:: STIRL2_1 semantic presentation begin theorem :: STIRL2_1:1 (Bool "for" (Set (Var "N")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_seq_4 :::"min"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "N"))))) ; theorem :: STIRL2_1:2 (Bool "for" (Set (Var "K")) "," (Set (Var "N")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_nat_1 :::"min"::: ) "(" (Set "(" ($#k6_seq_4 :::"min"::: ) (Set (Var "K")) ")" ) "," (Set "(" ($#k6_seq_4 :::"min"::: ) (Set (Var "N")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_seq_4 :::"min"::: ) (Set "(" (Set (Var "K")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "N")) ")" )))) ; theorem :: STIRL2_1:3 (Bool "for" (Set (Var "Ke")) "," (Set (Var "Ne")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_nat_1 :::"min"::: ) "(" (Set "(" ($#k5_nat_1 :::"min*"::: ) (Set (Var "Ke")) ")" ) "," (Set "(" ($#k5_nat_1 :::"min*"::: ) (Set (Var "Ne")) ")" ) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "Ke")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Ne")) ")" )))) ; theorem :: STIRL2_1:4 (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "Ne"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "Ne")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Ke")))))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "Ne"))) ($#r1_hidden :::"="::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "Ne")) ($#k7_subset_1 :::"\"::: ) (Set (Var "Ke")) ")" )))) ; theorem :: STIRL2_1:5 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "n")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k6_seq_4 :::"min"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "n")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )) ; theorem :: STIRL2_1:6 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "n")) "," (Set (Var "k")) ($#k7_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_nat_1 :::"min"::: ) "(" (Set (Var "n")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k6_seq_4 :::"min"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "n")) "," (Set (Var "k")) ($#k7_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_nat_1 :::"min"::: ) "(" (Set (Var "n")) "," (Set (Var "k")) ")" )) ")" )) ; theorem :: STIRL2_1:7 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set ($#k8_domain_1 :::"{"::: ) (Set (Var "n")) "," (Set (Var "k")) "," (Set (Var "l")) ($#k8_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_nat_1 :::"min"::: ) "(" (Set (Var "n")) "," (Set "(" ($#k6_nat_1 :::"min"::: ) "(" (Set (Var "k")) "," (Set (Var "l")) ")" ")" ) ")" ))) ; theorem :: STIRL2_1:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Var "n")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster -> ($#v7_ordinal1 :::"natural"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" "n"; end; theorem :: STIRL2_1:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1)) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )))) ; theorem :: STIRL2_1:10 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1))) & (Bool (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1)) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" )) ; theorem :: STIRL2_1:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: STIRL2_1:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "N"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1))))) ; theorem :: STIRL2_1:13 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set (Var "n"))) & (Bool (Set (Var "N")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ($#k1_seq_4 :::"}"::: ) ))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "N"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1))))) ; theorem :: STIRL2_1:14 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "Ne")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "Ne")) ($#r1_tarski :::"c="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set (Var "Ne"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1))))) ; definitionlet "n" be ($#m1_hidden :::"Nat":::); let "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "n")) "," (Set (Const "X")); let "x" be ($#m1_hidden :::"set"::: ) ; :: original: :::"""::: redefine func "f" :::"""::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "k" be ($#m1_hidden :::"Nat":::); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "k")); let "x" be ($#m1_hidden :::"set"::: ) ; :: original: :::"."::: redefine func "f" :::"."::: "x" -> ($#m1_subset_1 :::"Element"::: ) "of" "k"; end; definitionlet "n", "k" be ($#m1_hidden :::"Nat":::); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "n")) "," (Set (Const "k")); attr "f" is :::""increasing"::: means :: STIRL2_1:def 1 (Bool "(" "(" (Bool (Bool "n" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "n" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "l")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) "f")) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) "f")) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" "f" ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "l")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" "f" ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::""increasing"::: STIRL2_1:def 1 : (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) "iff" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "l")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "l")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" ))) ")" ) ")" ) ")" ))); theorem :: STIRL2_1:15 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ))) ; theorem :: STIRL2_1:16 