:: CARD_3 semantic presentation begin definitionlet "IT" be ($#m1_hidden :::"Function":::); attr "IT" is :::"Cardinal-yielding"::: means :: CARD_3:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "IT"))) "holds" (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#m1_hidden :::"Cardinal":::))); end; :: deftheorem defines :::"Cardinal-yielding"::: CARD_3:def 1 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_card_3 :::"Cardinal-yielding"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "IT"))))) "holds" (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) "is" ($#m1_hidden :::"Cardinal":::))) ")" )); registration cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v1_card_3 :::"Cardinal-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_card_3 :::"Cardinal-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionmode Cardinal-Function is ($#v1_card_3 :::"Cardinal-yielding"::: ) ($#m1_hidden :::"Function":::); end; registrationlet "ff" be ($#m1_hidden :::"Cardinal-Function":::); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "ff" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_card_3 :::"Cardinal-yielding"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "K" be ($#m1_hidden :::"Cardinal":::); cluster (Set "X" ($#k2_funcop_1 :::"-->"::: ) "K") -> ($#v1_card_3 :::"Cardinal-yielding"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "K" be ($#m1_hidden :::"Cardinal":::); cluster (Set "X" ($#k16_funcop_1 :::".-->"::: ) "K") -> ($#v1_card_3 :::"Cardinal-yielding"::: ) ; end; scheme :: CARD_3:sch 1 CFLambda{ F1() -> ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"Cardinal":::) } : (Bool "ex" (Set (Var "ff")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "ff"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool (Set (Set (Var "ff")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "x")) ")" )) ")" ) ")" )) proof end; definitionlet "f" be ($#m1_hidden :::"Function":::); func :::"Card"::: "f" -> ($#m1_hidden :::"Cardinal-Function":::) means :: CARD_3:def 2 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ); func :::"disjoin"::: "f" -> ($#m1_hidden :::"Function":::) means :: CARD_3:def 3 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" ) ")" ); func :::"Union"::: "f" -> ($#m1_hidden :::"set"::: ) equals :: CARD_3:def 4 (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) "f" ")" )); func :::"product"::: "f" -> ($#m1_hidden :::"set"::: ) means :: CARD_3:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"Card"::: CARD_3:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_card_3 :::"Card"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"disjoin"::: CARD_3:def 3 : (Bool "for" (Set (Var "f")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )) ")" ) ")" ) ")" )); :: deftheorem defines :::"Union"::: CARD_3:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )))); :: deftheorem defines :::"product"::: CARD_3:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) ")" ) ")" )) ")" )) ")" ))); theorem :: CARD_3:1 (Bool "for" (Set (Var "ff")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool (Set ($#k1_card_3 :::"Card"::: ) (Set (Var "ff"))) ($#r1_hidden :::"="::: ) (Set (Var "ff")))) ; theorem :: CARD_3:2 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_card_3 :::"Card"::: ) (Set "(" (Set (Var "X")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k7_funcop_1 :::"-->"::: ) (Set "(" ($#k1_card_1 :::"card"::: ) (Set (Var "Y")) ")" )))) ; theorem :: CARD_3:3 (Bool (Set ($#k2_card_3 :::"disjoin"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: CARD_3:4 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_card_3 :::"disjoin"::: ) (Set "(" (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )))) ; theorem :: CARD_3:5 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))))) ; theorem :: CARD_3:6 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" (Set (Var "X")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "Y")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) ; theorem :: CARD_3:7 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" (Set (Var "X")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) ; theorem :: CARD_3:8 (Bool "for" (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) ; theorem :: CARD_3:9 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ) ")" )) ; theorem :: CARD_3:10 (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: CARD_3:11 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "X")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "Y")) ")" )))) ; definitionlet "x", "X" be ($#m1_hidden :::"set"::: ) ; func :::"pi"::: "(" "X" "," "x" ")" -> ($#m1_hidden :::"set"::: ) means :: CARD_3:def 6 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) "x")) ")" )) ")" )); end; :: deftheorem defines :::"pi"::: CARD_3:def 6 : (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" )) "iff" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) ")" )) ")" )); theorem :: CARD_3:12 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set "(" ($#k4_card_3 :::"product"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: CARD_3:13 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: CARD_3:14 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set ($#k1_tarski :::"{"::: ) (Set (Var "g")) ($#k1_tarski :::"}"::: ) ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: CARD_3:15 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set ($#k2_tarski :::"{"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k2_tarski :::"}"::: ) ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k2_tarski :::"}"::: ) )))) ; theorem :: CARD_3:16 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "Y")) "," (Set (Var "x")) ")" ")" )))) ; theorem :: CARD_3:17 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_card_3 :::"pi"::: ) "(" (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "x")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "Y")) "," (Set (Var "x")) ")" ")" )))) ; theorem :: CARD_3:18 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "Y")) "," (Set (Var "x")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_card_3 :::"pi"::: ) "(" (Set "(" (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "x")) ")" ))) ; theorem :: CARD_3:19 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "Y")) "," (Set (Var "x")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_card_3 :::"pi"::: ) "(" (Set "(" (Set (Var "X")) ($#k5_xboole_0 :::"\+\"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "x")) ")" ))) ; theorem :: CARD_3:20 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "x")) ")" ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) ; theorem :: CARD_3:21 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ))))) ; theorem :: CARD_3:22 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f")) ")" ))) "iff" (Bool "(" (Bool (Set (Set (Var "x")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "x")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k2_xtuple_0 :::"`2"::: ) ")" ))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "x")) ($#k1_xtuple_0 :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "x")) ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ))) ; theorem :: CARD_3:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_tarski :::"c="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k2_card_3 :::"disjoin"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_card_3 :::"disjoin"::: ) (Set (Var "g"))))) ; theorem :: CARD_3:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_tarski :::"c="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_card_3 :::"Union"::: ) (Set (Var "g"))))) ; theorem :: CARD_3:25 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set "(" (Set (Var "Y")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "X")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) ))) ; theorem :: CARD_3:26 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) ")" )) ; theorem :: CARD_3:27 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "g"))))) ; theorem :: CARD_3:28 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: CARD_3:29 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "F")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; definitionlet "F" be ($#m1_hidden :::"Cardinal-Function":::); func :::"Sum"::: "F" -> ($#m1_hidden :::"Cardinal":::) equals :: CARD_3:def 7 (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k2_card_3 :::"disjoin"::: ) "F" ")" ) ")" )); func :::"Product"::: "F" -> ($#m1_hidden :::"Cardinal":::) equals :: CARD_3:def 8 (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) "F" ")" )); end; :: deftheorem defines :::"Sum"::: CARD_3:def 7 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set "(" ($#k2_card_3 :::"disjoin"::: ) (Set (Var "F")) ")" ) ")" )))); :: deftheorem defines :::"Product"::: CARD_3:def 8 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) (Set (Var "F")) ")" )))); theorem :: CARD_3:30 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#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 ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "F"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "G"))))) ; theorem :: CARD_3:31 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool "(" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")))) "iff" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" )) ; theorem :: CARD_3:32 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#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 ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "F"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "G"))))) ; theorem :: CARD_3:33 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "F"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "G"))))) ; theorem :: CARD_3:34 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Bool "not" (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "G")))))) "holds" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "F"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "G"))))) ; theorem :: CARD_3:35 (Bool "for" (Set (Var "K")) "being" ($#m1_hidden :::"Cardinal":::) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) ; theorem :: CARD_3:36 (Bool "for" (Set (Var "K")) "being" ($#m1_hidden :::"Cardinal":::) "holds" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: CARD_3:37 (Bool "for" (Set (Var "K")) "being" ($#m1_hidden :::"Cardinal":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set "(" (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "K"))))) ; theorem :: CARD_3:38 (Bool "for" (Set (Var "K")) "being" ($#m1_hidden :::"Cardinal":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_card_3 :::"Product"::: ) (Set "(" (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "K"))))) ; theorem :: CARD_3:39 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set (Var "f")) ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k6_card_3 :::"Sum"::: ) (Set "(" ($#k1_card_3 :::"Card"::: ) (Set (Var "f")) ")" )))) ; theorem :: CARD_3:40 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Cardinal-Function":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_card_3 :::"Union"::: ) (Set (Var "F")) ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "F"))))) ; theorem :: CARD_3:41 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Cardinal-Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#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 ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set ($#k6_card_3 :::"Sum"::: ) (Set (Var "F"))) ($#r2_hidden :::"in"::: ) (Set ($#k7_card_3 :::"Product"::: ) (Set (Var "G"))))) ; scheme :: CARD_3:sch 2 FuncSeparation{ F1() -> ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set F2 "(" (Set (Var "x")) ")" )) & (Bool P1[(Set (Var "x")) "," (Set (Var "y"))]) ")" ) ")" )) ")" ) ")" )) proof end; theorem :: CARD_3:42 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_1 :::"card"::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )))) ; theorem :: CARD_3:43 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) ; theorem :: CARD_3:44 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ; theorem :: CARD_3:45 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) )) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "X")))) & (Bool "(" "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "Z1")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "Z1")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "Z2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Z2")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "Z1")) ($#r1_tarski :::"c="::: ) (Set (Var "Z2"))) ")" ) ")" )) ")" ) ")" ))) ; theorem :: CARD_3:46 (Bool "for" (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Z")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "Z"))) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) ")" ) & (Bool (Set (Var "X")) "is" ($#v6_ordinal1 :::"c=-linear"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "X")) ")" )) ($#r1_ordinal1 :::"c="::: ) (Set (Var "M"))))) ; begin registrationlet "f" be ($#m1_hidden :::"Function":::); cluster (Set ($#k4_card_3 :::"product"::: ) "f") -> ($#v4_funct_1 :::"functional"::: ) ; end; registrationlet "A" be ($#m1_hidden :::"set"::: ) ; let "B" be ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) -> ($#v2_relat_1 :::"non-empty"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," "B" ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "f" be ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::); cluster (Set ($#k4_card_3 :::"product"::: ) "f") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CARD_3:47 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k4_funct_4 :::"-->"::: ) "(" (Set ($#k1_tarski :::"{"::: ) (Set (Var "c")) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "d")) ($#k1_tarski :::"}"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k4_funct_4 :::"-->"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: CARD_3:48 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"Function":::)))) ; begin definitionlet "f" be ($#m1_hidden :::"Function":::); func :::"sproduct"::: "f" -> ($#m1_hidden :::"set"::: ) means :: CARD_3:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )) ")" )); end; :: deftheorem defines :::"sproduct"::: CARD_3:def 9 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )) ")" )) ")" ))); registrationlet "f" be ($#m1_hidden :::"Function":::); cluster (Set ($#k8_card_3 :::"sproduct"::: ) "f") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ; end; theorem :: CARD_3:49 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )) ; theorem :: CARD_3:50 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) ; registrationlet "f" be ($#m1_hidden :::"Function":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k8_card_3 :::"sproduct"::: ) "f"); end; theorem :: CARD_3:51 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) ; theorem :: CARD_3:52 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ")" )))) ; theorem :: CARD_3:53 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "h"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))) ; theorem :: CARD_3:54 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))) & (Bool (Set (Var "g")) ($#r1_tarski :::"c="::: ) (Set (Var "h"))) ")" )) ")" )) ; theorem :: CARD_3:55 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_partfun1 :::"PFuncs"::: ) "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ; theorem :: CARD_3:56 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_tarski :::"c="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "g"))))) ; theorem :: CARD_3:57 (Bool (Set ($#k8_card_3 :::"sproduct"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ; theorem :: CARD_3:58 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_partfun1 :::"PFuncs"::: ) "(" (Set (Var "A")) "," (Set (Var "B")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "A")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "B")) ")" )))) ; theorem :: CARD_3:59 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "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 "A")) "," (Set (Var "B")) "holds" (Bool (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "}" ")" ))))) ; theorem :: CARD_3:60 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "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 (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; theorem :: CARD_3:61 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" )) ; theorem :: CARD_3:62 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "h1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "h2")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "h1")) ($#r1_partfun1 :::"tolerates"::: ) (Set (Var "h2"))) ")" )) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; theorem :: CARD_3:63 (Bool "for" (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r1_partfun1 :::"tolerates"::: ) (Set (Var "h"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "h"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) ; theorem :: CARD_3:64 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "h")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "h"))) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; theorem :: CARD_3:65 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; theorem :: CARD_3:66 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")) ")" ))))) ; theorem :: CARD_3:67 (Bool "for" (Set (Var "h")) "," (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )))) "holds" (Bool "ex" (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "f9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool (Set (Var "g9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "g")))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f9")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g9")))) ")" ))) ; theorem :: CARD_3:68 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f9")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g9")) ")" ))) & (Bool (Set (Var "f9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool (Set (Var "g9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "f9")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g9"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )))) ; theorem :: CARD_3:69 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "f9")) "," (Set (Var "g9")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f9"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g9")) ")" ))) & (Bool (Set (Var "f9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool (Set (Var "g9")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "f9")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g9"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )))) ; theorem :: CARD_3:70 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))) & (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "h"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))) ; theorem :: CARD_3:71 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x1")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y2")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x2"))))) "holds" (Bool (Set "(" (Set (Var "x1")) "," (Set (Var "x2")) ")" ($#k4_funct_4 :::"-->"::: ) "(" (Set (Var "y1")) "," (Set (Var "y2")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; begin definitionlet "IT" be ($#m1_hidden :::"set"::: ) ; attr "IT" is :::"with_common_domain"::: means :: CARD_3:def 10 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "IT") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))); end; :: deftheorem defines :::"with_common_domain"::: CARD_3:def 10 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_card_3 :::"with_common_domain"::: ) ) "iff" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "IT"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) ")" )); registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#m1_hidden :::"Function":::); cluster (Set ($#k1_tarski :::"{"::: ) "f" ($#k1_tarski :::"}"::: ) ) -> ($#v2_card_3 :::"with_common_domain"::: ) ; end; definitionlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; func :::"DOM"::: "X" -> ($#m1_hidden :::"set"::: ) equals :: CARD_3:def 11 (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) where f "is" ($#m1_subset_1 :::"Element"::: ) "of" "X" : (Bool verum) "}" ); end; :: deftheorem defines :::"DOM"::: CARD_3:def 11 : (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_card_3 :::"DOM"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) "{" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) where f "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) : (Bool verum) "}" ))); theorem :: CARD_3:72 (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k9_card_3 :::"DOM"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; registrationlet "X" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k9_card_3 :::"DOM"::: ) "X") -> ($#v1_xboole_0 :::"empty"::: ) ; end; begin definitionlet "S" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; func :::"product""::: "S" -> ($#m1_hidden :::"Function":::) means :: CARD_3:def 12 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_card_3 :::"DOM"::: ) "S")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_3 :::"pi"::: ) "(" "S" "," (Set (Var "i")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"product""::: CARD_3:def 12 : (Bool "for" (Set (Var "S")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_card_3 :::"product""::: ) (Set (Var "S")))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k9_card_3 :::"DOM"::: ) (Set (Var "S")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_3 :::"pi"::: ) "(" (Set (Var "S")) "," (Set (Var "i")) ")" )) ")" ) ")" ) ")" ))); theorem :: CARD_3:73 canceled; theorem :: CARD_3:74 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" )))) "holds" (Bool (Set (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) where f "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) : (Bool verum) "}" ))) ; definitionlet "S" be ($#m1_hidden :::"set"::: ) ; attr "S" is :::"product-like"::: means :: CARD_3:def 13 (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "S" ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))); end; :: deftheorem defines :::"product-like"::: CARD_3:def 13 : (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_card_3 :::"product-like"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))) ")" )); registrationlet "f" be ($#m1_hidden :::"Function":::); cluster (Set ($#k4_card_3 :::"product"::: ) "f") -> ($#v3_card_3 :::"product-like"::: ) ; end; registration cluster ($#v3_card_3 :::"product-like"::: ) -> ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_card_3 :::"product-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: CARD_3:75 canceled; theorem :: CARD_3:76 canceled; theorem :: CARD_3:77 (Bool "for" (Set (Var "S")) "being" ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Var "S")) ($#r1_tarski :::"c="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" )))) ; theorem :: CARD_3:78 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_card_3 :::"product-like"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" )))) ; theorem :: CARD_3:79 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "s")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "t")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")) ")" )) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))))) ; theorem :: CARD_3:80 (Bool "for" (Set (Var "f")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))) (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) "st" (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "s")))))) ; theorem :: CARD_3:81 (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "g")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f")))))) ; definitionlet "f" be ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Const "f"))); let "X" be ($#m1_hidden :::"set"::: ) ; :: original: :::"|"::: redefine