
theorem Th16:
  for M being non empty non void subset-closed finite-degree
  SubsetFamilyStr holds M is Matroid iff for C being Subset of M, A,B being
  independent Subset of M st A is_maximal_independent_in C & B
  is_maximal_independent_in C holds card A = card B
proof
  let M be non empty non void subset-closed finite-degree SubsetFamilyStr;
  hereby
    assume
A1: M is Matroid;
    let C be Subset of M;
A2: now
      let A,B be independent Subset of M such that
A3:   A is_maximal_independent_in C and
A4:   B is_maximal_independent_in C and
A5:   card A < card B;
A6:   A c= C by A3;
      (card A)+1 <= card B by A5,NAT_1:13;
      then Segm((card A)+1) c= Segm card B by NAT_1:39;
      then consider D being set such that
A7:   D c= B and
A8:   card D = (card A)+1 by CARD_FIL:36;
      reconsider D as finite Subset of M by A7,XBOOLE_1:1;
      D is independent by A7,Th3;
      then consider e being Element of M such that
A9:   e in D \ A and
A10:  A \/ {e} is independent by A1,A8,Th4;
      D \ A c= D by XBOOLE_1:36;
      then
A11:  D \ A c= B by A7;
      B c= C by A4;
      then D \ A c= C by A11;
      then {e} c= C by A9,ZFMISC_1:31;
      then
A12:  A \/ {e} c= C by A6,XBOOLE_1:8;
      A c= A \/ {e} by XBOOLE_1:7;
      then A \/ {e} = A by A3,A10,A12;
      then {e} c= A by XBOOLE_1:7;
      then e in A by ZFMISC_1:31;
      hence contradiction by A9,XBOOLE_0:def 5;
    end;
    let A,B be independent Subset of M such that
A13: A is_maximal_independent_in C and
A14: B is_maximal_independent_in C;
    card A < card B or card B < card A or card A = card B by XXREAL_0:1;
    hence card A = card B by A2,A13,A14;
  end;
  assume
A15: for C being Subset of M, A,B being independent Subset of M st A
is_maximal_independent_in C & B is_maximal_independent_in C holds card A = card
  B;
  M is with_exchange_property
  proof
    let A,B be finite Subset of M;
    reconsider C = A \/ B as Subset of M;
    assume that
A16: A in the_family_of M and
A17: B in the_family_of M and
A18: card B = (card A)+1;
    B is independent by A17;
    then consider B9 being independent Subset of M such that
A19: B c= B9 and
A20: B9 is_maximal_independent_in C by Th14,XBOOLE_1:7;
A21: card B <= card B9 by A19,NAT_1:43;
    assume
A22: for e be Element of M st e in B \ A holds not A \/ {e} in the_family_of M;
    reconsider A as independent Subset of M by A16,Def2;
    A is_maximal_independent_in C
    proof
      thus A in the_family_of M by A16;
      thus A c= C by XBOOLE_1:7;
      let D be Subset of M;
      assume that
A23:  D is independent and
A24:  D c= C and
A25:  A c= D;
      assume not (A c= D & D c= A);
      then consider e being object such that
A26:  e in D and
A27:  not e in A by A25;
      reconsider e as Element of M by A26;
      e in B by A24,A26,A27,XBOOLE_0:def 3;
      then e in B \ A by A27,XBOOLE_0:def 5;
      then not A \/ {e} in the_family_of M by A22;
      then
A28:  A \/ {e} is not independent;
      {e} c= D by A26,ZFMISC_1:31;
      then A \/ {e} c= D by A25,XBOOLE_1:8;
      hence contradiction by A23,A28,Th3;
    end;
    then card A = card B9 by A15,A20;
    hence contradiction by A18,A21,NAT_1:13;
  end;
  hence thesis;
end;
