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

theorem Th34:
  e is_dependent_on Span A implies e is_dependent_on A
proof
  assume
A1: Rnk ((Span A)\/{e}) = Rnk Span A;
A2: Rnk A = Rnk Span A by Th33;
  consider Ca being independent Subset of M such that
A3: Ca c= A and
A4: card Ca = Rnk A by Th18;
A5: Rnk A = Rnk Ca by A4,Th21;
A6: Rnk Ca <= Rnk(A\/{e}) by A3,Th24,XBOOLE_1:10;
  A c= Span A by Th31;
  then Rnk(A\/{e}) <= Rnk A by A1,A2,Th24,XBOOLE_1:9;
  hence Rnk (A\/{e}) = Rnk A by A5,A6,XXREAL_0:1;
end;
