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;
reserve Q for Quantale,
  a,a9,b,b9,c,d,d1,d2,D for Element of Q;

theorem Th21:
  for Q being non empty Girard-QuantaleStr st the LattStr of Q =
  BooleLatt {} holds Q is cyclic dualized
proof
  let Q be non empty Girard-QuantaleStr;
  set c = the absurd of Q;
  assume the LattStr of Q = BooleLatt {};
  then
A1: carr(Q) = {{}} by LATTICE3:def 1,ZFMISC_1:1;
  thus Q is cyclic
  proof
    let a be Element of Q;
    a -r> c = {} by A1,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let a be Element of Q;
  (a-r>c)-l>c = {} & (a-l>c)-r>c = {} by A1,TARSKI:def 1;
  hence thesis by A1,TARSKI:def 1;
end;
