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

theorem Th29:
  A c= B & e is_dependent_on A implies e is_dependent_on B
proof
  assume that
A1: A c= B and
A2: Rnk (A \/ {e}) = Rnk A;
  consider Ca being independent Subset of M such that
A3: Ca c= A and
A4: card Ca = Rnk A by Th18;
A5: Ca c= B by A1,A3;
  B c= B\/{e} by XBOOLE_1:7;
  then Ca c= B\/{e} by A5;
  then consider E being independent Subset of M such that
A6: Ca c= E and
A7: E is_maximal_independent_in B\/{e} by Th14;
A8: now
    E c= B\/{e} by A7;
    then
A9: E = E/\(B\/{e}) by XBOOLE_1:28
      .= E/\B\/E/\{e} by XBOOLE_1:23;
    E/\{e} c= {e} by XBOOLE_1:17;
    then
A10: E/\{e} = {} & card {} = 0 or E/\{e} = {e} & card {e} = 1 by CARD_1:30
,ZFMISC_1:33;
    card (E/\B) <= Rnk B by Th17,XBOOLE_1:17;
    then
A11: card (E/\B)+1 <= Rnk B + 1 by XREAL_1:6;
    Ca c= A\/{e} by A3,XBOOLE_1:10;
    then
A12: Ca is_maximal_independent_in A\/{e} by A2,A4,Th19;
A13: Ca c= Ca\/{e} by XBOOLE_1:10;
    assume
A14: Rnk (B\/{e}) = Rnk B + 1;
    then card E = Rnk B + 1 by A7,Th19;
    then Rnk B + 1 <= card (E/\B) + card (E/\{e}) by A9,CARD_2:43;
    then card (E/\B)+1 <= card (E/\B) + card (E/\{e}) by A11,XXREAL_0:2;
    then e in E/\{e} by A10,TARSKI:def 1,XREAL_1:6;
    then e in E by XBOOLE_0:def 4;
    then {e} c= E by ZFMISC_1:31;
    then Ca\/{e} c= E by A6,XBOOLE_1:8;
    then
A15: Ca\/{e} is independent by Th3;
    Ca\/{e} c= A\/{e} by A3,XBOOLE_1:9;
    then Ca = Ca\/{e} by A13,A15,A12;
    then {e} c= Ca by XBOOLE_1:7;
    then B = B\/{e} by A5,XBOOLE_1:1,12;
    hence contradiction by A14;
  end;
A16: Rnk (B\/{e}) <= Rnk B + 1 by Th26;
  Rnk B <= Rnk (B\/{e}) by Th26;
  hence Rnk (B\/{e}) = Rnk B by A16,A8,NAT_1:9;
end;
