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
  "\/"(X,Q) [*] a = "\/"({b [*] a: b in X}, Q) & "/\"(X,Q) delta a =
  "/\"({c delta a: c in X}, Q)
proof
  deffunc F(Element of Q) = $1 [*] Bottom a;
  deffunc G(Element of Q) = Bottom $1;
  deffunc H(Element of Q) = Bottom $1 [*] Bottom a;
  defpred P[set] means $1 in X;
  deffunc F1(Element of Q) = Bottom (Bottom $1 [*] Bottom a);
  deffunc F2(Element of Q) = $1 delta a;
  thus "\/"(X,Q) [*] a = "\/"({b [*] a: b in X}, Q) by Def6;
A1: {F(c): c in {G(d): P[d]}} = {F(G(b)): P[b]} from DenestFraenkel;
A2: {G(c): c in {H(d): P[d]}} = {G(H(b)): P[b]} from DenestFraenkel;
A3: F1(b) = F2(b);
A4: {F1(b): P[b]} = {F2(c): P[c]} from FRAENKEL:sch 5(A3);
  thus "/\"(X,Q) delta a = Bottom ("\/"({Bottom b: b in X}, Q) [*] Bottom a)
  by Th25
    .= Bottom "\/"({Bottom b [*] Bottom a: b in X}, Q) by A1,Def6
    .= "/\"({b delta a: b in X}, Q) by A2,A4,Th24;
end;
