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 Th18:
  for F,Ch st dom F/\Ch"{x} is non empty holds y in Intersection(F
  ,Ch,x) iff for z st z in dom Ch & Ch.z=x holds y in F.z
proof
  let F,Ch such that
A1: dom F/\Ch"{x} is non empty;
  thus y in Intersection(F,Ch,x) implies for z st z in dom Ch & Ch.z=x holds y
  in F.z by Def2;
  thus (for z st z in dom Ch & Ch.z=x holds y in F.z) implies y in
  Intersection(F,Ch,x)
  proof
    consider z being object such that
A2: z in dom F/\Ch"{x} by A1;
A3: z in Ch"{x} by A2,XBOOLE_0:def 4;
    then Ch.z in {x} by FUNCT_1:def 7;
    then
A4: Ch.z=x by TARSKI:def 1;
    z in dom F by A2,XBOOLE_0:def 4;
    then
A5: F.z in rng F by FUNCT_1:def 3;
    assume
A6: for z st z in dom Ch & Ch.z=x holds y in F.z;
    z in dom Ch by A3,FUNCT_1:def 7;
    then y in F.z by A6,A4;
    then y in union rng F by A5,TARSKI:def 4;
    hence thesis by A6,Def2;
  end;
end;
