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 Th39:
  G is independent implies 'not' All(Ex(a,A,G),B,G) '<' Ex('not'
  All(a,B,G),A,G)
proof
  assume G is independent;
  then
A1: Ex('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th31;
  'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) & Ex('not' Ex(a,A,G)
  ,B,G ) = Ex(All('not' a,A,G),B,G) by Th19,BVFUNC_2:19;
  hence thesis by A1,BVFUNC_2:18;
end;
