:: ALTCAT_4 semantic presentation begin registrationlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; let "o" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "C")); cluster (Set ($#k1_altcat_1 :::"<^"::: ) "o" "," "o" ($#k1_altcat_1 :::"^>"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: ALTCAT_4:1 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "," (Set (Var "o3")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o3")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o3")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "v")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k1_altcat_3 :::"""::: ) ")" ) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o2")))) & (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 ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o3")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_altcat_3 :::"""::: ) ")" ) ($#k5_altcat_1 :::"*"::: ) (Set (Var "u"))))))))) ; theorem :: ALTCAT_4:2 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o2")) "," (Set (Var "o3")) "," (Set (Var "o1")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o3")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o3")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k5_altcat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_altcat_3 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o2")))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#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 (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k5_altcat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_altcat_3 :::"""::: ) ")" )))))))) ; theorem :: ALTCAT_4:3 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) )) "holds" (Bool (Set (Set (Var "m")) ($#k1_altcat_3 :::"""::: ) ) "is" ($#v3_altcat_3 :::"iso"::: ) )))) ; theorem :: ALTCAT_4:4 (Bool "for" (Set (Var "C")) "being" ($#~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 "(" (Bool (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o"))) "is" ($#v5_altcat_3 :::"epi"::: ) ) & (Bool (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o"))) "is" ($#v4_altcat_3 :::"mono"::: ) ) ")" ))) ; registrationlet "C" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) ; let "o" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "C")); cluster (Set ($#k8_altcat_1 :::"idm"::: ) "o") -> ($#v1_altcat_3 :::"retraction"::: ) ($#v2_altcat_3 :::"coretraction"::: ) ($#v4_altcat_3 :::"mono"::: ) ($#v5_altcat_3 :::"epi"::: ) ; end; registrationlet "C" be ($#l2_altcat_1 :::"category":::); let "o" be ($#m1_subset_1 :::"object":::) "of" (Set (Const "C")); cluster (Set ($#k8_altcat_1 :::"idm"::: ) "o") -> ($#v3_altcat_3 :::"iso"::: ) ; end; theorem :: ALTCAT_4:5 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o1")) "st" (Bool (Bool (Set (Set (Var "h")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o1")))) & (Bool (Set (Set (Var "f")) ($#k5_altcat_1 :::"*"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set (Var "o2")))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "h"))))))) ; theorem :: ALTCAT_4:6 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool (Set (Var "f")) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) ")" )) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "b")) "," (Set (Var "a")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "g")) "is" ($#v3_altcat_3 :::"iso"::: ) )))) ; begin theorem :: ALTCAT_4:7 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "," (Set (Var "m9")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "m")) "is" ($#v9_altcat_3 :::"_zero"::: ) ) & (Bool (Set (Var "m9")) "is" ($#v9_altcat_3 :::"_zero"::: ) ) & (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Set (Var "O")) "is" ($#v8_altcat_3 :::"_zero"::: ) ))) "holds" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "m9")))))) ; theorem :: ALTCAT_4:8 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "O")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "O")) "," (Set (Var "A")) "st" (Bool (Bool (Set (Var "O")) "is" ($#v7_altcat_3 :::"terminal"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v4_altcat_3 :::"mono"::: ) )))) ; theorem :: ALTCAT_4:9 (Bool "for" (Set (Var "C")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "O")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "A")) "," (Set (Var "O")) "st" (Bool (Bool (Set (Var "O")) "is" ($#v6_altcat_3 :::"initial"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v5_altcat_3 :::"epi"::: ) )))) ; theorem :: ALTCAT_4:10 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o2")) "," (Set (Var "o1")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "o2")) "is" ($#v7_altcat_3 :::"terminal"::: ) ) & (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Var "o1")) "is" ($#v7_altcat_3 :::"terminal"::: ) ))) ; theorem :: ALTCAT_4:11 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "o1")) "is" ($#v6_altcat_3 :::"initial"::: ) ) & (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Var "o2")) "is" ($#v6_altcat_3 :::"initial"::: ) ))) ; theorem :: ALTCAT_4:12 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "o1")) "is" ($#v6_altcat_3 :::"initial"::: ) ) & (Bool (Set (Var "o2")) "is" ($#v7_altcat_3 :::"terminal"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "o2")) "is" ($#v6_altcat_3 :::"initial"::: ) ) & (Bool (Set (Var "o1")) "is" ($#v7_altcat_3 :::"terminal"::: ) ) ")" ))) ; begin theorem :: ALTCAT_4:13 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set "(" ($#k8_altcat_1 :::"idm"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_altcat_1 :::"idm"::: ) (Set "(" (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "a")) ")" )))))) ; theorem :: ALTCAT_4:14 (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v11_functor0 :::"Contravariant"::: ) ($#l2_functor0 :::"FunctorStr"::: ) "over" (Set (Var "C1")) "," (Set (Var "C2")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) "iff" (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C1")) "holds" (Bool (Set ($#k7_functor0 :::"Morph-Map"::: ) "(" (Set (Var "F")) "," (Set (Var "o2")) "," (Set (Var "o1")) ")" ) "is" ($#v2_funct_2 :::"onto"::: ) )) ")" ))) ; theorem :: ALTCAT_4:15 (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v11_functor0 :::"Contravariant"::: ) ($#l2_functor0 :::"FunctorStr"::: ) "over" (Set (Var "C1")) "," (Set (Var "C2")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) "iff" (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C1")) "holds" (Bool (Set ($#k7_functor0 :::"Morph-Map"::: ) "(" (Set (Var "F")) "," (Set (Var "o2")) "," (Set (Var "o1")) ")" ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ")" ))) ; theorem :: ALTCAT_4:16 (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v10_functor0 :::"Covariant"::: ) ($#l2_functor0 :::"FunctorStr"::: ) "over" (Set (Var "C1")) "," (Set (Var "C2")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C1")) (Bool "for" (Set (Var "Fm")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set "(" (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1")) ")" ) "," (Set "(" (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2")) ")" ) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v8_functor0 :::"feasible"::: ) )) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Set (Var "Fm")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "m"))))))))) ; theorem :: ALTCAT_4:17 (Bool "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v11_functor0 :::"Contravariant"::: ) ($#l2_functor0 :::"FunctorStr"::: ) "over" (Set (Var "C1")) "," (Set (Var "C2")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C1")) (Bool "for" (Set (Var "Fm")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set "(" (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2")) ")" ) "," (Set "(" (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1")) ")" ) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v8_functor0 :::"feasible"::: ) )) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Set (Var "Fm")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "m"))))))))) ; theorem :: ALTCAT_4:18 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v1_altcat_3 :::"retraction"::: ) ))))) ; theorem :: ALTCAT_4:19 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v2_altcat_3 :::"coretraction"::: ) ))))) ; theorem :: ALTCAT_4:20 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v3_altcat_3 :::"iso"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v3_altcat_3 :::"iso"::: ) ))))) ; theorem :: ALTCAT_4:21 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1"))) "," (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2"))) ($#r2_altcat_3 :::"are_iso"::: ) )))) ; theorem :: ALTCAT_4:22 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v2_altcat_3 :::"coretraction"::: ) ))))) ; theorem :: ALTCAT_4:23 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v1_altcat_3 :::"retraction"::: ) ))))) ; theorem :: ALTCAT_4:24 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "a")) "is" ($#v3_altcat_3 :::"iso"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v3_altcat_3 :::"iso"::: ) ))))) ; theorem :: ALTCAT_4:25 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2"))) "," (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1"))) ($#r2_altcat_3 :::"are_iso"::: ) )))) ; theorem :: ALTCAT_4:26 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v1_altcat_3 :::"retraction"::: ) ))))) ; theorem :: ALTCAT_4:27 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ))))) ; theorem :: ALTCAT_4:28 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k6_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v3_altcat_3 :::"iso"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v3_altcat_3 :::"iso"::: ) ))))) ; theorem :: ALTCAT_4:29 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v15_functor0 :::"covariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1"))) "," (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2"))) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )))) ; theorem :: ALTCAT_4:30 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ))))) ; theorem :: ALTCAT_4:31 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v12_altcat_1 :::"with_units"::: ) ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v1_altcat_3 :::"retraction"::: ) ))))) ; theorem :: ALTCAT_4:32 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k8_functor0 :::"."::: ) (Set (Var "a"))) "is" ($#v3_altcat_3 :::"iso"::: ) )) "holds" (Bool (Set (Var "a")) "is" ($#v3_altcat_3 :::"iso"::: ) ))))) ; theorem :: ALTCAT_4:33 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "F")) "being" ($#v16_functor0 :::"contravariant"::: ) ($#m2_functor0 :::"Functor"::: ) "of" (Set (Var "A")) "," (Set (Var "B")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v18_functor0 :::"full"::: ) ) & (Bool (Set (Var "F")) "is" ($#v17_functor0 :::"faithful"::: ) ) & (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o2"))) "," (Set (Set (Var "F")) ($#k3_functor0 :::"."::: ) (Set (Var "o1"))) ($#r2_altcat_3 :::"are_iso"::: ) )) "holds" (Bool (Set (Var "o1")) "," (Set (Var "o2")) ($#r2_altcat_3 :::"are_iso"::: ) )))) ; begin theorem :: ALTCAT_4:34 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"AltCatStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "D")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "D"))))) "holds" (Bool (Set (Var "D")) "is" ($#v2_altcat_2 :::"full"::: ) ))) ; theorem :: ALTCAT_4:35 (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 "D")) "being" ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "D")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "D"))))) "holds" (Bool (Set (Var "D")) "is" ($#v3_altcat_2 :::"id-inheriting"::: ) ))) ; registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v2_altcat_2 :::"full"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) for ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C"; end; theorem :: ALTCAT_4:36 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "B")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "A")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "B")) "holds" (Bool (Set (Var "A")) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "C")))))) ; theorem :: ALTCAT_4:37 (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 "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "p1")) "," (Set (Var "p2")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "o1"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "m")) "is" ($#v4_altcat_3 :::"mono"::: ) )) "implies" (Bool (Set (Var "n")) "is" ($#v4_altcat_3 :::"mono"::: ) ) ")" & "(" (Bool (Bool (Set (Var "m")) "is" ($#v5_altcat_3 :::"epi"::: ) )) "implies" (Bool (Set (Var "n")) "is" ($#v5_altcat_3 :::"epi"::: ) ) ")" ")" ))))))) ; theorem :: ALTCAT_4:38 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "m1")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o2")) "," (Set (Var "o1")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "p1")) "," (Set (Var "p2")) (Bool "for" (Set (Var "n1")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "p2")) "," (Set (Var "p1")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "o1"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "m1")) ($#r1_hidden :::"="::: ) (Set (Var "n1"))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p2")) "," (Set (Var "p1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "m")) ($#r1_altcat_3 :::"is_left_inverse_of"::: ) (Set (Var "m1")))) "implies" (Bool (Set (Var "n")) ($#r1_altcat_3 :::"is_left_inverse_of"::: ) (Set (Var "n1"))) ")" & "(" (Bool (Bool (Set (Var "n")) ($#r1_altcat_3 :::"is_left_inverse_of"::: ) (Set (Var "n1")))) "implies" (Bool (Set (Var "m")) ($#r1_altcat_3 :::"is_left_inverse_of"::: ) (Set (Var "m1"))) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_altcat_3 :::"is_right_inverse_of"::: ) (Set (Var "m1")))) "implies" (Bool (Set (Var "n")) ($#r1_altcat_3 :::"is_right_inverse_of"::: ) (Set (Var "n1"))) ")" & "(" (Bool (Bool (Set (Var "n")) ($#r1_altcat_3 :::"is_right_inverse_of"::: ) (Set (Var "n1")))) "implies" (Bool (Set (Var "m")) ($#r1_altcat_3 :::"is_right_inverse_of"::: ) (Set (Var "m1"))) ")" ")" ))))))))) ; theorem :: ALTCAT_4:39 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_2 :::"full"::: ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "p1")) "," (Set (Var "p2")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "o1"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p2")) "," (Set (Var "p1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "m")) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "implies" (Bool (Set (Var "n")) "is" ($#v1_altcat_3 :::"retraction"::: ) ) ")" & "(" (Bool (Bool (Set (Var "m")) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "implies" (Bool (Set (Var "n")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ) ")" & "(" (Bool (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) )) "implies" (Bool (Set (Var "n")) "is" ($#v3_altcat_3 :::"iso"::: ) ) ")" ")" ))))))) ; theorem :: ALTCAT_4:40 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "p1")) "," (Set (Var "p2")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set (Var "o1"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set (Var "o2"))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "p2")) "," (Set (Var "p1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) "is" ($#v1_altcat_3 :::"retraction"::: ) )) "implies" (Bool (Set (Var "m")) "is" ($#v1_altcat_3 :::"retraction"::: ) ) ")" & "(" (Bool (Bool (Set (Var "n")) "is" ($#v2_altcat_3 :::"coretraction"::: ) )) "implies" (Bool (Set (Var "m")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ) ")" & "(" (Bool (Bool (Set (Var "n")) "is" ($#v3_altcat_3 :::"iso"::: ) )) "implies" (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) ) ")" ")" ))))))) ; definitionlet "C" be ($#l2_altcat_1 :::"category":::); func :::"AllMono"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C" means :: ALTCAT_4:def 1 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "C")) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v4_altcat_3 :::"mono"::: ) ) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"AllMono"::: ALTCAT_4:def 1 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C")))) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v4_altcat_3 :::"mono"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" ))); registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster (Set ($#k1_altcat_4 :::"AllMono"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) ; end; definitionlet "C" be ($#l2_altcat_1 :::"category":::); func :::"AllEpi"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C" means :: ALTCAT_4:def 2 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "C")) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v5_altcat_3 :::"epi"::: ) ) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"AllEpi"::: ALTCAT_4:def 2 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C")))) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v5_altcat_3 :::"epi"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" ))); registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster (Set ($#k2_altcat_4 :::"AllEpi"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) ; end; definitionlet "C" be ($#l2_altcat_1 :::"category":::); func :::"AllRetr"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C" means :: ALTCAT_4:def 3 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "C")) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v1_altcat_3 :::"retraction"::: ) ) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"AllRetr"::: ALTCAT_4:def 3 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_altcat_4 :::"AllRetr"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C")))) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v1_altcat_3 :::"retraction"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" ))); registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster (Set ($#k3_altcat_4 :::"AllRetr"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) ; end; definitionlet "C" be ($#l2_altcat_1 :::"category":::); func :::"AllCoretr"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C" means :: ALTCAT_4:def 4 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "C")) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"AllCoretr"::: ALTCAT_4:def 