What is the Shariah ruling regarding the children from a mixed marriage of a Muslim male and a Christian wife?

Question

Shariah has granted a Muslim male leeway to marry a Christian female in order to prevail over her and lead her to Islam. The children in such a marriage will be subordinates of the father, not the mother. If the children were not brought Islamically that is the weakness of the father. If he was negligent in their upbringing, then he will be sinful for his children becoming Christians. It is precisely for such unforeseen but possible circumstances that Shariah discourages mixed marriages.