reserve V, C for set;

theorem Th2:
  { {} } in SubstitutionSet (V, C)
proof
A1: for s, t being Element of PFuncs (V,C) holds ( s in { {} } & t in { {} }
  & s c= t implies s = t )
  proof
    let s, t be Element of PFuncs (V,C);
    assume that
A2: s in { {} } and
A3: t in { {} } and
    s c= t;
    s = {} by A2,TARSKI:def 1;
    hence thesis by A3,TARSKI:def 1;
  end;
  {} is PartFunc of V,C by RELSET_1:12;
  then {} in PFuncs (V,C) by PARTFUN1:45;
  then { {} } c= PFuncs (V,C) by ZFMISC_1:31;
  then
A4: { {} } in Fin PFuncs (V,C) by FINSUB_1:def 5;
  for u being set st u in { {} } holds u is finite;
  hence thesis by A4,A1;
end;
