reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem
  All(All(a,A,G),B,G) '<' Ex(Ex(a,B,G),A,G)
proof
A1: All(All(a,A,G),B,G) '<' All(a,A,G) by BVFUNC_2:11;
  All(a,A,G) '<' Ex(Ex(a,B,G),A,G) by Th8,BVFUNC_2:12;
  hence thesis by A1,BVFUNC_1:15;
end;
