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

theorem Th25:
  Rnk (A\/B) + Rnk (A/\B) <= Rnk A + Rnk B
proof
  consider C being independent Subset of M such that
A1: C c= A/\B and
A2: card C = Rnk (A/\B) by Th18;
  A/\B c= A by XBOOLE_1:17;
  then C c= A by A1;
  then consider Ca being independent Subset of M such that
A3: C c= Ca and
A4: Ca is_maximal_independent_in A by Th14;
A5: Ca c= A by A4;
A6: Ca/\B c= C
  proof
    let x be object;
    assume
A7: x in Ca/\B;
    then
A8: x in Ca by XBOOLE_0:def 4;
    then {x} c= Ca by ZFMISC_1:31;
    then C\/{x} c= Ca by A3,XBOOLE_1:8;
    then reconsider Cx = C\/{x} as independent Subset of M by Th3,XBOOLE_1:1;
    x in B by A7,XBOOLE_0:def 4;
    then x in A/\B by A5,A8,XBOOLE_0:def 4;
    then {x} c= A/\B by ZFMISC_1:31;
    then
A9: Cx c= A/\B by A1,XBOOLE_1:8;
A10: C c= Cx by XBOOLE_1:7;
    C is_maximal_independent_in A/\B by A1,A2,Th19;
    then C = Cx by A9,A10;
    then {x} c= C by XBOOLE_1:7;
    hence thesis by ZFMISC_1:31;
  end;
  A/\B c= B by XBOOLE_1:17;
  then C c= B by A1;
  then C c= Ca /\B by A3,XBOOLE_1:19;
  then
A11: Ca/\B = C by A6;
  A c= A\/B by XBOOLE_1:7;
  then Ca c= A\/B by A5;
  then consider C9 being independent Subset of M such that
A12: Ca c= C9 and
A13: C9 is_maximal_independent_in A\/B by Th14;
A14: Ca/\(C9/\B) = Ca/\C9/\B by XBOOLE_1:16
    .= Ca/\B by A12,XBOOLE_1:28;
A15: C9 c= A\/B by A13;
A16: C9 = Ca \/ (C9/\B)
  proof
    thus C9 c= Ca \/ (C9/\B)
    proof
      let x be object;
      assume
A17:  x in C9;
      then {x} c= C9 by ZFMISC_1:31;
      then Ca\/{x} c= C9 by A12,XBOOLE_1:8;
      then reconsider Cax = Ca\/{x} as independent Subset of M by Th3,
XBOOLE_1:1;
A18:  now
        assume x in A;
        then {x} c= A by ZFMISC_1:31;
        then
A19:    Cax c= A by A5,XBOOLE_1:8;
        Ca c= Cax by XBOOLE_1:7;
        then Ca = Cax by A4,A19;
        then {x} c= Ca by XBOOLE_1:7;
        hence x in Ca by ZFMISC_1:31;
      end;
      x in B implies x in C9/\B by A17,XBOOLE_0:def 4;
      hence thesis by A15,A17,A18,XBOOLE_0:def 3;
    end;
    let x be object;
    assume x in Ca \/ (C9/\B);
    then x in Ca or x in C9/\B by XBOOLE_0:def 3;
    hence thesis by A12,XBOOLE_0:def 4;
  end;
  C9/\B c= B by XBOOLE_1:17;
  then consider Cb being independent Subset of M such that
A20: C9/\B c= Cb and
A21: Cb is_maximal_independent_in B by Th14;
  card Cb = Rnk B by A21,Th19;
  then
A22: card (C9/\B) <= Rnk B by A20,NAT_1:43;
A23: card C9 = Rnk (A\/B) by A13,Th19;
  card Ca = Rnk A by A4,Th19;
  then Rnk (A\/B) = Rnk A + card (C9/\B) - Rnk (A/\B) by A2,A23,A16,A14,A11,
CARD_2:45;
  hence thesis by A22,XREAL_1:6;
end;
