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 Th13:
  for Q being Quantale, s,a,b being Element of Q st a [= b holds b
  -r>s [= a-r>s & b-l>s [= a-l>s
proof
  let Q be Quantale, s,a,b be Element of Q such that
A1: a [= b;
  {d where d is Element of Q: d [*] b [= s} c= {c where c is Element of Q:
  c [*] a [= s}
  proof
    let x be object;
    assume x in {d where d is Element of Q: d [*] b [= s};
    then consider d being Element of Q such that
A2: x = d and
A3: d [*] b [= s;
    d [*] a [= d [*] b by A1,Th8;
    then d [*] a [= s by A3,LATTICES:7;
    hence thesis by A2;
  end;
  hence b-r>s [= a-r>s by LATTICE3:45;
  {d where d is Element of Q: b [*] d [= s} c= {c where c is Element of Q:
  a [*] c [= s}
  proof
    let x be object;
    assume x in {d where d is Element of Q: b [*] d [= s};
    then consider d being Element of Q such that
A4: x = d and
A5: b [*] d [= s;
    a [*] d [= b [*] d by A1,Th8;
    then a [*] d [= s by A5,LATTICES:7;
    hence thesis by A4;
  end;
  hence thesis by LATTICE3:45;
end;
