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

theorem Th43:
  A is cycle & B is cycle & A <> B & e in A /\ B implies ex C st C
  is cycle & C c= (A \/ B) \ {e}
proof
  assume that
A1: A is cycle and
A2: B is cycle and
A3: A <> B and
A4: e in A /\ B and
A5: for C st C is cycle holds C c/= (A \/ B) \ {e};
A6: e in A by A4,XBOOLE_0:def 4;
  A/\B c= B by XBOOLE_1:17;
  then A c/= A/\B by A1,A2,A3,Th41,XBOOLE_1:1;
  then consider a being object such that
A7: a in A and
A8: a nin A/\B;
  reconsider a as Element of M by A7;
  {a} misses A/\B by A8,ZFMISC_1:50;
  then
A9: A/\B c= A\{a} by XBOOLE_1:17,86;
  reconsider A9 = A, B9 = B as finite Subset of M by A1,A2;
  Rnk(A\/B)+Rnk(A/\B) <= Rnk A + Rnk B by Th25;
  then
A10: Rnk(A\/B)+Rnk(A/\B)+1 <= Rnk A + Rnk B+1 by XREAL_1:6;
  A\{a} is independent by A1,A7;
  then A/\B is independent by A9,Th3;
  then
A11: card (A9/\B9) = Rnk (A/\B) by Th21;
  for C st C c= (A \/ B) \ {e} holds C is not cycle by A5;
  then reconsider C = (A\/B)\{e} as independent Subset of M by Th42;
A12: e in {e} by TARSKI:def 1;
  then
A13: e nin C by XBOOLE_0:def 5;
A14: e in B by A4,XBOOLE_0:def 4;
  then reconsider
  Ae = A\{e}, Be = B\{e} as independent Subset of M by A1,A2,A6;
A15: e nin Be by A12,XBOOLE_0:def 5;
  B = Be\/{e} by A14,ZFMISC_1:116;
  then
A16: card B9 = card Be+1 by A15,CARD_2:41;
  then
A17: Rnk B + 1 = card Be+1 by A2,Th39;
A18: e nin Ae by A12,XBOOLE_0:def 5;
  A = Ae\/{e} by A6,ZFMISC_1:116;
  then
A19: card A9 = card Ae+1 by A18,CARD_2:41;
  then Rnk A + 1 = card Ae+1 by A1,Th39;
  then card(A9\/B9)+card(A9/\B9) = Rnk A+1 + (Rnk B+1) by A19,A16,A17,
HALLMAR1:1
    .= Rnk A + Rnk B+1+1;
  then
A20: Rnk(A\/B)+Rnk(A/\B)+1+1 <= card(A9\/B9)+card(A9/\B9) by A10,XREAL_1:6;
  e in A\/B by A6,XBOOLE_0:def 3;
  then
A21: C\/{e} = A9\/B9 by ZFMISC_1:116;
  C is_maximal_independent_in A\/B
  proof
    thus C is independent & C c= A\/B by XBOOLE_1:36;
    let D be Subset of M;
A22: A c= A\/B by XBOOLE_1:7;
    A is dependent by A1;
    then A\/B is dependent by A22,Th3;
    hence thesis by A21,ZFMISC_1:138;
  end;
  then Rnk(A\/B)+1 = card C+1 by Th19
    .= card(A9\/B9) by A13,A21,CARD_2:41;
  hence contradiction by A20,A11,NAT_1:13;
end;
