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

theorem
  A is independent & B is cycle & C is cycle & B c= A\/{e} & C c= A\/{e}
  implies B = C
proof
  assume that
A1: A is independent and
A2: B is cycle and
A3: C is cycle and
A4: B c= A\/{e} and
A5: C c= A\/{e};
  not C c= A by A1,Th3,A3;
  then consider c being object such that
A6: c in C and
A7: c nin A;
  c in {e} by A5,A6,A7,XBOOLE_0:def 3;
  then
A8: c = e by TARSKI:def 1;
  not B c= A by A1,Th3,A2;
  then consider b being object such that
A9: b in B and
A10: b nin A;
  assume
A11: B <> C;
  b in {e} by A4,A9,A10,XBOOLE_0:def 3;
  then b = e by TARSKI:def 1;
  then e in B/\C by A9,A6,A8,XBOOLE_0:def 4;
  then consider D being Subset of M such that
A12: D is cycle and
A13: D c= (B \/ C) \ {e} by A2,A3,A11,Th43;
  D c= A
  proof
    let x be object;
    assume
A14: x in D;
    then x in B\/C by A13,XBOOLE_0:def 5;
    then
A15: x in B or x in C by XBOOLE_0:def 3;
    x nin {e} by A13,A14,XBOOLE_0:def 5;
    hence thesis by A4,A5,A15,XBOOLE_0:def 3;
  end;
  then D is independent by A1,Th3;
  hence thesis by A12;
end;
