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 Th15:
  c is cyclic iff for a,b st a [*] b [= c holds b [*] a [= c
proof
  thus c is cyclic implies for a,b st a [*] b [= c holds b [*] a [= c
  proof
    assume
A1: a -r> c = a -l> c;
    let a,b;
    assume a [*] b [= c;
    then a [= b-r>c by Th12;
    then a [= b-l>c by A1;
    hence thesis by Th11;
  end;
  assume
A2: a [*] b [= c implies b [*] a [= c;
  let a;
  set X1 = {d1: d1 [*] a [= c}, X2 = {d2: a [*] d2 [= c};
  X1 = X2
  proof
    thus X1 c= X2
    proof
      let x be object;
      assume x in X1;
      then consider d such that
A3:   x = d and
A4:   d [*] a [= c;
      a [*] d [= c by A2,A4;
      hence thesis by A3;
    end;
    let x be object;
    assume x in X2;
    then consider d such that
A5: x = d and
A6: a [*] d [= c;
    d [*] a [= c by A2,A6;
    hence thesis by A5;
  end;
  hence thesis;
end;
