Of course there were people in France who did not ant to abide by the Treaty of Troyes, giving Charles VI's mental instability at the signing. They considered the Dauphin—Charles VI's eldest surviving son Charles—the rightful heir.
Even so, there were some who questioned the legitimacy of the Dauphin's birth, and supported the claim of his cousin Charles, the Duke of Orléans. One problem with that choice: the Duke of Orléans had been a captive of the English since the Battle of Agincourt. Henry had disallowed a ransom for Orléans' release, considering him too close to the line of succession to the French throne and (rightly) assuming he would be a threat to the terms of Troyes. (Spoiler: he would not be released until 1440!) The illustration represents him in the Tower of London.
Because of Agincourt, England already occupied northern France, including Paris. Henry V's brother, John Duke of Bedford, was regent for the child Henry and was put in control of France.
Charles and his people decided to ignore Troyes and continue the Hundred Years War. He gained some land back, executing any Frenchman who sided with the English. In December 1422 he married Marie of Anjou, the daughter of Yolande of Aragon, Duchess of Anjou and Countess of Provence, giving him an alliance that might prove useful.
Although he claimed the title King of France, he did not achieve any significant progress driving the English out of France. He operated out of Chinon. By 1429, Charles' future was looking particularly bleak because of advances by the Duke of Bedford.
Then, on 23 February 1429, a young girl arrived in Chinon to meet with the Dauphin. She was about 16 years old, from the village of Domrémy, and she had demanded of a garrison commander that he assemble an escort to take her to the Dauphin, because she had been given a mission in a vision.
Her name was Jeanne, and we will begin her part of the story next time.