Hi there...hypothyroidism is the medical term for the condition where the thyroid, for some reason or another, is not pumping out enough thyroid hormone.
"Hashi's" is short for "Hashimoto's thyroiditis" which is a term which refers to inflammation and subsequent damaging of one's thyroid gland by the immune system. You can probably guess that, if the thyroid is damaged as in the immune attack in Hashimoto's thyroiditis, you would not be able to pump out thyroid hormone and would thus become hypothyroid.
There is an exception where you can be hyperthyroid with a variant of Hashi's...but the example I gave above is much more common.
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Is hypo and hashi's the same 
