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 Th11:
  a [*] b [= c iff b [= a -l> c
proof
  set X = {d: a [*] d [= c};
  X c= carr(Q)
  proof
    let x be object;
    assume x in X;
    then ex d st x = d & a [*] d [= c;
    hence thesis;
  end;
  then reconsider X as Subset of Q;
  thus a [*] b [= c implies b [= a -l> c
  proof
    assume a [*] b [= c;
    then b in X;
    hence thesis by LATTICE3:38;
  end;
  deffunc F(Element of Q) = a [*] $1;
  defpred P1[set] means $1 in X;
  defpred P2[Element of Q] means a [*] $1 [= c;
  assume b [= a -l> c;
  then
A1: a [*] b [= a [*] "\/"X by Th8;
  now
    let d;
    assume d in X;
    then ex d1 st d = d1 & a [*] d1 [= c;
    hence a [*] d [= c;
  end;
  then
A2: P1[d] iff P2[d];
A3: {F(d1): P1[d1]} = {F(d2): P2[d2]} from FRAENKEL:sch 3(A2);
A4: {a [*] d: d in X} is_less_than c
  proof
    let d1;
    assume d1 in {a [*] d: d in X};
    then ex d2 st d1 = a [*] d2 & a [*] d2 [= c by A3;
    hence thesis;
  end;
  a [*] "\/"X = "\/"({a [*] d: d in X}, Q) by Def5;
  then a [*] "\/"X [= c by A4,LATTICE3:def 21;
  hence thesis by A1,LATTICES:7;
end;
