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,b being Element of Q holds ((a-r>s)-r>s)[*](
  (b-r>s)-r>s) [= ((a[*]b)-r>s)-r>s
proof
  let Q;
  let s,a,b be Element of Q;
  deffunc NEG(Element of Q) = {c: c[*]($1-r>s) [= s};
A1: {a9[*]b9: a9 in NEG(a) & b9 in NEG(b)} c= NEG(a[*]b)
  proof
    defpred P[Element of Q] means $1[*](a[*]b) [= s;
    deffunc G(Element of Q) = $1;
    let x be object;
    set A = {G(c): P[c]};
    assume x in {a9[*]b9: a9 in NEG(a) & b9 in NEG(b)};
    then consider a9, b9 such that
A2: x = a9[*]b9 and
A3: a9 in NEG(a) and
A4: b9 in NEG(b);
    deffunc F(Element of Q) = a9[*]b9[*]$1;
    set B = {F(G(c)): P[c]};
A5: ex c st b9 = c & c[*] (b-r>s) [= s by A4;
A6: ex c st a9 = c & c[*](a-r>s) [= s by A3;
A7: B is_less_than s
    proof
      let d;
      assume d in B;
      then consider c such that
A8:   d = a9[*]b9[*]c and
A9:   c[*](a[*]b) [= s;
A10:  b-r>s [= b9-l>s by A5,Th11;
      c[*]a[*]b [= s by A9,GROUP_1:def 3;
      then c[*]a [= b-r>s by Th12;
      then c[*]a [= b9-l>s by A10,LATTICES:7;
      then b9[*](c[*]a) [= s by Th11;
      then b9[*]c[*]a [= s by GROUP_1:def 3;
      then
A11:  b9[*]c [= a-r>s by Th12;
      a-r>s [= a9-l>s by A6,Th11;
      then b9[*]c [= a9-l>s by A11,LATTICES:7;
      then a9[*](b9[*]c) [= s by Th11;
      hence d [= s by A8,GROUP_1:def 3;
    end;
    {F(c): c in A} = B from DenestFraenkel;
    then a9[*]b9[*]((a[*]b)-r>s) = "\/"(B, Q) by Def5;
    then a9[*]b9[*]((a[*]b)-r>s) [= s by A7,LATTICE3:def 21;
    hence thesis by A2;
  end;
  ((a-r>s)-r>s)[*]((b-r>s)-r>s) = "\/"({a9[*] b9: a9 in NEG(a) & b9 in NEG
  (b)}, Q) by Th5;
  hence thesis by A1,LATTICE3:45;
end;
