theorem
  X is finite implies nextcard X is finite
proof
  assume X is finite;
  then reconsider X as finite set;
  card X = card card X;
  then
A1: card(succ card X) = nextcard card X by Th42;
  nextcard card X = nextcard X by Def2,Th16;
  hence thesis by A1,Th38;
end;
