:: ALTCAT_1 semantic presentation begin theorem :: ALTCAT_1:1 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "A")) ")" ))) ; theorem :: ALTCAT_1:2 (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" ) ($#k1_tarski :::"}"::: ) )) ; theorem :: ALTCAT_1:3 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" )) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ))) "holds" (Bool (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "C")) ")" )))) ; theorem :: ALTCAT_1:4 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "C")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: ALTCAT_1:5 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "A")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "B")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "F")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "y")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "D")) "," (Set (Var "C")) ($#k8_mcart_1 :::":]"::: ) ) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "y")) "," (Set (Var "x")) ")" ))))))) ; scheme :: ALTCAT_1:sch 1 MSSLambda2{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool (Set (Set (Var "M")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "i")) "," (Set (Var "j")) ")" )))) proof end; scheme :: ALTCAT_1:sch 2 MSSLambda2D{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) "holds" (Bool (Set (Set (Var "M")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))))) proof end; scheme :: ALTCAT_1:sch 3 MSSLambda3{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_hidden :::"set"::: ) , F4( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ($#k3_zfmisc_1 :::":]"::: ) ) "st" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set F3 "(" ")" ))) "holds" (Bool (Set (Set (Var "M")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" )))) proof end; scheme :: ALTCAT_1:sch 4 MSSLambda3D{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F4( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ($#k3_zfmisc_1 :::":]"::: ) ) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ) "holds" (Bool (Set (Set (Var "M")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" )))))) proof end; theorem :: ALTCAT_1:6 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "N")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) "holds" (Bool (Set (Var "M")) ($#r6_pboole :::"="::: ) (Set (Var "N"))))) ; theorem :: ALTCAT_1:7 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "B")) "holds" (Bool (Set (Set (Var "N")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) ")" )) "holds" (Bool (Set (Var "M")) ($#r8_pboole :::"="::: ) (Set (Var "N"))))) ; theorem :: ALTCAT_1:8 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set (Var "A")) "," (Set (Var "A")) "," (Set (Var "A")) ($#k3_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "N")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" )) ")" )) "holds" (Bool (Set (Var "M")) ($#r6_pboole :::"="::: ) (Set (Var "N"))))) ; theorem :: ALTCAT_1:9 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "(" (Set (Var "i")) "," (Set (Var "j")) ")" ($#k17_funcop_1 :::":->"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "k"))))) ; theorem :: ALTCAT_1:10 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" "(" (Set (Var "i")) "," (Set (Var "j")) ")" ($#k17_funcop_1 :::":->"::: ) (Set (Var "k")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "k")))) ; begin definitionattr "c1" is :::"strict"::: ; struct :::"AltGraph"::: -> ($#l1_struct_0 :::"1-sorted"::: ) ; aggr :::"AltGraph":::(# :::"carrier":::, :::"Arrows"::: #) -> ($#l1_altcat_1 :::"AltGraph"::: ) ; sel :::"Arrows"::: "c1" -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1") ($#k2_zfmisc_1 :::":]"::: ) ); end; definitionlet "G" be ($#l1_altcat_1 :::"AltGraph"::: ) ; mode object of "G" is ($#m1_subset_1 :::"Element":::) "of" "G"; end; definitionlet "G" be ($#l1_altcat_1 :::"AltGraph"::: ) ; let "o1", "o2" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "G")); func :::"<^":::"o1" "," "o2":::"^>"::: -> ($#m1_hidden :::"set"::: ) equals :: ALTCAT_1:def 1 (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "G") ($#k1_binop_1 :::"."::: ) "(" "o1" "," "o2" ")" ); end; :: deftheorem defines :::"<^"::: ALTCAT_1:def 1 : (Bool "for" (Set (Var "G")) "being" ($#l1_altcat_1 :::"AltGraph"::: ) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "G")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "G"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )))); definitionlet "G" be ($#l1_altcat_1 :::"AltGraph"::: ) ; let "o1", "o2" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "G")); mode Morphism of "o1" "," "o2" is ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_altcat_1 :::"<^"::: ) "o1" "," "o2" ($#k1_altcat_1 :::"^>"::: ) ); end; definitionlet "G" be ($#l1_altcat_1 :::"AltGraph"::: ) ; attr "G" is :::"transitive"::: means :: ALTCAT_1:def 2 (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" "G" "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))); end; :: deftheorem defines :::"transitive"::: ALTCAT_1:def 2 : (Bool "for" (Set (Var "G")) "being" ($#l1_altcat_1 :::"AltGraph"::: ) "holds" (Bool "(" (Bool (Set (Var "G")) "is" ($#v2_altcat_1 :::"transitive"::: ) ) "iff" (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" )); begin definitionlet "I" be ($#m1_hidden :::"set"::: ) ; let "G" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "I")) "," (Set (Const "I")) ($#k2_zfmisc_1 :::":]"::: ) ); func :::"{|":::"G":::"|}"::: -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) "I" "," "I" "," "I" ($#k3_zfmisc_1 :::":]"::: ) ) means :: ALTCAT_1:def 3 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) "I")) "holds" (Bool (Set it ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ))); let "H" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "I")) "," (Set (Const "I")) ($#k2_zfmisc_1 :::":]"::: ) ); func :::"{|":::"G" "," "H":::"|}"::: -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) "I" "," "I" "," "I" ($#k3_zfmisc_1 :::":]"::: ) ) means :: ALTCAT_1:def 4 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) "I")) "holds" (Bool (Set it ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "H" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))); end; :: deftheorem defines :::"{|"::: ALTCAT_1:def 3 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) "," (Set (Var "I")) ($#k3_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_altcat_1 :::"{|"::: ) (Set (Var "G")) ($#k2_altcat_1 :::"|}"::: ) )) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ))) ")" )))); :: deftheorem defines :::"{|"::: ALTCAT_1:def 4 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "b4")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) "," (Set (Var "I")) ($#k3_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k3_altcat_1 :::"{|"::: ) (Set (Var "G")) "," (Set (Var "H")) ($#k3_altcat_1 :::"|}"::: ) )) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_multop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "H")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" )))); definitionlet "I" be ($#m1_hidden :::"set"::: ) ; let "G" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "I")) "," (Set (Const "I")) ($#k2_zfmisc_1 :::":]"::: ) ); mode BinComp of "G" is ($#m2_pboole :::"ManySortedFunction"::: ) "of" (Set ($#k3_altcat_1 :::"{|"::: ) "G" "," "G" ($#k3_altcat_1 :::"|}"::: ) ) "," (Set ($#k2_altcat_1 :::"{|"::: ) "G" ($#k2_altcat_1 :::"|}"::: ) ); end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "G" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "I")) "," (Set (Const "I")) ($#k2_zfmisc_1 :::":]"::: ) ); let "o" be ($#m2_pboole :::"BinComp":::) "of" (Set (Const "G")); let "i", "j", "k" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); :: original: :::"."::: redefine func "o" :::"."::: "(" "i" "," "j" "," "k" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "G" ($#k1_binop_1 :::"."::: ) "(" "j" "," "k" ")" ")" ) "," (Set "(" "G" ($#k1_binop_1 :::"."::: ) "(" "i" "," "j" ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" "G" ($#k1_binop_1 :::"."::: ) "(" "i" "," "k" ")" ")" ); end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "G" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "I")) "," (Set (Const "I")) ($#k2_zfmisc_1 :::":]"::: ) ); let "IT" be ($#m2_pboole :::"BinComp":::) "of" (Set (Const "G")); attr "IT" is :::"associative"::: means :: ALTCAT_1:def 5 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) ")" )) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "k")) "," (Set (Var "l")) ")" ))) "holds" (Bool (Set (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "h")) "," (Set "(" (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "f")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "h")) "," (Set (Var "g")) ")" ")" ) "," (Set (Var "f")) ")" )))); attr "IT" is :::"with_right_units"::: means :: ALTCAT_1:def 6 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" )) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "e")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ")" ) ")" ))); attr "IT" is :::"with_left_units"::: means :: ALTCAT_1:def 7 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "j")) ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "G" ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Set "(" "IT" ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "j")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "e")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ")" ) ")" ))); end; :: deftheorem defines :::"associative"::: ALTCAT_1:def 5 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "IT")) "being" ($#m2_pboole :::"BinComp":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_altcat_1 :::"associative"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) ")" )) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "k")) "," (Set (Var "l")) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "h")) "," (Set "(" (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "f")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "l")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "h")) "," (Set (Var "g")) ")" ")" ) "," (Set (Var "f")) ")" )))) ")" )))); :: deftheorem defines :::"with_right_units"::: ALTCAT_1:def 6 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "IT")) "being" ($#m2_pboole :::"BinComp":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_altcat_1 :::"with_right_units"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" )) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "e")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ")" ) ")" ))) ")" )))); :: deftheorem defines :::"with_left_units"::: ALTCAT_1:def 7 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "I")) "," (Set (Var "I")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "IT")) "being" ($#m2_pboole :::"BinComp":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v5_altcat_1 :::"with_left_units"::: ) ) "iff" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "j")) "," (Set (Var "j")) ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "IT")) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "j")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "e")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ")" ) ")" ))) ")" )))); begin definitionattr "c1" is :::"strict"::: ; struct :::"AltCatStr"::: -> ($#l1_altcat_1 :::"AltGraph"::: ) ; aggr :::"AltCatStr":::(# :::"carrier":::, :::"Arrows":::, :::"Comp"::: #) -> ($#l2_altcat_1 :::"AltCatStr"::: ) ; sel :::"Comp"::: "c1" -> ($#m2_pboole :::"BinComp":::) "of" (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "c1"); end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; definitionlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; let "o1", "o2", "o3" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "C")); assume (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Const "o1")) "," (Set (Const "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Const "o2")) "," (Set (Const "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ; let "f" be ($#m1_subset_1 :::"Morphism":::) "of" (Set (Const "o1")) "," (Set (Const "o2")); let "g" be ($#m1_subset_1 :::"Morphism":::) "of" (Set (Const "o2")) "," (Set (Const "o3")); func "g" :::"*"::: "f" -> ($#m1_subset_1 :::"Morphism":::) "of" "o1" "," "o3" equals :: ALTCAT_1:def 8 (Set (Set "(" (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" "C") ($#k4_altcat_1 :::"."::: ) "(" "o1" "," "o2" "," "o3" ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" "g" "," "f" ")" ); end; :: deftheorem defines :::"*"::: ALTCAT_1:def 8 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o3")) "holds" (Bool (Set (Set (Var "g")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "f")) ")" )))))); definitionlet "IT" be ($#m1_hidden :::"Function":::); attr "IT" is :::"compositional"::: means :: ALTCAT_1:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "IT"))) "holds" (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "g")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) )) & (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")))) ")" ))); end; :: deftheorem defines :::"compositional"::: ALTCAT_1:def 9 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v7_altcat_1 :::"compositional"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "IT"))))) "holds" (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "g")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")))) ")" ))) ")" )); registrationlet "A", "B" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "B" ($#k2_zfmisc_1 :::":]"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "B" ($#k2_zfmisc_1 :::":]"::: ) )) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() ($#v7_altcat_1 :::"compositional"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: ALTCAT_1:11 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#v7_altcat_1 :::"compositional"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "F")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))))))) ; definitionlet "A", "B" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; func :::"FuncComp"::: "(" "A" "," "B" ")" -> ($#v7_altcat_1 :::"compositional"::: ) ($#m1_hidden :::"ManySortedFunction":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "B" "," "A" ($#k2_zfmisc_1 :::":]"::: ) ) means :: ALTCAT_1:def 10 (Bool verum); end; :: deftheorem defines :::"FuncComp"::: ALTCAT_1:def 10 : (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#v7_altcat_1 :::"compositional"::: ) ($#m1_hidden :::"ManySortedFunction":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "A")) ($#k2_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" )) "iff" (Bool verum) ")" ))); theorem :: ALTCAT_1:12 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ")" ) "," (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ")" ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "A")) "," (Set (Var "C")) ")" ))) ; theorem :: ALTCAT_1:13 (Bool "for" (Set (Var "o")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#r8_pboole :::"="::: ) (Set "(" (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) "," (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) ")" ($#k17_funcop_1 :::":->"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" )))) ; theorem :: ALTCAT_1:14 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "B1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "B")) "holds" (Bool (Set ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set (Var "A1")) "," (Set (Var "B1")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ")" ) ($#k5_relat_1 :::"|"::: ) (Set ($#k8_mcart_1 :::"[:"::: ) (Set (Var "B1")) "," (Set (Var "A1")) ($#k8_mcart_1 :::":]"::: ) )))))) ; definitionlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; attr "C" is :::"quasi-functional"::: means :: ALTCAT_1:def 11 (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) ")" ))); attr "C" is :::"semi-functional"::: means :: ALTCAT_1:def 12 (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a2")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a1")) "," (Set (Var "a2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a2")) "," (Set (Var "a3")) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f9"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g9")))) "holds" (Bool (Set (Set (Var "g")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g9")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f9")))))))); attr "C" is :::"pseudo-functional"::: means :: ALTCAT_1:def 13 (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "holds" (Bool (Set (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" "C") ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" ")" ) "," (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "o2")) "," (Set (Var "o3")) ")" ")" ) ")" ")" ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) "," (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )))); end; :: deftheorem defines :::"quasi-functional"::: ALTCAT_1:def 11 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v8_altcat_1 :::"quasi-functional"::: ) ) "iff" (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) ")" ))) ")" )); :: deftheorem defines :::"semi-functional"::: ALTCAT_1:def 12 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v9_altcat_1 :::"semi-functional"::: ) ) "iff" (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a2")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a1")) "," (Set (Var "a2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a2")) "," (Set (Var "a3")) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f9"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g9")))) "holds" (Bool (Set (Set (Var "g")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g9")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f9")))))))) ")" )); :: deftheorem defines :::"pseudo-functional"::: ALTCAT_1:def 13 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v10_altcat_1 :::"pseudo-functional"::: ) ) "iff" (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_altcat_1 :::"FuncComp"::: ) "(" (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" ")" ) "," (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "o2")) "," (Set (Var "o3")) ")" ")" ) ")" ")" ) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) "," (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )))) ")" )); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "X")) "," (Set (Const "X")) ($#k2_zfmisc_1 :::":]"::: ) ); let "C" be ($#m2_pboole :::"BinComp":::) "of" (Set (Const "A")); cluster (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" "X" "," "A" "," "C" "#)" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; theorem :: ALTCAT_1:15 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) ")" ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a2")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) "," (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) )))) ; theorem :: ALTCAT_1:16 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a2")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a1")) "," (Set (Var "a2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a2")) "," (Set (Var "a3")) (Bool "for" (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f9"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "g9")))) "holds" (Bool (Set (Set (Var "g")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g9")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f9"))))))))) ; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"EnsCat"::: "A" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) means :: ALTCAT_1:def 14 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "A") & (Bool "(" "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"object":::) "of" it "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"EnsCat"::: ALTCAT_1:def 14 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_altcat_1 :::"EnsCat"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "b2")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) ")" )) ")" ) ")" ) ")" ))); definitionlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; attr "C" is :::"associative"::: means :: ALTCAT_1:def 15 (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" "C") "is" ($#v3_altcat_1 :::"associative"::: ) ); attr "C" is :::"with_units"::: means :: ALTCAT_1:def 16 (Bool "(" (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" "C") "is" ($#v5_altcat_1 :::"with_left_units"::: ) ) & (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" "C") "is" ($#v4_altcat_1 :::"with_right_units"::: ) ) ")" ); end; :: deftheorem defines :::"associative"::: ALTCAT_1:def 15 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v11_altcat_1 :::"associative"::: ) ) "iff" (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "is" ($#v3_altcat_1 :::"associative"::: ) ) ")" )); :: deftheorem defines :::"with_units"::: ALTCAT_1:def 16 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v12_altcat_1 :::"with_units"::: ) ) "iff" (Bool "(" (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "is" ($#v5_altcat_1 :::"with_left_units"::: ) ) & (Bool (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "is" ($#v4_altcat_1 :::"with_right_units"::: ) ) ")" ) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; theorem :: ALTCAT_1:17 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a2")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) ) "," (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "a1")) "," (Set (Var "a2")) "," (Set (Var "a3")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a1")) "," (Set (Var "a3")) ($#k1_altcat_1 :::"^>"::: ) )) ")" ))) ; theorem :: ALTCAT_1:18 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o")) "," (Set (Var "o")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; registrationlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_altcat_1 :::"EnsCat"::: ) "A") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v8_altcat_1 :::"quasi-functional"::: ) ($#v9_altcat_1 :::"semi-functional"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_altcat_1 :::"pseudo-functional"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#v12_altcat_1 :::"with_units"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_altcat_1 :::"quasi-functional"::: ) ($#v9_altcat_1 :::"semi-functional"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; definitionmode category is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; begin definitionlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; let "o" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "C")); func :::"idm"::: "o" -> ($#m1_subset_1 :::"Morphism":::) "of" "o" "," "o" means :: ALTCAT_1:def 17 (Bool "for" (Set (Var "o9")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) "o" "," (Set (Var "o9")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" "o" "," (Set (Var "o9")) "holds" (Bool (Set (Set (Var "a")) ($#k5_altcat_1 :::"*"::: ) it) ($#r1_hidden :::"="::: ) (Set (Var "a"))))); end; :: deftheorem defines :::"idm"::: ALTCAT_1:def 17 : (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o")) "," (Set (Var "o")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o")))) "iff" (Bool "for" (Set (Var "o9")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o")) "," (Set (Var "o9")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o")) "," (Set (Var "o9")) "holds" (Bool (Set (Set (Var "a")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ")" )))); theorem :: ALTCAT_1:19 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o")) "," (Set (Var "o")) ($#k1_altcat_1 :::"^>"::: ) )))) ; theorem :: ALTCAT_1:20 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool (Set (Set "(" ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o2")) ")" ) ($#k5_altcat_1 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a")))))) ; theorem :: ALTCAT_1:21 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "," (Set (Var "o4")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o3")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o3")) "," (Set (Var "o4")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o3")) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o3")) "," (Set (Var "o4")) "holds" (Bool (Set (Set (Var "c")) ($#k5_altcat_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "c")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "b")) ")" ) ($#k5_altcat_1 :::"*"::: ) (Set (Var "a"))))))))) ; begin definitionlet "C" be ($#l2_altcat_1 :::"AltCatStr"::: ) ; attr "C" is :::"quasi-discrete"::: means :: ALTCAT_1:def 18 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))); attr "C" is :::"pseudo-discrete"::: means :: ALTCAT_1:def 19 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"object":::) "of" "C" "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_altcat_1 :::"^>"::: ) ) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )); end; :: deftheorem defines :::"quasi-discrete"::: ALTCAT_1:def 18 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v13_altcat_1 :::"quasi-discrete"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) ")" )); :: deftheorem defines :::"pseudo-discrete"::: ALTCAT_1:def 19 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v14_altcat_1 :::"pseudo-discrete"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_altcat_1 :::"^>"::: ) ) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) ")" )); theorem :: ALTCAT_1:22 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v14_altcat_1 :::"pseudo-discrete"::: ) ) "iff" (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o")) "," (Set (Var "o")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o")) ")" ) ($#k1_tarski :::"}"::: ) ))) ")" )) ; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) -> ($#v13_altcat_1 :::"quasi-discrete"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; theorem :: ALTCAT_1:23 (Bool "(" (Bool (Set ($#k7_altcat_1 :::"EnsCat"::: ) (Num 1)) "is" ($#v14_altcat_1 :::"pseudo-discrete"::: ) ) & (Bool (Set ($#k7_altcat_1 :::"EnsCat"::: ) (Num 1)) "is" (Num 1) ($#v13_struct_0 :::"-element"::: ) ) ")" ) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#v14_altcat_1 :::"pseudo-discrete"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#v13_altcat_1 :::"quasi-discrete"::: ) ($#v14_altcat_1 :::"pseudo-discrete"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; definitionmode discrete_category is ($#v13_altcat_1 :::"quasi-discrete"::: ) ($#v14_altcat_1 :::"pseudo-discrete"::: ) ($#l2_altcat_1 :::"category":::); end; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"DiscrCat"::: "A" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v13_altcat_1 :::"quasi-discrete"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) means :: ALTCAT_1:def 20 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "A") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"object":::) "of" it "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "i")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"DiscrCat"::: ALTCAT_1:def 20 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v13_altcat_1 :::"quasi-discrete"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_altcat_1 :::"DiscrCat"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "b2")) "holds" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "i")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" ) ")" ) ")" ))); registration cluster ($#v13_altcat_1 :::"quasi-discrete"::: ) -> ($#v2_altcat_1 :::"transitive"::: ) for ($#l2_altcat_1 :::"AltCatStr"::: ) ; end; theorem :: ALTCAT_1:24 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set "(" ($#k9_altcat_1 :::"DiscrCat"::: ) (Set (Var "A")) ")" ) "st" (Bool (Bool "(" (Bool (Set (Var "o1")) ($#r1_hidden :::"<>"::: ) (Set (Var "o2"))) "or" (Bool (Set (Var "o2")) ($#r1_hidden :::"<>"::: ) (Set (Var "o3"))) ")" )) "holds" (Bool (Set (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set "(" ($#k9_altcat_1 :::"DiscrCat"::: ) (Set (Var "A")) ")" )) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: ALTCAT_1:25 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "o")) "being" ($#m1_subset_1 :::"object":::) "of" (Set "(" ($#k9_altcat_1 :::"DiscrCat"::: ) (Set (Var "A")) ")" ) "holds" (Bool (Set (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set "(" ($#k9_altcat_1 :::"DiscrCat"::: ) (Set (Var "A")) ")" )) ($#k4_altcat_1 :::"."::: ) "(" (Set (Var "o")) "," (Set (Var "o")) "," (Set (Var "o")) ")" ) ($#r1_hidden :::"="::: ) (Set "(" (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) "," (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ) ")" ($#k17_funcop_1 :::":->"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "o")) ")" ))))) ; registrationlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k9_altcat_1 :::"DiscrCat"::: ) "A") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_altcat_1 :::"strict"::: ) ($#v10_altcat_1 :::"pseudo-functional"::: ) ($#v11_altcat_1 :::"associative"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#v13_altcat_1 :::"quasi-discrete"::: ) ($#v14_altcat_1 :::"pseudo-discrete"::: ) ; end;