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;
reserve Q for Girard-Quantale,
  a,a1,a2,b,b1,b2,c,d for Element of Q,
  X for set;

theorem
  a delta (b"/\"c) = (a delta b)"/\"(a delta c) & (b"/\"c) delta a = (b
  delta a)"/\"(c delta a)
proof
  thus a delta (b"/\"c) = Bottom (Bottom a [*] (Bottom b"\/"Bottom c)) by Th26
    .= Bottom ((Bottom a [*] Bottom b)"\/"(Bottom a [*] Bottom c)) by Th6
    .= (a delta b)"/\"(a delta c) by Th26;
  thus (b"/\"c) delta a = Bottom ((Bottom b"\/"Bottom c) [*] Bottom a) by Th26
    .= Bottom ((Bottom b [*] Bottom a)"\/"(Bottom c [*] Bottom a)) by Th6
    .= (b delta a)"/\"(c delta a) by Th26;
end;
