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
  a1 [= b1 & a2 [= b2 implies a1 delta a2 [= b1 delta b2
proof
  assume a1 [= b1 & a2 [= b2;
  then Bottom b1 [= Bottom a1 & Bottom b2 [= Bottom a2 by Th13;
  then Bottom b1 [*] Bottom b2 [= Bottom a1 [*] Bottom a2 by Th9;
  hence thesis by Th13;
end;
