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 Th23:
 for y being object holds
  F|Ch"{y}=(Ch"{y}-->union rng F)
    implies Intersection(F,Ch,y) = union rng F
proof let y be object;
  set ChF=Ch"{y}-->union rng F;
  assume
A1: F|Ch"{y}=ChF;
  union rng F c= Intersection(F,Ch,y)
  proof
    let z be object such that
A2: z in union rng F;
    now
      let x such that
A3:   x in dom Ch and
A4:   Ch.x=y;
      Ch.x in {y} by A4,TARSKI:def 1;
      then
A5:   x in Ch"{y} by A3,FUNCT_1:def 7;
      then dom ChF= Ch"{y} & ChF.x=union rng F by FUNCOP_1:7;
      hence z in F.x by A1,A2,A5,FUNCT_1:47;
    end;
    hence thesis by A2,Def2;
  end;
  hence thesis;
end;
