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;

theorem Th5:
  for Q for X,Y being set holds "\/"(X,Q) [*] "\/"(Y,Q) = "\/"({a
  [*] b: a in X & b in Y}, Q)
proof
  let Q;
  let X,Y be set;
  deffunc F(Element of Q) = $1[*]"\/"(Y,Q);
  deffunc G(Element of Q) = "\/"({$1[*]b: b in Y}, Q);
  defpred P[set] means $1 in X;
  deffunc H(Element of Q,Element of Q) = $1[*]$2;
A1: for a holds F(a) = G(a) by Def5;
  {F(c): P[c]} = {G(a): P[a]} from FRAENKEL:sch 5(A1);
  hence
  "\/"(X,Q) [*] "\/"(Y,Q) = "\/"({"\/"({H(a,b) where b is Element of Q: b
  in Y}, Q) where a is Element of Q: a in X}, Q) by Def6
    .= "\/"({H(c,d) where c is Element of Q, d is Element of Q: c in X & d
  in Y}, Q) from LUBFraenkelDistr;
end;
