reserve n,m,k for Nat,
  X,Y,Z for set,
  f for Function of X,Y,
  H for Subset of X;

theorem Th14:
  for F being Function of the_subsets_of_card(n,X),k st k<>0 & X is infinite
  ex H st H is infinite & F|the_subsets_of_card(n,H) is constant
proof
  let F be Function of the_subsets_of_card(n,X),k;
  assume that
A1: k<>0 and
A2: X is infinite;
  F in Funcs(the_subsets_of_card(n,X),k) by A1,FUNCT_2:8;
  then
A3: ex g1 be Function st F=g1 & dom g1=the_subsets_of_card(n,X) & rng g1 c=
  k by FUNCT_2:def 2;
  consider Y be set such that
A4: Y c= X and
A5: card Y = omega by A2,CARD_3:87;
  reconsider Y as non empty set by A5;
  Y,omega are_equipotent by A5,CARD_1:5,47;
  then consider f be Function such that
A6: f is one-to-one and
A7: dom f = omega and
A8: rng f = Y by WELLORD2:def 4;
  reconsider f as Function of omega,Y by A7,A8,FUNCT_2:1;
  not card Y c= card n by A5;
  then the_subsets_of_card(n,Y) is non empty by GROUP_10:1;
  then f||^n in Funcs(the_subsets_of_card(n,omega), the_subsets_of_card(n,Y))
  by FUNCT_2:8;
  then
A9: ex g2 be Function st f||^n=g2 & dom g2= the_subsets_of_card(n,omega) &
  rng g2 c= the_subsets_of_card(n,Y) by FUNCT_2:def 2;
  set F9 = F * (f||^n);
  the_subsets_of_card(n,Y) c= the_subsets_of_card(n,X) by A4,Lm1;
  then
A10: dom F9 = the_subsets_of_card(n,omega) by A3,A9,RELAT_1:27,XBOOLE_1:1;
A11: rng F9 c= rng F by RELAT_1:26;
  then
A12: rng F9 c= k by A3;
  reconsider F9 as Function of the_subsets_of_card(n,omega),k by A3,A10,A11,
FUNCT_2:2,XBOOLE_1:1;
  consider H9 be Subset of omega such that
A13: H9 is infinite and
A14: F9|the_subsets_of_card(n,H9) is constant by A1,Lm4,CARD_1:47;
A15: rng(F9|the_subsets_of_card(n,H9)) c= rng F9 by RELAT_1:70;
  set H = f .: H9;
  f .: H9 c= rng f by RELAT_1:111;
  then reconsider H as Subset of X by A4,A8,XBOOLE_1:1;
  take H;
  H9,f.:H9 are_equipotent by A6,A7,CARD_1:33;
  hence H is infinite by A13,CARD_1:38;
  dom(F9|the_subsets_of_card(n,H9)) = the_subsets_of_card(n,H9) by A10,Lm1,
RELAT_1:62;
  then
  F9|the_subsets_of_card(n,H9) is Function of the_subsets_of_card(n,H9),k
  by A12,A15,FUNCT_2:2,XBOOLE_1:1;
  then consider y be Element of k such that
A16: rng(F9|the_subsets_of_card(n,H9)) = {y} by A1,A13,A14,FUNCT_2:111;
A17: not card omega c= card n;
A18: ex y being Element of k st rng(F|the_subsets_of_card(n,H)) = {y}
  proof
    take y;
    thus rng(F|the_subsets_of_card(n,H)) = F.:the_subsets_of_card(n,H) by
RELAT_1:115
      .= F.:((f||^n) .: the_subsets_of_card(n,H9)) by A6,A17,Th1
      .= F9 .: the_subsets_of_card(n,H9) by RELAT_1:126
      .= {y} by A16,RELAT_1:115;
  end;
  thus thesis by A18;
end;
