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 being Element of Q, j being UnOp of Q st for a
  being Element of Q holds j.a = (a-r>s)-r>s holds j is monotone
proof
  let Q be Quantale, s be Element of Q;
  let j be UnOp of Q such that
A1: for a being Element of Q holds j.a = (a-r>s)-r>s;
  thus j is monotone
  proof
    let a,b be Element of Q;
    assume a [= b;
    then b-r>s [= a-r>s by Th13;
    then
A2: (a-r>s)-r>s [= (b-r>s)-r>s by Th13;
    (a-r>s)-r>s = j.a by A1;
    hence thesis by A1,A2;
  end;
end;
