Securing the succession for Eustace

In January 1153 Count Henry of Anjou invaded England in pursuance of his mother Matilda's claim to the throne.
King Stephen fought on in the hope of securing the succession for his son Eustace.
When Eustace died in August 1153, Stephen signed a treaty with Henry. Stephen named Henry as his heir.
If Prince Eustace had lived, would King Stephen have named Henry as his heir?
 
At a council held in London on April 6, 1152, Stephen induced a small number of barons to pay homage to Eustace as their future king.
However the Archbishop of Canterbury and the other bishops declined to perform the coronation ceremony on the grounds that the Roman curia had declared against the claim of Eustace.
 
Yes Stephen would have fought on and the Roman Curia would in the end have backed the winner just like they always did. See King John, King Henry III and "King" Louis of England. As for who the winner would be God only knows. The Anarchy swung back and forth so many times it's impossible to guess who would end up triumphant.
 
King Stephen and his wife, Queen Matilda of Boulogne took steps to have Eustace crowned in his father's lifetime.
It required the consent of Theobald, the Archbishop of Canterbury,
In 1149 Theobald refused to consent to Eustace's coronation. The curia had accepted Stephen as de facto king, but did not necessarily recognize him as the rightful ruler.
 
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