Medical Expense Subsidy System for Infants and Children of Compulsory Education

I would like to know about the medical expenses system for children.
I would like to know how to apply for a medical certificate for infants and children attending compulsory education when I move to Inagi City or when a child is born.
When I visited a medical institution outside of Tokyo, I was unable to use my infant/toddler/compulsory education student medical card. I would like to know how to claim medical expenses subsidy.

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