reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem Th1:
  for a, b being set st b in SubstitutionSet (V, C) & a in b holds
  a is finite Function
proof
  let a, b be set;
  assume that
A1: b in SubstitutionSet (V, C) and
A2: a in b;
  b in { A where A is Element of Fin PFuncs (V,C) : ( for u being set st u
in A holds u is finite ) & for s1, t being Element of PFuncs (V, C) holds ( s1
  in A & t in A & s1 c= t implies s1 = t ) } by A1,SUBSTLAT:def 1;
  then
  ex A being Element of Fin PFuncs (V,C) st A = b &( for u being set st u
in A holds u is finite)& for s, t being Element of PFuncs (V, C ) holds ( s in
  A & t in A & s c= t implies s = t );
  hence thesis by A1,A2;
end;
