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 Th44:
 for x being object holds
  dom Fy=X implies Card_Intersection(Fy,card X)=card Intersection( Fy,X-->x,x)
proof let x be object;
  set Ch=Choose(X,card X,x,{x});
  consider P be Function of card Ch,Ch such that
A1: P is one-to-one by Lm2;
S: x in {x} by TARSKI:def 1;
  then
  reconsider x as set;
  not x in x; then
A2: x<>{x} by S;
  assume dom Fy=X;
  then consider XFS be XFinSequence of NAT such that
A3: dom XFS=dom P and
A4: ( for z,f st z in dom XFS & f=P.z holds XFS.z=card(Intersection(Fy,f
  ,x)))& Card_Intersection(Fy,card X)=Sum XFS by A1,A2,Def3;
A5: card Ch=1 by Th11;
  then consider ch being object such that
A6: Ch={ch} by CARD_2:42;
  x in {x} by TARSKI:def 1;
  then X\/{}=X & {x}<>x;
  then ({}-->{x})+*(X-->x) in Ch by Th16;
  then {}+*(X-->x) in Ch;
  then X-->x in Ch;
  then
A7: X-->x=ch by A6,TARSKI:def 1;
A8: Ch={} implies card Ch={};
  then
A9: dom P=card Ch by FUNCT_2:def 1;
  then 0 in dom P by A5,CARD_1:49,TARSKI:def 1;
  then P.0 in rng P by FUNCT_1:def 3;
  then
A10: P.0=ch by A6,TARSKI:def 1;
  len XFS=1 by A3,A8,A5,FUNCT_2:def 1;
  then XFS=<%XFS.0%> by AFINSQ_1:34;
  then addnat "**" XFS=XFS.0 by AFINSQ_2:37;
  then
A11: Sum XFS=XFS.0 by AFINSQ_2:51;
  0 in dom XFS by A3,A5,A9,CARD_1:49,TARSKI:def 1;
  hence thesis by A4,A11,A10,A7;
end;
