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 Ex(All('not' a,A,G),B,G) '<' 'not' All(All(a,
  B,G),A,G)
proof
  assume
A1: G is independent;
  then
  Ex(All('not' a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) & All(Ex('not' a,B
  ,G), A,G) '<' 'not' All(All(a,A,G),B,G) by Th29,PARTIT_2:17;
  then Ex(All('not' a,A,G),B,G) '<' 'not' All(All(a,A,G),B,G) by BVFUNC_1:15;
  hence thesis by A1,PARTIT_2:15;
end;
