Integrity is defined as the adherence to moral and ethical principles, soundness of moral character, honesty. To get a respected level of integrity one must be constant in showing a set of actions and ideals. You cannot be called a person of integrity by just performing a couple of good deeds one day and being mean the other.
Integrity needs to be constant honourable action, and one good part of it is keeping one's word.
The quotation that best shapes the theme that integrity is established by keeping one's word is:
A "'It is impossible for me to doubt my friend's constancy,' replied Damon."
Damon had been shown time after time that his friend's constancy had been worthy of trust, that is why there is no need for more proof of it, it had already been proven before.