reserve a,b,c for set;

theorem
  for X being infinite set holds card Fin X = card X
proof
  deffunc f(set) = proj2 $1;
  let X be infinite set;
  set FX = Fin X;
  consider f being Function such that
A1: dom f = X* & for a st a in X* holds f.a = f(a) from FUNCT_1:sch 5;
  FX c= rng f
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
    assume
A2: a in FX;
    then aa is finite by FINSUB_1:def 5;
    then consider p being FinSequence such that
A3: a = rng p by FINSEQ_1:52;
    aa c= X by A2,FINSUB_1:def 5;
    then p is FinSequence of X by A3,FINSEQ_1:def 4;
    then
A4: p in X* by FINSEQ_1:def 11;
    then f.p in rng f by A1,FUNCT_1:def 3;
    hence thesis by A1,A3,A4;
  end;
  then card FX c= card (X*) by A1,CARD_1:12;
  hence card FX c= card X by CARD_4:24;
  set SX = SmallestPartition X;
  SX c= FX
  proof
    let a be object;
    assume a in SX;
    then a in the set of all {x} where x is Element of X by EQREL_1:37;
    then ex x being Element of X st a = {x};
    hence thesis by FINSUB_1:def 5;
  end;
  then card SX c= card FX by CARD_1:11;
  hence thesis by Th12;
end;
