theorem Th41:
  lfp f is_a_fixpoint_of f & ex O st card O c= card the carrier of
  L & (f, O)+.Bottom L = lfp f
proof
  reconsider ze = Bottom L as Element of L;
A1: Bottom L [= f.ze by LATTICES:16;
  then consider O such that
A2: card O c= card the carrier of L and
A3: (f, O)+.Bottom L is_a_fixpoint_of f by Th30;
  card the carrier of L in nextcard the carrier of L by CARD_1:def 3;
  then card O in nextcard the carrier of L by A2,ORDINAL1:12;
  then O in nextcard the carrier of L by CARD_3:44;
  then
A4: O c= nextcard the carrier of L by ORDINAL1:def 2;
  hence lfp f is_a_fixpoint_of f by A1,A3,Th28;
  take O;
  thus card O c= card the carrier of L by A2;
  thus thesis by A1,A3,A4,Th28;
end;
