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 Th31:
  G is independent implies Ex('not' Ex(a,A,G),B,G) '<' 'not' All(
  All(a,B,G),A,G)
proof
  All(a,A,G) '<' Ex(a,A,G) by Th8;
  then
A1: All(All(a,A,G),B,G) '<' All(Ex(a,A,G),B,G) by PARTIT_2:12;
  assume G is independent;
  then
  Ex('not' Ex(a,A,G),B,G) = 'not' All(Ex(a,A,G),B,G) & All(All(a,B,G),A,G)
  = All(All(a,A,G),B,G) by BVFUNC_2:18,PARTIT_2:15;
  hence thesis by A1,PARTIT_2:11;
end;
