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
  G is independent implies All('not' All(a,A,G),B,G) '<' Ex(Ex('not' a,B
  ,G),A,G)
proof
  assume G is independent;
  then
A1: Ex(Ex('not' a,B,G),A,G) = Ex(Ex('not' a,A,G),B,G) by PARTIT_2:16;
  'not' All(a,A,G) = Ex('not' a,A,G) by BVFUNC_2:18;
  hence thesis by A1,Th8;
end;
