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 Th21:
  z in Intersection(F,Ch,y) & y in rng Ch implies ex x st x in dom
  Ch & Ch.x=y & z in F.x
proof
  assume that
A1: z in Intersection(F,Ch,y) and
A2: y in rng Ch;
  Ch"{y} <> {} by A2,FUNCT_1:72;
  then consider x being object such that
A3: x in Ch"{y} by XBOOLE_0:def 1;
  Ch.x in {y} by A3,FUNCT_1:def 7;
  then
A4: Ch.x=y by TARSKI:def 1;
A5: x in dom Ch by A3,FUNCT_1:def 7;
  x in dom Ch by A3,FUNCT_1:def 7;
  then z in F.x by A1,A4,Def2;
  hence thesis by A4,A5;
end;
