card A <= $1;
  consider n being Nat such that
A3: for A being finite Subset of M st A is independent holds card A <= n
  by Def6;
  reconsider n as Element of NAT by ORDINAL1:def 12;
  P[n] by A3;
  then
A4: ex n being Nat st P[n];
  consider n0 being Nat such that
A5: P[n0] & for m being Nat st P[m] holds n0 <= m from NAT_1:sch 5(A4);
  now
    0 <= card AA by NAT_1:2;
    then
A6: (card AA)+1 >= 0+1 by XREAL_1:6;
    assume
A7: for A being independent Subset of M st A0 c= A & A c= C holds card A < n0;
    then card AA < n0 by A1;
    then (card AA)+1 <= n0 by NAT_1:13;
    then consider n being Nat such that
A8: n0 = 1+n by A6,NAT_1:10,XXREAL_0:2;
    reconsider n as Element of NAT by ORDINAL1:def 12;
    P[n]
    proof
      let A be finite Subset of M;
      assume that
A9:   A0 c= A and
A10:  A c= C and
A11:  A is independent;
      card A < n+1 by A7,A8,A9,A10,A11;
      hence thesis by NAT_1:13;
    end;
    then n+1 <= n by A5,A8;
    hence contradiction by NAT_1:13;
  end;
  then consider A being independent Subset of M such that
A12: A0 c= A and
A13: A c= C and
A14: card A >= n0;
A15: card A <= n0 by A5,A12,A13;
  take A;
  thus A0 c= A & A is independent & A c= C by A12,A13;
  let B be Subset of M;
  assume that
A16: B is independent and
A17: B c= C and
A18: A c= B;
  reconsider B9 = B as independent Subset of M by A16;
  card A <= card B9 by A18,NAT_1:43;
  then
A19: n0 <= card B9 by A14,XXREAL_0:2;
  A0 c= B by A12,A18;
  then card B9 <= n0 by A5,A17;
  then card B9 = n0 by A19,XXREAL_0:1;
  hence thesis by A14,A18,A15,CARD_2:102,XXREAL_0:1;
end;
