theorem Th29:
  for x st x in dom F holds card ((disjoin F).x) = F.x
proof
  let x;
  assume
A1: x in dom F;
  then reconsider M = F.x as Cardinal by Def1;
  M,[:M,{x}:] are_equipotent by CARD_1:69;
  then M = card [:M,{x}:] by CARD_1:def 2;
  hence thesis by A1,Def3;
end;
