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 Th27:
  Ch"{y}=(Ch|X9)"{y} implies Intersection(F,Ch,y)=Intersection(F, Ch|X9,y)
proof
  assume
A1: Ch"{y}=(Ch|X9)"{y};
A2: Intersection(F,Ch|X9,y) c=Intersection(F,Ch,y)
  proof
    let z be object such that
A3: z in Intersection(F,Ch|X9,y);
    now
      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|X9)"{y} by A1,A4,FUNCT_1:def 7;
      then (Ch|X9).x in {y} by FUNCT_1:def 7;
      then
A7:   (Ch|X9).x = y by TARSKI:def 1;
      x in dom (Ch|X9) by A6,FUNCT_1:def 7;
      hence z in F.x by A3,A7,Def2;
    end;
    hence thesis by A3,Def2;
  end;
  Intersection(F,Ch,y) c= Intersection(F,Ch|X9,y) by Th26;
  hence thesis by A2;
end;
