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
  y in rng Ch implies Intersection(Fy,Ch,y) is finite
proof
  assume y in rng Ch;
  then consider x being object such that
A1: x in dom Ch and
A2: Ch.x=y by FUNCT_1:def 3;
  Ch.x in {y} by A2,TARSKI:def 1;
  then x in Ch"{y} by A1,FUNCT_1:def 7;
  then Intersection(Fy,Ch,y) c= Fy.x by Th30;
  hence thesis;
end;
