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

theorem
  Rnk (A\/{e}) = Rnk (A\/{f}) & Rnk (A\/{f}) = Rnk A implies Rnk (A \/ {
  e,f}) = Rnk A
proof
  assume that
A1: Rnk (A\/{e}) = Rnk (A\/{f}) and
A2: Rnk (A\/{f}) = Rnk A;
  consider C being independent Subset of M such that
A3: C c= A and
A4: card C = Rnk A by Th18;
A5: C is_maximal_independent_in A by A3,A4,Th19;
  A c= A\/{f} by XBOOLE_1:7;
  then C c= A\/{f} by A3;
  then
A6: C is_maximal_independent_in A\/{f} by A4,A2,Th19;
  A c= A\/{e} by XBOOLE_1:7;
  then C c= A\/{e} by A3;
  then
A7: C is_maximal_independent_in A\/{e} by A4,A1,A2,Th19;
  A c= A\/{e,f} by XBOOLE_1:7;
  then C c= A\/{e,f} by A3;
  then consider C9 being independent Subset of M such that
A8: C c= C9 and
A9: C9 is_maximal_independent_in A\/{e,f} by Th14;
A10: C9 c= A\/{e,f} by A9;
  now
    assume C9 <> C;
    then consider x being object such that
A11: not (x in C9 iff x in C) by TARSKI:2;
    {x} c= C9 by A8,A11,ZFMISC_1:31;
    then C\/{x} c= C9 by A8,XBOOLE_1:8;
    then reconsider Cx = C\/{x} as independent Subset of M by Th3,XBOOLE_1:1;
    now
      assume x in A;
      then {x} c= A by ZFMISC_1:31;
      then
A12:  Cx c= A by A3,XBOOLE_1:8;
      C c= Cx by XBOOLE_1:7;
      then C = Cx by A5,A12;
      then {x} c= C by XBOOLE_1:7;
      hence contradiction by A8,A11,ZFMISC_1:31;
    end;
    then x in {e,f} by A8,A10,A11,XBOOLE_0:def 3;
    then x = e or x = f by TARSKI:def 2;
    then {x} c= A\/{e} & C c= A\/{e} or {x} c= A\/{f} & C c= A\/{f} by A3,
XBOOLE_1:10;
    then
A13: Cx c= A\/{e} or Cx c= A\/{f} by XBOOLE_1:8;
    C c= Cx by XBOOLE_1:7;
    then C = Cx by A7,A6,A13;
    then {x} c= C by XBOOLE_1:7;
    hence contradiction by A8,A11,ZFMISC_1:31;
  end;
  hence thesis by A4,A9,Th19;
end;