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1))))) ; theorem :: STIRL2_1:17 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) )) "holds" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "k"))))) ; theorem :: STIRL2_1:18 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) )) "holds" (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" )))))) ; theorem :: STIRL2_1:19 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) )) "holds" (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Set (Var "k")) ")" ) ($#k20_binop_2 :::"+"::: ) (Set (Var "m"))))))) ; theorem :: STIRL2_1:20 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Var "k")))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set (Var "n")))))) ; theorem :: STIRL2_1:21 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set (Var "n")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ))) ; theorem :: STIRL2_1:22 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "not" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "holds" (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) )) ")" ) ")" ))) ; theorem :: STIRL2_1:23 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "not" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "k"))) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "holds" (Bool "(" "not" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) "or" "not" (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) ")" ) ")" ))) ; scheme :: STIRL2_1:sch 1 Sch1{ F1() -> ($#m1_hidden :::"Nat":::), F2() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) : (Bool P1[(Set (Var "f"))]) "}" "is" ($#v1_finset_1 :::"finite"::: ) ) proof end; theorem :: STIRL2_1:24 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" "is" ($#v1_finset_1 :::"finite"::: ) )) ; theorem :: STIRL2_1:25 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" ) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ))) ; definitionlet "n", "k" be ($#m1_hidden :::"Nat":::); func "n" :::"block"::: "k" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: STIRL2_1:def 2 (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" "n" "," "k" : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" ); end; :: deftheorem defines :::"block"::: STIRL2_1:def 2 : (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" ))); theorem :: STIRL2_1:26 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: STIRL2_1:27 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: STIRL2_1:28 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: STIRL2_1:29 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: STIRL2_1:30 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: STIRL2_1:31 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: STIRL2_1:32 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: STIRL2_1:33 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool "(" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ) "or" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ) "iff" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; scheme :: STIRL2_1:sch 2 Sch2{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_hidden :::"set"::: ) , F4() -> ($#m1_hidden :::"set"::: ) , F5() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ), F6( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set F3 "(" ")" ) "," (Set F4 "(" ")" ) "st" (Bool "(" (Bool (Set (Set (Var "h")) ($#k2_partfun1 :::"|"::: ) (Set F1 "(" ")" )) ($#r1_hidden :::"="::: ) (Set F5 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F3 "(" ")" ) ($#k6_subset_1 :::"\"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set F6 "(" (Set (Var "x")) ")" )) ")" ) ")" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F3 "(" ")" ) ($#k6_subset_1 :::"\"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set F6 "(" (Set (Var "x")) ")" ) ($#r2_hidden :::"in"::: ) (Set F4 "(" ")" ))) and (Bool "(" (Bool (Set F1 "(" ")" ) ($#r1_tarski :::"c="::: ) (Set F3 "(" ")" )) & (Bool (Set F2 "(" ")" ) ($#r1_tarski :::"c="::: ) (Set F4 "(" ")" )) ")" ) and "(" (Bool (Bool (Set F2 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set F1 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" proof end; scheme :: STIRL2_1:sch 3 Sch3{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_hidden :::"set"::: ) , F4() -> ($#m1_hidden :::"set"::: ) , F5( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) : (Bool P1[(Set (Var "f")) "," (Set F1 "(" ")" ) "," (Set F2 "(" ")" )]) "}" ) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set F3 "(" ")" ) "," (Set F4 "(" ")" ) : (Bool "(" (Bool P1[(Set (Var "f")) "," (Set F3 "(" ")" ) "," (Set F4 "(" ")" )]) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set F1 "(" ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F3 "(" ")" ) ($#k6_subset_1 :::"\"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set F5 "(" (Set (Var "x")) ")" )) ")" ) ")" ) "}" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F3 "(" ")" ) ($#k6_subset_1 :::"\"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set F5 "(" (Set (Var "x")) ")" ) ($#r2_hidden :::"in"::: ) (Set F4 "(" ")" ))) and (Bool "(" (Bool (Set F1 "(" ")" ) ($#r1_tarski :::"c="::: ) (Set F3 "(" ")" )) & (Bool (Set F2 "(" ")" ) ($#r1_tarski :::"c="::: ) (Set F4 "(" ")" )) ")" ) and "(" (Bool (Bool (Set F2 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set F1 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" and (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set F3 "(" ")" ) "," (Set F4 "(" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F3 "(" ")" ) ($#k6_subset_1 :::"\"::: ) (Set F1 "(" ")" )))) "holds" (Bool (Set F5 "(" (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" )) "holds" (Bool "(" (Bool P1[(Set (Var "f")) "," (Set F3 "(" ")" ) "," (Set F4 "(" ")" )]) "iff" (Bool P1[(Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set F1 "(" ")" )) "," (Set F1 "(" ")" ) "," (Set F2 "(" ")" )]) ")" )) proof end; scheme :: STIRL2_1:sch 4 Sch4{ F1() -> ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_hidden :::"set"::: ) , F4() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) : (Bool P1[(Set (Var "f")) "," (Set F1 "(" ")" ) "," (Set F2 "(" ")" )]) "}" ) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set F1 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F3 "(" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" (Set F2 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F4 "(" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool P1[(Set (Var "f")) "," (Set (Set F1 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F3 "(" ")" ) ($#k1_tarski :::"}"::: ) )) "," (Set (Set F2 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F4 "(" ")" ) ($#k1_tarski :::"}"::: ) ))]) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set F1 "(" ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set F2 "(" ")" )) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set F3 "(" ")" )) ($#r1_hidden :::"="::: ) (Set F4 "(" ")" )) ")" ) "}" )) provided "(" (Bool (Bool (Set F2 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set F1 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" and (Bool (Bool "not" (Set F3 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) and (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set F1 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F3 "(" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" (Set F2 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F4 "(" ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set F3 "(" ")" )) ($#r1_hidden :::"="::: ) (Set F4 "(" ")" ))) "holds" (Bool "(" (Bool P1[(Set (Var "f")) "," (Set (Set F1 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F3 "(" ")" ) ($#k1_tarski :::"}"::: ) )) "," (Set (Set F2 "(" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set F4 "(" ")" ) ($#k1_tarski :::"}"::: ) ))]) "iff" (Bool P1[(Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set F1 "(" ")" )) "," (Set F1 "(" ")" ) "," (Set F2 "(" ")" )]) ")" )) proof end; theorem :: STIRL2_1:34 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))))) ; theorem :: STIRL2_1:35 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) ))) & (Bool (Set (Var "m")) ($#r1_hidden :::"<>"::: ) (Set (Var "n"))) ")" )))) ; theorem :: STIRL2_1:36 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "," (Set (Var "m")) "," (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Set (Var "m")) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Set (Var "l")) ")" ) "st" (Bool (Bool (Set (Var "g")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "i")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "j")) ($#k1_seq_4 :::"}"::: ) ) ")" ))))))) ; theorem :: STIRL2_1:37 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "k"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n"))))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) ")" ) ")" ))) ; theorem :: STIRL2_1:38 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" )))) ; theorem :: STIRL2_1:39 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "g")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "i")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "g")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "j")) ($#k1_seq_4 :::"}"::: ) ) ")" ))))))) ; theorem :: STIRL2_1:40 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "g")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "k")))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "g")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) ")" )))) ; theorem :: STIRL2_1:41 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "g")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "g")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) ")" )))) ; theorem :: STIRL2_1:42 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) ")" ) "}" ) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" ))) ; theorem :: STIRL2_1:43 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "l"))) ")" ) "}" ) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "n")) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) ")" ) "}" ))) ; theorem :: STIRL2_1:44 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))))) ; scheme :: STIRL2_1:sch 5 Sch6{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"XFinSequence":::) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set (Var "k")) "," (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool P1[(Set (Var "k")) "," (Set (Var "x"))]))) proof end; scheme :: STIRL2_1:sch 6 Sch8{ F1() -> ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) , F2() -> ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) , F3() -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) ], F4() -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_card_1 :::"card"::: ) (Set F2 "(" ")" ) ")" ) "," (Set F2 "(" ")" ) } : (Bool "ex" (Set (Var "F")) "being" ($#m1_hidden :::"XFinSequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set F2 "(" ")" ))) & (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) : (Bool P1[(Set (Var "g"))]) "}" ) ($#r1_hidden :::"="::: ) (Set ($#k7_afinsq_2 :::"Sum"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set F2 "(" ")" ) : (Bool "(" (Bool P1[(Set (Var "g"))]) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set F3 "(" ")" )) ($#r1_hidden :::"="::: ) (Set (Set F4 "(" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) "}" )) ")" ) ")" )) provided (Bool "(" (Bool (Set F4 "(" ")" ) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set F4 "(" ")" ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) and (Bool (Bool "not" (Set F2 "(" ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) )) and (Bool (Set F3 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" )) proof end; theorem :: STIRL2_1:45 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "k")) ($#k24_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_stirl2_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_4 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "n")) ($#k1_seq_4 :::"}"::: ) )) ")" ) "}" ))) ; theorem :: STIRL2_1:46 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k3_stirl2_1 :::"block"::: ) (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k24_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set "(" (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k23_binop_2 :::"+"::: ) (Set "(" (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "k")) ")" )))) ; theorem :: STIRL2_1:47 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 2) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k10_binop_2 :::"-"::: ) (Num 2) ")" )))) ; theorem :: STIRL2_1:48 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 6) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Num 3) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Num 3) ($#k11_binop_2 :::"*"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Num 3) ")" )))) ; theorem :: STIRL2_1:49 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 3))) "holds" (Bool (Set (Set (Var "n")) ($#k3_stirl2_1 :::"block"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k18_binop_2 :::"/"::: ) (Num 24) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Num 4) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Num 4) ($#k11_binop_2 :::"*"::: ) (Set "(" (Num 3) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Num 6) ($#k11_binop_2 :::"*"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Num 4) ")" )))) ; theorem :: STIRL2_1:50 (Bool "(" (Bool (Set (Num 3) ($#k9_newton :::"!"::: ) ) ($#r1_hidden :::"="::: ) (Num 6)) & (Bool (Set (Num 4) ($#k9_newton :::"!"::: ) ) ($#r1_hidden :::"="::: ) (Num 24)) ")" ) ; theorem :: STIRL2_1:51 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k18_binop_2 :::"/"::: ) (Num 2))) & (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k18_binop_2 :::"/"::: ) (Num 6))) & (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k22_binop_2 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k21_binop_2 :::"-"::: ) (Num 3) ")" ) ")" ) ($#k18_binop_2 :::"/"::: ) (Num 24))) ")" )) ; theorem :: STIRL2_1:52 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k6_newton :::"choose"::: ) (Num 2)))) ; theorem :: STIRL2_1:53 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 2) ")" ) ($#k3_stirl2_1 :::"block"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 3) ($#k24_binop_2 :::"*"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 2) ")" ) ($#k6_newton :::"choose"::: ) (Num 4) ")" ) ")" ) ($#k23_binop_2 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 2) ")" ) ($#k6_newton :::"choose"::: ) (Num 3) ")" )))) ; theorem :: STIRL2_1:54 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set "(" (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ")" ) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" ) ")" ))) ; theorem :: STIRL2_1:55 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k23_binop_2 :::"+"::: ) (Num 1))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "k"))))) ; scheme :: STIRL2_1:sch 7 Sch9{ P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"Function":::)] } : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool P1[(Set (Var "F"))])) provided (Bool P1[(Set ($#k1_xboole_0 :::"{}"::: ) )]) and (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")))) & (Bool P2[(Set (Var "x")) "," (Set (Var "F"))])) "holds" (Bool P1[(Set (Set (Var "F")) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "F")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))]) ")" )) "holds" (Bool P1[(Set (Var "F"))])) proof end; theorem :: STIRL2_1:56 (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))))) ; theorem :: STIRL2_1:57 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set (Var "X")) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" (Set (Var "Y")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Set (Var "G")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" )))) ; theorem :: STIRL2_1:58 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Bool (Set (Var "Y")) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set (Var "X")) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" ) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" (Set (Var "Y")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" ) "st" (Bool (Bool (Set (Set (Var "G")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Set (Var "G")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "F")) "is" ($#v2_funct_2 :::"onto"::: ) )) "implies" (Bool (Set (Var "G")) "is" ($#v2_funct_2 :::"onto"::: ) ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) & (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "implies" (Bool (Set (Var "G")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ")" )))) ; theorem :: STIRL2_1:59 (Bool "for" (Set (Var "N")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "Order")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "N")) "," (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "N")) ")" ) "st" (Bool "(" (Bool (Set (Var "Order")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "Order")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "Order")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "Order")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "Order")) ($#k2_stirl2_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ))) ; theorem :: STIRL2_1:60 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Y"))))) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_funct_2 :::"onto"::: ) ) "iff" (Bool (Set (Var "F")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: STIRL2_1:61 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "G")) ($#k3_relat_1 :::"*"::: ) (Set (Var "F")) ")" ))) & (Bool (Set (Var "G")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "G")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")))) & (Bool (Set (Set (Var "G")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_relat_1 :::"*"::: ) (Set (Var "F")) ")" ) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ))) ")" )))) ; definitionlet "Ne", "Ke" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "Ne")) "," (Set (Const "Ke")); attr "f" is :::""increasing"::: means :: STIRL2_1:def 3 (Bool "for" (Set (Var "l")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) "f")) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) "f")) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" "f" ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "l")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" "f" ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" )))); end; :: deftheorem defines :::""increasing"::: STIRL2_1:def 3 : (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_stirl2_1 :::""increasing"::: ) ) "iff" (Bool "for" (Set (Var "l")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "l")) ($#k1_seq_4 :::"}"::: ) ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_seq_4 :::"{"::: ) (Set (Var "m")) ($#k1_seq_4 :::"}"::: ) ) ")" )))) ")" ))); theorem :: STIRL2_1:62 (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_stirl2_1 :::""increasing"::: ) )) "holds" (Bool (Set ($#k5_nat_1 :::"min*"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k5_nat_1 :::"min*"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: STIRL2_1:63 (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke"))(Bool "ex" (Set (Var "P")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "F")) ")" ) "st" (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I")))) & (Bool (Set (Var "I")) "is" ($#v2_stirl2_1 :::""increasing"::: ) ) ")" ))))) ; theorem :: STIRL2_1:64 (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "," (Set (Var "Me")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Me")) (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "P1")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "P2")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I2")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "P1")))) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "P2")))) & (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P1")) ($#k3_relat_1 :::"*"::: ) (Set (Var "I1")))) & (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P2")) ($#k3_relat_1 :::"*"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I1")) "is" ($#v2_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "I2")) "is" ($#v2_stirl2_1 :::""increasing"::: ) )) "holds" (Bool "(" (Bool (Set (Var "P1")) ($#r1_hidden :::"="::: ) (Set (Var "P2"))) & (Bool (Set (Var "I1")) ($#r2_funct_2 :::"="::: ) (Set (Var "I2"))) ")" ))))) ; theorem :: STIRL2_1:65 (Bool "for" (Set (Var "Ne")) "," (Set (Var "Ke")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Ne")) "," (Set (Var "Ke")) (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "F")) ")" ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P1")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I1")))) & (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I2")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I1")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "I2")))) & (Bool (Set (Var "I1")) "is" ($#v2_stirl2_1 :::""increasing"::: ) ) & (Bool (Set (Var "I2")) "is" ($#v2_stirl2_1 :::""increasing"::: ) )) "holds" (Bool "(" (Bool (Set (Var "P1")) ($#r2_funct_2 :::"="::: ) (Set (Var "P2"))) & (Bool (Set (Var "I1")) ($#r2_funct_2 :::"="::: ) (Set (Var "I2"))) ")" ))))) ; theorem :: STIRL2_1:66 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"XFinSequence":::) "of" (Set (Var "D")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#v1_finset_1 :::"finite"::: ) ) ")" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" )) "holds" (Bool "ex" (Set (Var "CardF")) "being" ($#m1_hidden :::"XFinSequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "CardF"))) ($#r1_hidden :::"="::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "F")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_afinsq_1 :::"dom"::: ) (Set (Var "CardF"))))) "holds" (Bool (Set (Set (Var "CardF")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ))) ")" ) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_afinsq_2 :::"Sum"::: ) (Set (Var "CardF")))) ")" )))) ;