
theorem Th29:
  for k being Element of NAT, V, X being set, a being Element of
  SubstPoset (V, {k}) st X in a holds X is finite Subset of [:V, {k}:]
proof
  let k be Element of NAT;
  let V, X be set;
  let a be Element of SubstPoset (V, {k});
  assume
A1: X in a;
A2: PFuncs (V, {k}) c= bool [:V, {k}:] by PRE_POLY:16;
A3: SubstitutionSet (V, {k}) = the carrier of SubstPoset (V, {k}) by
SUBSTLAT:def 4;
  then a in SubstitutionSet (V, {k});
  then a c= PFuncs (V, {k}) by FINSUB_1:def 5;
  then X in PFuncs (V, {k}) by A1;
  hence thesis by A3,A1,A2,HEYTING2:1;
end;
