reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;

theorem
  G is independent implies Ex(Ex(a,P,G),Q,G) = Ex(Ex(a,Q,G),P,G)
proof
  assume
A1: G is independent;
  thus Ex(Ex(a,P,G),Q,G) = 'not' 'not' Ex(Ex(a,P,G),Q,G)
    .= 'not' All('not' Ex(a,P,G),Q,G) by BVFUNC_2:19
    .= 'not' All(All('not' a,P,G),Q,G) by BVFUNC_2:19
    .= 'not' All(All('not' a,Q,G),P,G) by A1,Th15
    .= 'not' All('not' Ex(a,Q,G),P,G) by BVFUNC_2:19
    .= 'not' 'not' Ex(Ex(a,Q,G),P,G) by BVFUNC_2:19
    .= Ex(Ex(a,Q,G),P,G);
end;
