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 Th19:
  Intersection(F,Ch,y) is non empty implies Ch"{y} c= dom F
proof
  assume Intersection(F,Ch,y) is non empty;
  then consider z being object such that
A1: z in Intersection(F,Ch,y);
  assume not Ch"{y} c= dom F;
  then consider x being object such that
A2: x in Ch"{y} and
A3: not x in dom F;
  Ch.x in {y} by A2,FUNCT_1:def 7;
  then
A4: Ch.x=y by TARSKI:def 1;
  x in dom Ch by A2,FUNCT_1:def 7;
  then z in F.x by A1,A4,Def2;
  hence thesis by A3,FUNCT_1:def 2;
end;
