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 Th29:
  y in rng Ch & Ch"{y} c= X9 implies Intersection(F|X9,Ch,y) =
  Intersection(F,Ch,y)
proof
  assume that
A1: y in rng Ch and
A2: Ch"{y} c= X9;
A3: Intersection(F,Ch,y) c=Intersection(F|X9,Ch,y)
  proof
    let z be object such that
A4: z in Intersection(F,Ch,y);
A5: now
      let x such that
A6:   x in dom Ch and
A7:   Ch.x=y;
      Ch.x in {y} by A7,TARSKI:def 1;
      then
A8:   x in Ch"{y} by A6,FUNCT_1:def 7;
      z in F.x by A4,A6,A7,Def2;
      then x in dom F by FUNCT_1:def 2;
      then x in dom F/\X9 by A2,A8,XBOOLE_0:def 4;
      then
A9:   x in dom (F|X9) by RELAT_1:61;
      z in F.x by A4,A6,A7,Def2;
      hence z in F|X9.x by A9,FUNCT_1:47;
    end;
    consider x such that
A10: x in dom Ch and
A11: Ch.x=y and
A12: z in F.x by A1,A4,Th21;
    Ch.x in {y} by A11,TARSKI:def 1;
    then
A13: x in Ch"{y} by A10,FUNCT_1:def 7;
    x in dom F by A12,FUNCT_1:def 2;
    then x in dom F/\X9 by A2,A13,XBOOLE_0:def 4;
    then x in dom (F|X9) by RELAT_1:61;
    then
A14: F|X9.x in rng (F|X9) by FUNCT_1:def 3;
    z in F|X9.x by A5,A10,A11;
    then z in union rng (F|X9) by A14,TARSKI:def 4;
    hence thesis by A5,Def2;
  end;
  Intersection(F|X9,Ch,y) c=Intersection(F,Ch,y) by Th28;
  hence thesis by A3;
end;
