On the basis of blood type, you know that one of the children could not have been the son of the two adults. Does this prove that the other child is their son

No.

Explanation:

No, Blood test can't tell us that the which child is the son of two adults, it can only be find out by DNA testing of children with the two adults. If the DNA testing matches of children to the adults then we can say that these children are the sons of two adults while on the other hand, if the DNA of two children not match with the two adults then we can say that they are not their child.