, ▁denoted ▁by ▁inf ix ▁" =" , ▁is ▁part ▁of ▁the ▁background ▁logic , ▁the ▁binary ▁relation ▁Pro per ▁Part , ▁denoted ▁by ▁inf ix ▁"< ", ▁is ▁defined ▁as : ▁ ▁The ▁ax i oms ▁are : ▁In clusion ▁partially ▁orders ▁the ▁domain . ▁G 1 . ▁▁▁▁▁▁▁ ▁( ref lex ive ) ▁G 2 . ▁▁▁ ▁( trans itive ) ▁W P 4 . ▁G 3 . ▁▁▁ ▁( anti - sym metric ) ▁ ▁Given ▁any ▁two ▁regions , ▁there ▁exists ▁a ▁region ▁that ▁includes ▁both ▁of ▁them . ▁W P 6 . ▁G 4 . ▁▁ ▁Pro per ▁Part ▁dens ely ▁orders ▁the ▁domain . ▁W P 5 . ▁G 5 . ▁▁ ▁Both ▁atomic ▁regions ▁and ▁a ▁universal ▁region ▁do ▁not ▁exist . ▁Hence ▁the ▁domain ▁has ▁neither ▁an ▁upper ▁nor ▁a ▁lower ▁bound . ▁W P 2 . ▁G 6 . ▁▁▁ ▁Pro per ▁Part s ▁Princi ple . ▁If ▁all ▁the ▁proper ▁parts ▁of ▁x ▁are ▁proper ▁parts ▁of ▁y , ▁then ▁x ▁is ▁included ▁in ▁y . ▁W P 3 . ▁ ▁G 7 . ▁▁ ▁A ▁model ▁of ▁G 1 – G 7 ▁is ▁an ▁inclusion ▁space . ▁ ▁Definition ▁( G er la ▁and ▁Mir anda ▁ 2 0 0 8 : ▁Def . ▁ 4 . 1 ). ▁Given ▁some ▁inclusion ▁space ▁S , ▁an ▁abstract ive ▁class ▁is ▁a ▁class ▁G ▁of ▁regions ▁such ▁that ▁S \ G ▁is ▁totally ▁ordered ▁by ▁In clusion . ▁Moreover , ▁there ▁does ▁not