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
  Intersection(F,Ch,y) is non empty implies for x1,x2 st x1 in Ch"{y} &
  x2 in Ch"{y} holds F.x1 meets F.x2
proof
  assume Intersection(F,Ch,y) is non empty;
  then consider z being object such that
A1: z in Intersection(F,Ch,y);
  let x1,x2 such that
A2: x1 in Ch"{y} and
A3: x2 in Ch"{y};
  Ch.x2 in {y} by A3,FUNCT_1:def 7;
  then
A4: Ch.x2=y by TARSKI:def 1;
  Ch.x1 in {y} by A2,FUNCT_1:def 7;
  then
A5: Ch.x1=y by TARSKI:def 1;
  x2 in dom Ch by A3,FUNCT_1:def 7;
  then
A6: z in F.x2 by A1,A4,Def2;
  x1 in dom Ch by A2,FUNCT_1:def 7;
  then z in F.x1 by A1,A5,Def2;
  hence thesis by A6,XBOOLE_0:3;
end;
