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
  x<>y & Fy=(x,y)-->(X,Y) implies Card_Intersection(Fy,1)=card X + card
  Y & Card_Intersection(Fy,2)=card (X/\Y)
proof
  assume that
A1: x<>y and
A2: Fy=(x,y)-->(X,Y);
  set P=(0,1)-->(x,y);
A3: dom P={0,1} & rng P={x,y} by FUNCT_4:62,64;
  card {x,y}=2 by A1,CARD_2:57;
  then reconsider P as Function of card {x,y},{x,y} by A3,CARD_1:50,FUNCT_2:1;
A4: card card {x,y}=card {x,y};
A5: P.0=x & Fy.x=X by A1,A2,FUNCT_4:63;
A6: P.1=y & Fy.y=Y by A2,FUNCT_4:63;
A7: dom Fy={x,y} by A2,FUNCT_4:62;
  rng P={x,y} by FUNCT_4:64;
  then P is onto by FUNCT_2:def 3;
  then P is one-to-one by A4,FINSEQ_4:63;
  then consider XFS be XFinSequence of NAT such that
A8: dom XFS=card {x,y} and
A9: for z st z in dom XFS holds XFS.z=card ((Fy*P).z) and
A10: Card_Intersection(Fy,1)=Sum XFS by A7,Th43;
  len XFS=2 by A1,A8,CARD_2:57;
  then
A11: XFS=<%XFS.0,XFS.1%> by AFINSQ_1:38;
A12: dom P={0,1} by FUNCT_4:62;
  then 1 in dom P by TARSKI:def 2;
  then
A13: (Fy*P).1=Fy.(P.1) by FUNCT_1:13;
  0 in {0} by TARSKI:def 1;
  then
A14: ({x,y}-->0)"{0}={x,y} by FUNCOP_1:14;
  Fy.x=X & Fy.y=Y by A1,A2,FUNCT_4:63;
  then
A15: Intersection(Fy,{x,y}-->0,0)=X/\Y by A14,Th35;
  0 in dom P by A12,TARSKI:def 2;
  then
A16: (Fy*P).0=Fy.(P.0) by FUNCT_1:13;
A17: dom XFS=2 by A1,A8,CARD_2:57;
  then 1 in dom XFS by CARD_1:50,TARSKI:def 2;
  then
A18: XFS.1=card Y by A9,A6,A13;
  0 in dom XFS by A17,CARD_1:50,TARSKI:def 2;
  then XFS.0=card X by A9,A5,A16;
  then addnat "**" XFS=addnat.(card X,card Y) by A11,A18,AFINSQ_2:38;
  then
A19: addnat "**" XFS=card X + card Y by BINOP_2:def 23;
  card {x,y}=2 & dom Fy={x,y} by A1,A2,CARD_2:57,FUNCT_4:62;
  hence thesis by A10,A19,A15,Th44,AFINSQ_2:51;
end;
