▁obey ▁the ▁relation ▁x   <   z ▁there ▁necessarily ▁exists ▁another ▁rational ▁number ▁z ▁between ▁them ▁( for ▁instance , ▁their ▁average ) ▁with ▁x   <   y ▁and ▁y   <   z . ▁ ▁In ▁contrast , ▁the ▁same ▁ordering ▁relation   <   on ▁the ▁integers ▁is ▁not ▁ide mp ot ent . ▁It ▁is ▁still ▁trans itive , ▁but ▁does ▁not ▁obey ▁the ▁second ▁property ▁of ▁an ▁ide mp ot ent ▁relation . ▁For ▁instance , ▁ 1   <   2 ▁but ▁there ▁does ▁not ▁exist ▁any ▁other ▁integer ▁y ▁between ▁ 1 ▁and ▁ 2 . ▁ ▁An ▁outer ▁product ▁of ▁logical ▁vectors ▁forms ▁an ▁ide mp ot ent ▁relation . ▁Such ▁a ▁relation ▁may ▁be ▁contained ▁in ▁a ▁more ▁complex ▁relation , ▁in ▁which ▁case ▁it ▁is ▁called ▁a ▁concept ▁in ▁that ▁larger ▁context . ▁The ▁description ▁of ▁relations ▁in ▁terms ▁of ▁their ▁concepts ▁is ▁called ▁formal ▁concept ▁analysis . ▁ ▁Us es ▁Ide mp ot ent ▁relations ▁have ▁been ▁used ▁as ▁an ▁example ▁to ▁illustrate ▁the ▁application ▁of ▁Mechan ized ▁F ormal isation ▁of ▁mathematics ▁using ▁the ▁interactive ▁theorem ▁pro ver ▁Isabel le / H OL . ▁Besides ▁checking ▁the ▁mathematical ▁properties ▁of ▁finite ▁ide mp ot ent ▁relations , ▁an ▁algorithm ▁for ▁counting ▁the ▁number ▁of ▁ide mp ot ent ▁relations ▁has ▁been ▁derived ▁in ▁Isabel le / H OL . ▁ ▁Ide mp ot ent ▁relations ▁defined ▁on ▁weak ly ▁count ably ▁compact ▁spaces ▁have ▁also ▁been ▁shown ▁to ▁satisfy ▁"