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 'not' All(All(a,A,G),B,G) '<' Ex(Ex('not' a,A
  ,G),B,G)
proof
  assume
A1: G is independent;
  then Ex('not' All(a,B,G),A,G) '<' Ex(Ex('not' a,A,G),B,G) by Th14;
  hence thesis by A1,Th15;
end;
