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;

theorem
  for F st F is non empty holds y in Intersection(F,(dom F-->x),x) iff
  for z st z in dom F holds y in F.z
proof
  let F such that
A1: F is non empty;
  set Ch=dom F-->x;
  thus y in Intersection(F,Ch,x) implies for z st z in dom F holds y in F.z
  proof
    assume
A2: y in Intersection(F,Ch,x);
    let z;
    assume z in dom F;
    then z in dom Ch & Ch.z=x by FUNCOP_1:7;
    hence thesis by A2,Def2;
  end;
  Ch"{x}=dom F by FUNCOP_1:15;
  then
A3: dom F/\Ch"{x}=dom F;
  assume for z st z in dom F holds y in F.z;
  then for z st z in dom Ch & Ch.z=x holds y in F.z;
  hence thesis by A1,A3,Th18;
end;
