The correct answer is option 'D' - direct elections of all representatives.

Explanation:

Presence of Rule of Law:
The presence of the rule of law is vital for the establishment and success of a democracy. The rule of law ensures that all individuals, including those in positions of power, are subject to the law and are not above it. It provides a framework for the functioning of a democratic society, ensuring that laws are enforced impartially and consistently. Without the rule of law, there would be no guarantee of fairness, justice, or protection of individual rights, which are fundamental to a democratic system.

Granting fundamental rights to the population:
Granting fundamental rights to the population is another essential aspect of a democracy. Fundamental rights include rights such as freedom of speech, expression, assembly, religion, and the right to equality. These rights ensure that individuals have the freedom to express their opinions, participate in the political process, and live their lives without discrimination or repression. Without the protection of fundamental rights, democracy cannot thrive, as the rights and freedoms of individuals would be curtailed.

Political equality of citizens to run for public offices:
Political equality is a core principle of democracy. It means that all citizens, regardless of their background, have an equal opportunity to participate in the political process and run for public offices. Political equality ensures that no individual or group is excluded from participating in the democratic decision-making process. It promotes inclusivity and prevents the concentration of power in the hands of a few.

Direct elections of all representatives:
While direct elections of representatives are commonly associated with democracies, they are not necessarily vital for the establishment or success of a democracy. In some democratic systems, representatives may be elected indirectly through an electoral college or other means. The key aspect of democracy is not the specific method of electing representatives but the principles of representation, accountability, and the participation of citizens in the political process.

Conclusion:
In conclusion, while the presence of the rule of law, granting fundamental rights to the population, and political equality are all vital for the establishment and success of a democracy, direct elections of all representatives are not essential. Other methods of representative selection can also be compatible with democratic principles.