reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;
reserve Fy for finite-yielding Function;

theorem Th42:
  dom Fy=X & k > card X implies Card_Intersection(Fy,k)=0
proof
  assume that
A1: dom Fy=X and
A2: k > card X;
  set Ch=Choose(X,k,0,1);
  consider P be Function of card Ch,Ch such that
A3: P is one-to-one by Lm2;
  consider XFS be XFinSequence of NAT such that
A4: dom XFS=dom P and
  for z,f st z in dom XFS & f=P.z holds XFS.z=card(Intersection(Fy,f,0)) and
A5: Card_Intersection(Fy,k)=Sum XFS by A1,A3,Def3;
  Ch is empty by A2,Th9;
  then XFS=0 by A4;
  hence thesis by A5;
end;
