reserve M for finite-degree Matroid,
  A,B,C for Subset of M,
  e,f for Element of M;

theorem Th42:
  (for B st B c= A holds B is not cycle) implies A is independent
proof
  assume
A1: for B st B c= A holds B is not cycle;
  consider C being independent Subset of M such that
A2: C c= A and
A3: card C = Rnk A by Th18;
  per cases;
  suppose
    A c= C;
    hence thesis by A2,XBOOLE_0:def 10;
  end;
  suppose
    A c/= C;
    then consider x being object such that
A4: x in A and
A5: x nin C;
    reconsider x as Element of M by A4;
A6: C c= C\/{x} by ZFMISC_1:137;
    defpred P[Nat] means ex B being independent Subset of M st card B = $1 & B
    c= C & B\/{x} is dependent;
A7: C\/{x} c= A by A2,A4,ZFMISC_1:137;
A8: ex n being Nat st P[n]
    proof
      take n = Rnk A, C;
      thus card C = n & C c= C by A3;
      assume
A9:   C\/{x} is independent;
      C is_maximal_independent_in A by A2,A3,Th19;
      then C = C\/{x} by A7,A6,A9;
      then {x} c= C by XBOOLE_1:7;
      hence contradiction by A5,ZFMISC_1:31;
    end;
    consider n being Nat such that
A10: P[n] & for k being Nat st P[k] holds n <= k from NAT_1:sch 5(A8);
    consider B being independent Subset of M such that
A11: card B = n and
A12: B c= C and
A13: B\/{x} is dependent by A10;
A14: x nin B by A5,A12;
A15: B\/{x} is cycle
    proof
      thus B\/{x} is dependent by A13;
      let e be Element of M;
      set Be = B\{e};
A16:  Be c= B by XBOOLE_1:36;
      assume
A17:  e in B\/{x};
      per cases by A17,ZFMISC_1:136;
      suppose
A18:    e in B;
A19:    e nin Be by ZFMISC_1:56;
        B = Be\/{e} by A18,ZFMISC_1:116;
        then
A20:    n = card Be+1 by A11,A19,CARD_2:41;
        assume
A21:    (B\/{x}) \ {e} is dependent;
        (B\/{x}) \ {e} = Be\/{x} by A14,A18,XBOOLE_1:87,ZFMISC_1:11;
        then n <= card Be by A10,A12,A16,A21,XBOOLE_1:1;
        hence contradiction by A20,NAT_1:13;
      end;
      suppose
        e = x;
        hence (B\/{x}) \ {e} is independent by A14,ZFMISC_1:117;
      end;
    end;
    B c= A by A2,A12;
    then B\/{x} c= A by A4,ZFMISC_1:137;
    hence thesis by A1,A15;
  end;
end;
