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

theorem Th30:
  e in Span A iff Rnk (A \/ {e}) = Rnk A
proof
  hereby
    assume e in Span A;
    then ex x being Element of M st e = x & x is_dependent_on A;
    hence Rnk (A \/ {e}) = Rnk A;
  end;
  assume Rnk (A \/ {e}) = Rnk A;
  then e is_dependent_on A;
  hence thesis;
end;
