reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);

theorem Th5:
  for a, b be set holds B in SubstitutionSet (V, C) & a in B & b in
  B & a c= b implies a = b
proof
  let a, b be set;
  assume B in SubstitutionSet (V, C);
  then
A1: ex A1 be Element of Fin PFuncs (V,C) st A1 = B & ( for u being set st u
  in A1 holds u is finite ) & for s, t being Element of PFuncs (V, C) holds ( s
  in A1 & t in A1 & s c= t implies s = t );
  assume that
A2: a in B & b in B and
A3: a c= b;
  B c= PFuncs (V,C) by FINSUB_1:def 5;
  then reconsider a9 = a, b9 = b as Element of PFuncs (V, C) by A2;
  a9 = b9 by A1,A2,A3;
  hence thesis;
end;
