reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem
  for D be non empty set, F be XFinSequence of D st
  (for i st i in dom F holds F.i is finite) &
  (for i,j st i in dom F & j in dom F & i<>j holds F.i misses F.j)
  ex CardF be XFinSequence of NAT st dom CardF = dom F &
  (for i st i in dom CardF holds CardF.i=card (F.i)) &
  card union rng F = Sum(CardF) by Lm2;
