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
  for x st x in dom F holds card (F.x) = F.x
proof
  let x;
  assume x in dom F;
  then reconsider M = F.x as Cardinal by Def1;
  card M = M;
  hence thesis;
end;
