:: FUNCT_3 semantic presentation begin theorem :: FUNCT_3:1 (Bool "for" (Set (Var "A")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "Y")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))))) ; theorem :: FUNCT_3:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))))) ; theorem :: FUNCT_3:3 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FUNCT_3:4 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FUNCT_3:5 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" ))))) ; scheme :: FUNCT_3:sch 1 FuncEx3{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set (Var "x")) "," (Set (Var "y")) "," (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )]) ")" ) ")" )) provided (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) & (Bool P1[(Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z1"))]) & (Bool P1[(Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z2"))])) "holds" (Bool (Set (Var "z1")) ($#r1_hidden :::"="::: ) (Set (Var "z2")))) and (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool P1[(Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z"))]))) proof end; scheme :: FUNCT_3:sch 2 Lambda3{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) ")" ) ")" )) proof end; theorem :: FUNCT_3:6 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); func :::".:"::: "f" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 1 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "f" ")" ))) & (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))) ")" ) ")" ); end; :: deftheorem defines :::".:"::: FUNCT_3:def 1 : (Bool "for" (Set (Var "f")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_funct_3 :::".:"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))) ")" ) ")" ) ")" )); theorem :: FUNCT_3:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")))))) ; theorem :: FUNCT_3:8 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FUNCT_3:9 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )))) ; theorem :: FUNCT_3:10 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FUNCT_3:11 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FUNCT_3:12 (Bool "for" (Set (Var "X")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D")) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X"))))))) ; theorem :: FUNCT_3:13 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FUNCT_3:14 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FUNCT_3:15 (Bool "for" (Set (Var "X")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" )))))) ; theorem :: FUNCT_3:16 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "B")) ")" ))))) ; theorem :: FUNCT_3:17 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B")) ")" ))))) ; theorem :: FUNCT_3:18 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_funct_3 :::".:"::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "g")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" )))) ; theorem :: FUNCT_3:19 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_funct_3 :::".:"::: ) (Set (Var "f"))) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ")" ))) ; theorem :: FUNCT_3:20 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "holds" (Bool (Set ($#k1_funct_3 :::".:"::: ) (Set (Var "f"))) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "Y")) ")" )))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "D")); :: original: :::".:"::: redefine func :::".:"::: "f" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "D" ")" ); end; definitionlet "f" be ($#m1_hidden :::"Function":::); func :::"""::: "f" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 2 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) "f" ")" ))) & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k8_relat_1 :::"""::: ) (Set (Var "Y")))) ")" ) ")" ); end; :: deftheorem defines :::"""::: FUNCT_3:def 2 : (Bool "for" (Set (Var "f")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_funct_3 :::"""::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")))) ")" ) ")" ) ")" )); theorem :: FUNCT_3:21 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")))))) ; theorem :: FUNCT_3:22 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )))) ; theorem :: FUNCT_3:23 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FUNCT_3:24 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FUNCT_3:25 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "B")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "B")) ")" ))))) ; theorem :: FUNCT_3:26 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "B")) ")" ))))) ; theorem :: FUNCT_3:27 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FUNCT_3:28 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FUNCT_3:29 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B")))))) ; theorem :: FUNCT_3:30 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "B")))))) ; theorem :: FUNCT_3:31 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")))))) ; theorem :: FUNCT_3:32 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A")))))) ; theorem :: FUNCT_3:33 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_funct_3 :::".:"::: ) (Set (Var "f")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")))))) ; theorem :: FUNCT_3:34 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k3_funct_3 :::"""::: ) (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "f")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k3_funct_3 :::"""::: ) (Set (Var "g")) ")" )))) ; theorem :: FUNCT_3:35 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_funct_3 :::"""::: ) (Set (Var "f"))) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ")" ))) ; definitionlet "A", "X" be ($#m1_hidden :::"set"::: ) ; func :::"chi"::: "(" "A" "," "X" ")" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 3 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "A")) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) "A"))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"chi"::: FUNCT_3:def 3 : (Bool "for" (Set (Var "A")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "A")) "," (Set (Var "X")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "implies" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "implies" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ) ")" ) ")" ) ")" ))); theorem :: FUNCT_3:36 (Bool "for" (Set (Var "x")) "," (Set (Var "A")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set "(" ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "A")) "," (Set (Var "X")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ; theorem :: FUNCT_3:37 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "A")) "," (Set (Var "X")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FUNCT_3:38 (Bool "for" (Set (Var "A")) "," (Set (Var "X")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "A")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "B")) "," (Set (Var "X")) ")" ))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) ; theorem :: FUNCT_3:39 (Bool "for" (Set (Var "A")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k4_funct_3 :::"chi"::: ) "(" (Set (Var "A")) "," (Set (Var "X")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Num 1) ($#k2_tarski :::"}"::: ) ))) ; theorem :: FUNCT_3:40 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Num 1) ($#k2_tarski :::"}"::: ) ) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_funct_3 :::"chi"::: ) "(" (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Num 1) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set (Var "X")) ")" )))) ; definitionlet "A", "X" be ($#m1_hidden :::"set"::: ) ; :: original: :::"chi"::: redefine func :::"chi"::: "(" "A" "," "X" ")" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Num 1) ($#k2_tarski :::"}"::: ) ); end; notationlet "Y" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); synonym :::"incl"::: "A" for :::"id"::: "Y"; end; definitionlet "Y" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); :: original: :::"incl"::: redefine func :::"incl"::: "A" -> ($#m1_subset_1 :::"Function":::) "of" "A" "," "Y"; end; theorem :: FUNCT_3:41 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool (Set ($#k6_funct_3 :::"incl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "Y")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")))))) ; theorem :: FUNCT_3:42 (Bool "for" (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" ($#k6_funct_3 :::"incl"::: ) (Set (Var "A")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))))) ; definitionlet "X", "Y" be ($#m1_hidden :::"set"::: ) ; func :::"pr1"::: "(" "X" "," "Y" ")" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 4 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "Y" ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "Y")) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ); func :::"pr2"::: "(" "X" "," "Y" ")" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 5 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "Y" ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "Y")) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ) ")" ); end; :: deftheorem defines :::"pr1"::: FUNCT_3:def 4 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_funct_3 :::"pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"pr2"::: FUNCT_3:def 5 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_funct_3 :::"pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ) ")" ) ")" ))); theorem :: FUNCT_3:43 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k7_funct_3 :::"pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) ; theorem :: FUNCT_3:44 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k7_funct_3 :::"pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "X")))) ; theorem :: FUNCT_3:45 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k8_funct_3 :::"pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) ; theorem :: FUNCT_3:46 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k8_funct_3 :::"pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) ; definitionlet "X", "Y" be ($#m1_hidden :::"set"::: ) ; :: original: :::"pr1"::: redefine func :::"pr1"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "Y" ($#k2_zfmisc_1 :::":]"::: ) ) "," "X"; :: original: :::"pr2"::: redefine func :::"pr2"::: "(" "X" "," "Y" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "Y" ($#k2_zfmisc_1 :::":]"::: ) ) "," "Y"; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"delta"::: "X" -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 6 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"delta"::: FUNCT_3:def 6 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k11_funct_3 :::"delta"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ) ")" ))); theorem :: FUNCT_3:47 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k11_funct_3 :::"delta"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "X")) ($#k2_zfmisc_1 :::":]"::: ) ))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; :: original: :::"delta"::: redefine func :::"delta"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "X" ($#k2_zfmisc_1 :::":]"::: ) ); end; definitionlet "f", "g" be ($#m1_hidden :::"Function":::); func :::"<:":::"f" "," "g":::":>"::: -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 7 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "f" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "g" ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" "g" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"<:"::: FUNCT_3:def 7 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ) ")" )); registrationlet "f" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#m1_hidden :::"Function":::); cluster (Set ($#k13_funct_3 :::"<:"::: ) "f" "," "g" ($#k13_funct_3 :::":>"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; cluster (Set ($#k13_funct_3 :::"<:"::: ) "g" "," "f" ($#k13_funct_3 :::":>"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: FUNCT_3:48 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" )))) "holds" (Bool (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k4_tarski :::"]"::: ) )))) ; theorem :: FUNCT_3:49 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k4_tarski :::"]"::: ) )))) ; theorem :: FUNCT_3:50 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X"))))) ; theorem :: FUNCT_3:51 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) ; theorem :: FUNCT_3:52 (Bool "for" (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "Y")) "," (Set (Var "Z")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "Y")) "," (Set (Var "Z")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) ; theorem :: FUNCT_3:53 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: FUNCT_3:54 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "k")) "," (Set (Var "h")) ($#k13_funct_3 :::":>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "k"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "h"))) ")" )) ; theorem :: FUNCT_3:55 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h")) ")" ) "," (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h")) ")" ) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set (Var "h"))))) ; theorem :: FUNCT_3:56 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: FUNCT_3:57 (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#k8_relat_1 :::"""::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "C")) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "B")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "g")) ($#k8_relat_1 :::"""::: ) (Set (Var "C")) ")" )))))) ; theorem :: FUNCT_3:58 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Z")) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "holds" (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "Y")) "," (Set (Var "Z")) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "D1", "D2" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f1" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "D1")); let "f2" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "D2")); :: original: :::"<:"::: redefine func :::"<:":::"f1" "," "f2":::":>"::: -> ($#m1_subset_1 :::"Function":::) "of" "X" "," (Set ($#k2_zfmisc_1 :::"[:"::: ) "D1" "," "D2" ($#k2_zfmisc_1 :::":]"::: ) ); end; theorem :: FUNCT_3:59 (Bool "for" (Set (Var "C")) "," (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "C")) "," (Set (Var "D1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "C")) "," (Set (Var "D2")) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "holds" (Bool (Set (Set ($#k14_funct_3 :::"<:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k14_funct_3 :::":>"::: ) ) ($#k3_funct_2 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "c")) ")" ) "," (Set "(" (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "c")) ")" ) ($#k4_tarski :::"]"::: ) )))))) ; theorem :: FUNCT_3:60 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Z")) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "Y")) "," (Set (Var "Z")) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: FUNCT_3:61 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Z")) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "Y")) "," (Set (Var "Z")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "Y")) "," (Set (Var "Z")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" )))) ; theorem :: FUNCT_3:62 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D1")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D2")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "D1")) "," (Set (Var "D2")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k14_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k14_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "D1")) "," (Set (Var "D2")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k14_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k14_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))))) ; theorem :: FUNCT_3:63 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Z")) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f1")) "," (Set (Var "g1")) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f2")) "," (Set (Var "g2")) ($#k13_funct_3 :::":>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2"))) & (Bool (Set (Var "g1")) ($#r2_funct_2 :::"="::: ) (Set (Var "g2"))) ")" )))) ; theorem :: FUNCT_3:64 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D1")) (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "D2")) "st" (Bool (Bool (Set ($#k14_funct_3 :::"<:"::: ) (Set (Var "f1")) "," (Set (Var "g1")) ($#k14_funct_3 :::":>"::: ) ) ($#r2_funct_2 :::"="::: ) (Set ($#k14_funct_3 :::"<:"::: ) (Set (Var "f2")) "," (Set (Var "g2")) ($#k14_funct_3 :::":>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2"))) & (Bool (Set (Var "g1")) ($#r2_funct_2 :::"="::: ) (Set (Var "g2"))) ")" ))))) ; definitionlet "f", "g" be ($#m1_hidden :::"Function":::); func :::"[:":::"f" "," "g":::":]"::: -> ($#m1_hidden :::"Function":::) means :: FUNCT_3:def 8 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "f" ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "g" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "g"))) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" "g" ($#k1_funct_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"[:"::: FUNCT_3:def 8 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ) ")" )); theorem :: FUNCT_3:65 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) "holds" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k4_tarski :::"]"::: ) )))) ; theorem :: FUNCT_3:66 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ")" ")" ) ")" ) "," (Set "(" (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ")" ")" ) ")" ) ($#k13_funct_3 :::":>"::: ) ))) ; theorem :: FUNCT_3:67 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) ; theorem :: FUNCT_3:68 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k12_funct_3 :::"delta"::: ) (Set (Var "X")) ")" ))))) ; theorem :: FUNCT_3:69 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k15_funct_3 :::"[:"::: ) (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k4_relat_1 :::"id"::: ) (Set (Var "Y")) ")" ) ($#k15_funct_3 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: FUNCT_3:70 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "h")) ($#k15_funct_3 :::":]"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "g")) "," (Set (Var "k")) ($#k13_funct_3 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "h")) ($#k3_relat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k13_funct_3 :::":>"::: ) ))) ; theorem :: FUNCT_3:71 