
theorem Th5: :: Sinfset:
for X being infinite set, n being Nat
 ex Y being finite Subset of X st card Y > n
proof :: DICKSON:3
 let X be infinite set, n be Nat;
 consider f being sequence of X such that
A1: f is one-to-one by DICKSON:3;
   set Sn = Seg (n+1);
   reconsider ff = f|Sn as Function of Sn, X by FUNCT_2:32;
   ff is one-to-one by A1,FUNCT_1:52;
   then card ff = card rng ff by PRE_POLY:19;
   then A2: card dom ff = card rng ff by CARD_1:62;
   reconsider Y = rng ff as finite Subset of X by RELAT_1:def 19;
   take Y;
   dom ff = Sn by FUNCT_2:def 1;
   then card dom ff = n+1 by FINSEQ_1:57;
   hence card Y > n by A2,NAT_1:13;
end;
