
theorem Th9:
  for V, V9, C, C9 being set st V c= V9 & C c= C9 holds
  SubstitutionSet (V, C) c= SubstitutionSet (V9, C9)
proof
  let V, V9, C, C9 be set;
  assume V c= V9 & C c= C9;
  then
A1: PFuncs (V,C) c= PFuncs (V9,C9) by PARTFUN1:50;
  let x be object;
  assume x in SubstitutionSet (V, C);
  then
  x in { A where A is Element of Fin PFuncs (V,C) : ( 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 ) } by SUBSTLAT:def 1;
  then consider B being Element of Fin PFuncs (V,C) such that
A2: B = x & for u being set st u in B holds u is finite and
A3: for s, t being Element of PFuncs (V, C) holds ( s in B & t in B & s
  c= t implies s = t );
A4: B in Fin PFuncs (V,C) & Fin PFuncs (V,C) c= Fin PFuncs (V9,C9) by A1,
FINSUB_1:10;
A5: B c= PFuncs (V,C) by FINSUB_1:def 5;
  reconsider B as Element of Fin PFuncs (V9, C9) by A4;
  for s, t being Element of PFuncs (V9, C9) st s in B & t in B & s c= t
  holds s = t by A3,A5;
  then x in { D where D is Element of Fin PFuncs (V9, C9) : ( for u being set
st u in D holds u is finite ) & for s, t being Element of PFuncs (V9, C9) holds
  ( s in D & t in D & s c= t implies s = t ) } by A2;
  hence thesis by SUBSTLAT:def 1;
end;
