reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);
reserve s for Element of PFuncs (V,C);
reserve K, L, M for Element of SubstitutionSet (V,C);

theorem
  Bottom SubstLatt (V,C) = {}
proof
  {} in SubstitutionSet (V,C) by Th1;
  then reconsider Z = {} as Element of SubstLatt (V,C) by Def4;
  now
    let u be Element of SubstLatt (V,C);
    reconsider z = Z, u9 = u as Element of SubstitutionSet (V,C) by Def4;
    thus Z "\/" u = mi (z \/ u9) by Def4
      .= u by Th11;
  end;
  hence thesis by LATTICE2:14;
end;
