reserve x,y,z for set;
reserve Q for left-distributive right-distributive complete Lattice-like non
  empty QuantaleStr,
  a, b, c, d for Element of Q;

theorem
  for Q being unital non empty QuasiNetStr st Q is Quantale holds Q is
  BlikleNet
proof
  defpred P[set] means $1 in {};
  let Q be unital non empty QuasiNetStr;
  assume Q is Quantale;
  then reconsider Q9 = Q as Quantale;
A1: Bottom Q9 = "\/"({},Q9) by LATTICE3:49;
A2: not ex c being Element of Q9 st P[c];
  Q9 is with_zero
  proof
    hereby
      reconsider a = Bottom Q9 as Element of Q9;
      take a;
      let b be Element of Q9;
      deffunc F1(Element of Q9) = $1 [*] b;
      {F1(c) where c is Element of Q9: P[c]} = {} from EmptyFraenkel(A2 );
      hence a[*]b = a by A1,Def6;
    end;
    take Bottom Q9;
    let a be Element of Q9;
    deffunc F2(Element of Q9) = a [*] $1;
    {F2(c) where c is Element of Q9: P[c]} = {} from EmptyFraenkel(A2);
    hence thesis by A1,Def5;
  end;
  hence thesis;
end;
