reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;

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;
