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

theorem
  f nin Span A & f in Span (A \/ {e}) implies e in Span (A \/ {f})
proof
  assume that
A1: f nin Span A and
A2: f in Span (A \/ {e});
A3: Rnk A <= Rnk (A\/{f}) by Th26;
A4: Rnk (A\/{f}) <= Rnk A + 1 by Th26;
  Rnk A <> Rnk (A\/{f}) by A1,Th30;
  then
A5: Rnk (A\/{f}) = Rnk A + 1 by A3,A4,NAT_1:9;
A6: A\/{f}\/{e} = A\/({f}\/{e}) by XBOOLE_1:4;
A7: Rnk (A\/{e}) <= Rnk A + 1 by Th26;
A8: A\/{e}\/{f} = A\/({e}\/{f}) by XBOOLE_1:4;
A9: Rnk(A\/{e}\/{f}) = Rnk(A\/{e}) by A2,Th30;
  then Rnk(A\/{f}) <= Rnk(A\/{e}) by A6,A8,Th26;
  then Rnk (A\/{f}) = Rnk (A\/{f}\/{e}) by A9,A5,A6,A8,A7,XXREAL_0:1;
  hence thesis by Th30;
end;