4 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_altcat_4 :::"AllCoretr"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C")))) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" ))); registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster (Set ($#k4_altcat_4 :::"AllCoretr"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) ; end; definitionlet "C" be ($#l2_altcat_1 :::"category":::); func :::"AllIso"::: "C" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" "C" means :: ALTCAT_4:def 5 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "C")) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" "C")) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" "C" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) ) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"AllIso"::: ALTCAT_4:def 5 : (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#m1_altcat_2 :::"SubCatStr"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C")))) & (Bool (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#r2_altcat_2 :::"cc="::: ) (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C")))) & (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "o1")) "," (Set (Var "o2")) ")" )) "iff" (Bool "(" (Bool (Set ($#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 "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" ))); registrationlet "C" be ($#l2_altcat_1 :::"category":::); cluster (Set ($#k5_altcat_4 :::"AllIso"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_altcat_1 :::"transitive"::: ) ($#v6_altcat_1 :::"strict"::: ) ($#v3_altcat_2 :::"id-inheriting"::: ) ; end; theorem :: ALTCAT_4:41 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set ($#k3_altcat_4 :::"AllRetr"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:42 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set ($#k4_altcat_4 :::"AllCoretr"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:43 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k4_altcat_4 :::"AllCoretr"::: ) (Set (Var "C"))) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:44 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k3_altcat_4 :::"AllRetr"::: ) (Set (Var "C"))) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_altcat_2 :::"subcategory":::) "of" (Set ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:45 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool (Set (Var "m")) "is" ($#v4_altcat_3 :::"mono"::: ) )) ")" )) "holds" (Bool (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:46 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool (Set (Var "m")) "is" ($#v5_altcat_3 :::"epi"::: ) )) ")" )) "holds" (Bool (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:47 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) "is" ($#v1_altcat_3 :::"retraction"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ")" )) "holds" (Bool (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k3_altcat_4 :::"AllRetr"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:48 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) "is" ($#v2_altcat_3 :::"coretraction"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ")" )) "holds" (Bool (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k4_altcat_4 :::"AllCoretr"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:49 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "st" (Bool (Bool "(" "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set (Var "C")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "holds" (Bool "(" (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) ) & (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ")" )) "holds" (Bool (Set ($#g2_altcat_1 :::"AltCatStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u1_altcat_1 :::"Arrows"::: ) "of" (Set (Var "C"))) "," (Set "the" ($#u2_altcat_1 :::"Comp"::: ) "of" (Set (Var "C"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:50 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set "(" ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "m")) "is" ($#v4_altcat_3 :::"mono"::: ) )))) ; theorem :: ALTCAT_4:51 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set "(" ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "m")) "is" ($#v5_altcat_3 :::"epi"::: ) )))) ; theorem :: ALTCAT_4:52 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"object":::) "of" (Set "(" ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Morphism":::) "of" (Set (Var "o1")) "," (Set (Var "o2")) "st" (Bool (Bool (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o1")) "," (Set (Var "o2")) ($#k1_altcat_1 :::"^>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "m")) "is" ($#v3_altcat_3 :::"iso"::: ) ) & (Bool (Set (Set (Var "m")) ($#k1_altcat_3 :::"""::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_altcat_1 :::"<^"::: ) (Set (Var "o2")) "," (Set (Var "o1")) ($#k1_altcat_1 :::"^>"::: ) )) ")" )))) ; theorem :: ALTCAT_4:53 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k1_altcat_4 :::"AllMono"::: ) (Set "(" ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:54 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k2_altcat_4 :::"AllEpi"::: ) (Set "(" ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:55 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set "(" ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:56 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set "(" ($#k1_altcat_4 :::"AllMono"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:57 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set "(" ($#k2_altcat_4 :::"AllEpi"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:58 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set "(" ($#k3_altcat_4 :::"AllRetr"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ; theorem :: ALTCAT_4:59 (Bool "for" (Set (Var "C")) "being" ($#l2_altcat_1 :::"category":::) "holds" (Bool (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set "(" ($#k4_altcat_4 :::"AllCoretr"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_altcat_4 :::"AllIso"::: ) (Set (Var "C"))))) ;