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 Th45:
 for x being object holds
  Fy=x.-->X implies Card_Intersection(Fy,1)=card X
proof let x be object;
  assume
A1: Fy=x.-->X;
  then
A2: dom Fy={x};
A3: x in {x} by TARSKI:def 1;
  then
A4: (x.-->x)"{x}={x} by FUNCOP_1:14;
  Fy.x=X by A1,A3,FUNCOP_1:7;
  then 1=card {x} & Intersection(Fy,x.-->x,x)=X by A4,Th34,CARD_1:30;
  hence thesis by A2,Th44;
end;
