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

theorem Th26:
  Rnk A <= Rnk (A\/B) & Rnk (A \/ {e}) <= Rnk A + 1
proof
A1: card {e} = 1 by CARD_1:30;
  thus Rnk A <= Rnk (A\/B) by Th24,XBOOLE_1:7;
A2: Rnk(A\/{e}) + Rnk(A/\{e}) <= Rnk A + Rnk {e} by Th25;
  Rnk {e} <= card {e} by Th20;
  then Rnk A + Rnk {e} <= Rnk A + 1 by A1,XREAL_1:6;
  then
A3: Rnk(A\/{e}) + Rnk(A/\{e}) <= Rnk A + 1 by A2,XXREAL_0:2;
  Rnk(A\/{e}) <= Rnk(A\/{e}) + Rnk(A/\{e}) by NAT_1:11;
  hence thesis by A3,XXREAL_0:2;
end;
