Holy moly that is not at all what that paper says! It specifically argues that certain equational properties of a given total language continue to hold in the t

Holy moly that is not at all what that paper says! It specifically argues that certain equational properties of a given total language continue to hold in the total fragment of a given partial language. It is an embedding theorem combined with lifting properties, not a license to perform _any_ reasoning at all regarding the non-total portion of the language it considers!