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
  for Q being Quantale, s,a being Element of Q st s is cyclic holds a-r>
  s = ((a-r>s)-r>s)-r>s & a-l>s = ((a-l>s)-l>s)-l>s
proof
  let Q;
  let s,a be Element of Q;
  assume
A1: s is cyclic;
  then a [= (a-r>s)-r>s by Th16;
  then
A2: ((a-r>s)-r>s)-r>s [= a-r>s by Th13;
  a [= (a-l>s)-l>s by A1,Th16;
  then
A3: ((a-l>s)-l>s)-l>s [= a-l>s by Th13;
  a-r>s [= ((a-r>s)-r>s)-r>s & a-l>s [= ((a-l>s)-l>s)-l>s by A1,Th16;
  hence thesis by A2,A3,LATTICES:8;
end;
