 reserve T for Ternary_Boolean_Algebra;
 reserve a,b,c,d,e for Element of T;
 reserve x,y,z for Element of T;
reserve T for Ternary_Boolean_Algebra;
reserve x for Element of T;
reserve B for Boolean Lattice;
reserve v0,v1,v2,v3,v4,v5,v6,v103,v100,v102,v104,v105,v101 for
  Element of BA2TBAA B;

theorem BotTBA:
  for T being Ternary_Boolean_Algebra,
      p being Element of T holds
    Bottom the LattStr of TBA2BA (T,p) = p`
  proof
    let T be Ternary_Boolean_Algebra,
        p be Element of T;
    set L = the LattStr of TBA2BA (T,p);
    reconsider t = p` as Element of L;
    reconsider tt = t as Element of T;
    for a being Element of L holds t "/\" a = t & a "/\" t = t
    proof
      let a be Element of L;
      reconsider aa = a as Element of T;
      a "/\" t = Tern (aa,p`,tt) by MeetDef
         .= tt by TRIDef;
      hence thesis;
    end;
    hence thesis by LATTICES:def 16;
  end;
