reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

theorem
  for F be Function st dom F is infinite & rng F is finite ex x st
  x in rng F & F"{x} is infinite
proof
  let F be Function such that
A1: dom F is infinite and
A2: rng F is finite;
  deffunc f(object)=F"{$1};
  consider g be Function such that
A3: dom g = rng F &
for x being object st x in rng F holds g.x = f(x) from FUNCT_1:
  sch 3;
A4: dom F c= Union g
  proof
    let x be object such that
A5: x in dom F;
A6: F.x in rng F by A5,FUNCT_1:def 3;
    then
A7: g.(F.x) in rng g by A3,FUNCT_1:def 3;
    F.x in {F.x} by TARSKI:def 1;
    then
A8: x in F"{F.x} by A5,FUNCT_1:def 7;
    g.(F.x)=F"{F.x} by A3,A6;
    then x in union rng g by A8,A7,TARSKI:def 4;
    hence thesis;
  end;
  assume
A9: for x st x in rng F holds F"{x} is finite;
  now
    let x such that
A10: x in dom g;
    g.x=F"{x} by A3,A10;
    hence g.x is finite by A9,A3,A10;
  end;
  then Union g is finite by A2,A3,Th87;
  hence thesis by A1,A4;
end;
