
theorem Th35:
  for k being Element of NAT, a being Element of SubstPoset (NAT,
  {k}), a9 being Element of Fin PFuncs (NAT, {k}), B being finite non empty
Subset of NAT st B = Involved a9 & a9 = a holds for X being set st X in a for l
  being Element of NAT st l > (max B) + 1 holds not [l,k] in X
proof
  let k be Element of NAT, a be Element of SubstPoset (NAT, {k}), a9 be
  Element of Fin PFuncs (NAT, {k}), B be finite non empty Subset of NAT;
  assume that
A1: B = Involved a9 and
A2: a9 = a;
  let X be set;
  assume
A3: X in a;
  SubstitutionSet (NAT, {k}) = the carrier of SubstPoset (NAT, {k}) by
SUBSTLAT:def 4;
  then reconsider X9 = X as finite Function by A3,HEYTING2:1;
  let l be Element of NAT;
  assume
A4: l > max B + 1;
  assume [l,k] in X;
  then l in dom X9 by XTUPLE_0:def 12;
  then l in Involved a9 by A2,A3,HEYTING2:def 1;
  then max B + 1 >= max B & max B >= l by A1,NAT_1:11,XXREAL_2:def 8;
  hence thesis by A4,XXREAL_0:2;
end;
