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 Th16:
  for Q being Quantale, s,a being Element of Q st s is cyclic
  holds a [= (a-r>s)-r>s & a [= (a-l>s)-l>s
proof
  let Q;
  let s,a be Element of Q such that
A1: s is cyclic;
A2: {b: b [= a} c= {c: c[*](a-r>s) [= s}
  proof
    let x be object;
    assume x in {b: b [= a};
    then consider b such that
A3: x = b and
A4: b [= a;
    (b-r>s)[*]b [= s by Th12;
    then
A5: b[*](b-r>s) [= s by A1,Th15;
    a-r>s [= b-r>s by A4,Th13;
    then b[*](a-r>s) [= b[*](b-r>s) by Th8;
    then b[*](a-r>s) [= s by A5,LATTICES:7;
    hence thesis by A3;
  end;
A6: {b: b [= a} c= {c: (a-l>s)[*]c [= s}
  proof
    let x be object;
    assume x in {b: b [= a};
    then consider b such that
A7: x = b and
A8: b [= a;
    b[*](b-l>s) [= s by Th11;
    then
A9: (b-l>s)[*]b [= s by A1,Th15;
    a-l>s [= b-l>s by A8,Th13;
    then (a-l>s)[*]b [= (b-l>s)[*]b by Th8;
    then (a-l>s)[*]b [= s by A9,LATTICES:7;
    hence thesis by A7;
  end;
  a = "\/"({d: d [= a},Q) by LATTICE3:44;
  hence a [= (a-r>s)-r>s by A2,LATTICE3:45;
  a = "\/"({d: d [= a},Q) by LATTICE3:44;
  hence thesis by A6,LATTICE3:45;
end;
