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We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness. We then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.doi:10.4064/fm216-3-2 fatcat:qn4gv7zqrzgondlkdnhllagtb4