func "g" :::"|"::: "X" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k8_card_3 :::"sproduct"::: ) "f"); end; theorem :: CARD_3:82 (Bool "for" (Set (Var "f")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "s")) "," (Set (Var "ss")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "ss")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "s")) ($#k11_card_3 :::"|"::: ) (Set (Var "A")) ")" ) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k11_card_3 :::"|"::: ) (Set (Var "A"))))))) ; theorem :: CARD_3:83 (Bool "for" (Set (Var "M")) "," (Set (Var "x")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "x")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "M")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")) ")" )))) ; theorem :: CARD_3:84 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_finset_1 :::"finite"::: ) ) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )) ; theorem :: CARD_3:85 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"Ordinal":::) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"infinite"::: ) ) "iff" (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r1_ordinal1 :::"c="::: ) (Set (Var "A"))) ")" )) ; theorem :: CARD_3:86 (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) "st" (Bool (Bool (Set (Var "N")) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool (Bool "not" (Set (Var "M")) "is" ($#v1_finset_1 :::"finite"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) & (Bool (Set (Var "N")) ($#r1_ordinal1 :::"c="::: ) (Set (Var "M"))) ")" )) ; theorem :: CARD_3:87 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "X")) "is" ($#v1_finset_1 :::"finite"::: ) )) "iff" (Bool "ex" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )) ")" )) ; theorem :: CARD_3:88 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "Y")))) "iff" (Bool (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "Y")))) ")" )) ; theorem :: CARD_3:89 (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) "st" (Bool (Bool (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Var "N")))) ; theorem :: CARD_3:90 (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) "iff" (Bool (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "N"))) ($#r1_ordinal1 :::"c="::: ) (Set (Var "M"))) ")" )) ; theorem :: CARD_3:91 (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m1_hidden :::"Cardinal":::) "holds" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k2_card_1 :::"nextcard"::: ) (Set (Var "M")))) "iff" (Bool (Set (Var "N")) ($#r1_ordinal1 :::"c="::: ) (Set (Var "M"))) ")" )) ; theorem :: CARD_3:92 (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_hidden :::"Cardinal":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool "(" (Bool (Set (Var "N")) ($#r1_ordinal1 :::"c="::: ) (Set (Var "M"))) "or" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) ")" )) "holds" (Bool (Set (Var "N")) "is" ($#v1_finset_1 :::"finite"::: ) )) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is :::"countable"::: means :: CARD_3:def 14 (Bool (Set ($#k1_card_1 :::"card"::: ) "X") ($#r1_ordinal1 :::"c="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )); attr "X" is :::"denumerable"::: means :: CARD_3:def 15 (Bool (Set ($#k1_card_1 :::"card"::: ) "X") ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )); end; :: deftheorem defines :::"countable"::: CARD_3:def 14 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_card_3 :::"countable"::: ) ) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )); :: deftheorem defines :::"denumerable"::: CARD_3:def 15 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v5_card_3 :::"denumerable"::: ) ) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )); registration cluster ($#v5_card_3 :::"denumerable"::: ) -> ($#v1_finset_1 :::"infinite"::: ) ($#v4_card_3 :::"countable"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_finset_1 :::"infinite"::: ) ($#v4_card_3 :::"countable"::: ) -> ($#v5_card_3 :::"denumerable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_finset_1 :::"finite"::: ) -> ($#v4_card_3 :::"countable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster (Set ($#k4_ordinal1 :::"omega"::: ) ) -> ($#v5_card_3 :::"denumerable"::: ) ; end; registration cluster ($#v5_card_3 :::"denumerable"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: CARD_3:93 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_card_3 :::"countable"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) ")" )) ")" )) ; registrationlet "X" be ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v4_card_3 :::"countable"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; theorem :: CARD_3:94 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v4_card_3 :::"countable"::: ) )) "holds" (Bool (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y"))) "is" ($#v4_card_3 :::"countable"::: ) )) ; theorem :: CARD_3:95 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v4_card_3 :::"countable"::: ) )) "holds" (Bool (Set (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set (Var "Y"))) "is" ($#v4_card_3 :::"countable"::: ) )) ; theorem :: CARD_3:96 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "," (Set (Var "A")) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; theorem :: CARD_3:97 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "g"))) "holds" (Bool (Set (Set (Var "x")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )))))) ; theorem :: CARD_3:98 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )) "holds" (Bool (Set (Set (Var "x")) ($#k11_card_3 :::"|"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "g")))))) ; theorem :: CARD_3:99 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_partfun1 :::"tolerates"::: ) (Set (Var "g")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )) "holds" (Bool (Set (Set (Var "x")) ($#k11_card_3 :::"|"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))))) ; theorem :: CARD_3:100 (Bool "for" (Set (Var "S")) "being" ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" ))))) ; theorem :: CARD_3:101 (Bool "for" (Set (Var "S")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; theorem :: CARD_3:102 (Bool "for" (Set (Var "S")) "being" ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_card_3 :::"product""::: ) (Set (Var "S")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; registrationlet "X" be ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v2_card_3 :::"with_common_domain"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; definitionlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"proj"::: "(" "f" "," "x" ")" -> ($#m1_hidden :::"Function":::) means :: CARD_3:def 16 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) "f")) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_funct_1 :::"."::: ) "x")) ")" ) ")" ); end; :: deftheorem defines :::"proj"::: CARD_3:def 16 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k12_card_3 :::"proj"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ) ")" )))); registrationlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k12_card_3 :::"proj"::: ) "(" "f" "," "x" ")" ) -> (Set ($#k4_card_3 :::"product"::: ) "f") ($#v4_relat_1 :::"-defined"::: ) ; end; registrationlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k12_card_3 :::"proj"::: ) "(" "f" "," "x" ")" ) -> ($#v1_partfun1 :::"total"::: ) ; end; registrationlet "f" be ($#v2_relat_1 :::"non-empty"::: ) ($#m1_hidden :::"Function":::); cluster -> "f" ($#v5_funct_1 :::"-compatible"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) "f"); end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#v2_relat_1 :::"non-empty"::: ) (Set (Const "I")) ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Function":::); cluster -> "I" ($#v4_relat_1 :::"-defined"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) "f"); end; registrationlet "f" be ($#m1_hidden :::"Function":::); cluster -> "f" ($#v5_funct_1 :::"-compatible"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k8_card_3 :::"sproduct"::: ) "f"); end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be (Set (Const "I")) ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Function":::); cluster -> "I" ($#v4_relat_1 :::"-defined"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k8_card_3 :::"sproduct"::: ) "f"); end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#v2_relat_1 :::"non-empty"::: ) (Set (Const "I")) ($#v4_relat_1 :::"-defined"::: ) ($#v1_partfun1 :::"total"::: ) ($#m1_hidden :::"Function":::); cluster -> ($#v1_partfun1 :::"total"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) "f"); end; theorem :: CARD_3:103 (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_relat_1 :::"non-empty"::: ) (Set (Var "b1")) ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "p")) "being" (Set (Var "b1")) ($#v4_relat_1 :::"-defined"::: ) (Set (Var "b2")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k8_card_3 :::"sproduct"::: ) (Set (Var "f"))))))) ; theorem :: CARD_3:104 (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_relat_1 :::"non-empty"::: ) (Set (Var "b1")) ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "p")) "being" (Set (Var "b1")) ($#v4_relat_1 :::"-defined"::: ) (Set (Var "b2")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"Function":::) (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) "st" (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "s"))))))) ; registrationlet "X" be ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"set"::: ) ; let "a" be ($#m1_hidden :::"set"::: ) ; cluster (Set "X" ($#k2_funcop_1 :::"-->"::: ) "a") -> ($#v1_finset_1 :::"infinite"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"infinite"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "R" be ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Relation":::); cluster (Set ($#k1_relat_1 :::"field"::: ) "R") -> ($#v1_finset_1 :::"infinite"::: ) ; end; registrationlet "X" be ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_wellord2 :::"RelIncl"::: ) "X") -> ($#v1_finset_1 :::"infinite"::: ) ; end; theorem :: CARD_3:105 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#m1_hidden :::"Relation":::) "st" (Bool (Bool (Set (Var "R")) "," (Set (Var "S")) ($#r4_wellord1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "R")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool (Set (Var "S")) "is" ($#v1_finset_1 :::"finite"::: ) )) ; theorem :: CARD_3:106 (Bool (Set ($#k10_card_3 :::"product""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: CARD_3:107 (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "s")) "being" (Set (Var "b2")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "holds" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))))))) ; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster -> ($#v1_partfun1 :::"total"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) "f"); end; definitionlet "I" be ($#m1_hidden :::"set"::: ) ; let "f" be bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "M" be (Set (Const "f")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); func :::"down"::: "M" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_card_3 :::"product"::: ) "f") equals :: CARD_3:def 17 "M"; end; :: deftheorem defines :::"down"::: CARD_3:def 17 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" bbbadV2_RELAT_1() ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "M")) "being" (Set (Var "b2")) ($#v5_funct_1 :::"-compatible"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "holds" (Bool (Set ($#k13_card_3 :::"down"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "M")))))); theorem :: CARD_3:108 (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#v2_card_3 :::"with_common_domain"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k9_card_3 :::"DOM"::: ) (Set (Var "X")))))) ; theorem :: CARD_3:109 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) ")" )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_card_3 :::"DOM"::: ) (Set (Var "X")))))) ;