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "h")) ($#k15_funct_3 :::":]"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "g")) "," (Set (Var "k")) ($#k15_funct_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k15_funct_3 :::"[:"::: ) (Set "(" (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "h")) ($#k3_relat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k15_funct_3 :::":]"::: ) ))) ; theorem :: FUNCT_3:72 (Bool "for" (Set (Var "B")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "A")) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set (Var "B")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_relat_1 :::".:"::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: FUNCT_3:73 (Bool "for" (Set (Var "B")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k8_relat_1 :::"""::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "A")) ($#k2_zfmisc_1 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set (Var "B")) ")" ) "," (Set "(" (Set (Var "g")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: FUNCT_3:74 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "V")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "Z")) "holds" (Bool (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "V")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "Y")) "," (Set (Var "Z")) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; definitionlet "X1", "X2", "Y1", "Y2" be ($#m1_hidden :::"set"::: ) ; let "f1" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X1")) "," (Set (Const "Y1")); let "f2" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X2")) "," (Set (Const "Y2")); :: original: :::"[:"::: redefine func :::"[:":::"f1" "," "f2":::":]"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) "X1" "," "X2" ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) "Y1" "," "Y2" ($#k2_zfmisc_1 :::":]"::: ) ); end; theorem :: FUNCT_3:75 (Bool "for" (Set (Var "C1")) "," (Set (Var "D1")) "," (Set (Var "C2")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "C1")) "," (Set (Var "D1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "C2")) "," (Set (Var "D2")) (Bool "for" (Set (Var "c1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C1")) (Bool "for" (Set (Var "c2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C2")) "holds" (Bool (Set (Set ($#k16_funct_3 :::"[:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k16_funct_3 :::":]"::: ) ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "c1")) "," (Set (Var "c2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f1")) ($#k3_funct_2 :::"."::: ) (Set (Var "c1")) ")" ) "," (Set "(" (Set (Var "f2")) ($#k3_funct_2 :::"."::: ) (Set (Var "c2")) ")" ) ($#k4_tarski :::"]"::: ) ))))))) ; theorem :: FUNCT_3:76 (Bool "for" (Set (Var "X1")) "," (Set (Var "Y1")) "," (Set (Var "X2")) "," (Set (Var "Y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X1")) "," (Set (Var "Y1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X2")) "," (Set (Var "Y2")) "st" (Bool "(" (Bool (Bool (Set (Var "Y1")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X1")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "Y2")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "X2")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "holds" (Bool (Set ($#k16_funct_3 :::"[:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k16_funct_3 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" (Set (Var "f1")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "X1")) "," (Set (Var "X2")) ")" ")" ) ")" ) "," (Set "(" (Set (Var "f2")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "X1")) "," (Set (Var "X2")) ")" ")" ) ")" ) ($#k13_funct_3 :::":>"::: ) ))))) ; theorem :: FUNCT_3:77 (Bool "for" (Set (Var "X1")) "," (Set (Var "X2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X1")) "," (Set (Var "D1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X2")) "," (Set (Var "D2")) "holds" (Bool (Set ($#k16_funct_3 :::"[:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k16_funct_3 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k13_funct_3 :::"<:"::: ) (Set "(" (Set (Var "f1")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "X1")) "," (Set (Var "X2")) ")" ")" ) ")" ) "," (Set "(" (Set (Var "f2")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "X1")) "," (Set (Var "X2")) ")" ")" ) ")" ) ($#k13_funct_3 :::":>"::: ) )))))) ; theorem :: FUNCT_3:78 (Bool "for" (Set (Var "X")) "," (Set (Var "Y1")) "," (Set (Var "Y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y2")) "holds" (Bool (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k13_funct_3 :::":>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k16_funct_3 :::"[:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k16_funct_3 :::":]"::: ) ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k12_funct_3 :::"delta"::: ) (Set (Var "X")) ")" )))))) ; begin theorem :: FUNCT_3:79 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ")" ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) ; theorem :: FUNCT_3:80 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "B")) "," (Set (Var "C")) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")))) & (Bool (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "B")) "," (Set (Var "C")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "f")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))))) ; registrationlet "F", "G" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"Function":::); cluster (Set ($#k15_funct_3 :::"[:"::: ) "F" "," "G" ($#k15_funct_3 :::":]"::: ) ) -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; registrationlet "A" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "A" ($#k2_zfmisc_1 :::":]"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "A" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v3_binop_1 :::"idempotent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "A" ($#k2_zfmisc_1 :::":]"::: ) ) "," "A" ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "A" be ($#m1_hidden :::"set"::: ) ; let "b" be ($#v3_binop_1 :::"idempotent"::: ) ($#m1_subset_1 :::"BinOp":::) "of" (Set (Const "A")); let "a" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "A")); reduce ; end;