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 Th28:
  Intersection(F|X9,Ch,y) c= Intersection(F,Ch,y)
proof
  let z be object such that
A1: z in Intersection(F|X9,Ch,y);
A2: now
A3: Ch"{y} c= dom (F|X9) by A1,Th19;
    let x such that
A4: x in dom Ch and
A5: Ch.x=y;
    Ch.x in {y} by A5,TARSKI:def 1;
    then
A6: x in Ch"{y} by A4,FUNCT_1:def 7;
    z in F|X9.x by A1,A4,A5,Def2;
    hence z in F.x by A6,A3,FUNCT_1:47;
  end;
  union rng (F|X9) c= union rng F & z in union rng (F|X9) by A1,RELAT_1:70
,ZFMISC_1:77;
  hence thesis by A2,Def2;
end;
