:: SERIES_1 semantic presentation begin theorem :: SERIES_1:1 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: SERIES_1:2 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "a")) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ))))) ; theorem :: SERIES_1:3 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_membered :::"complex-membered"::: ) ($#v7_membered :::"add-closed"::: ) ($#m1_hidden :::"set"::: ) ; let "s1", "s2" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "X")); :: original: :::"+"::: redefine func "s1" :::"+"::: "s2" -> ($#m1_subset_1 :::"sequence":::) "of" "X"; end; definitionlet "s" be ($#v1_valued_0 :::"complex-valued"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"Partial_Sums"::: "s" -> ($#v1_valued_0 :::"complex-valued"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: SERIES_1:def 1 (Bool "(" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set "s" ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" it ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" "s" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"Partial_Sums"::: SERIES_1:def 1 : (Bool "for" (Set (Var "s")) "," (Set (Var "b2")) "being" ($#v1_valued_0 :::"complex-valued"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_series_1 :::"Partial_Sums"::: ) (Set (Var "s")))) "iff" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) ")" ) ")" )); registrationlet "s" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k2_series_1 :::"Partial_Sums"::: ) "s") -> ($#v1_valued_0 :::"complex-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ; end; definitionlet "s" be ($#m1_subset_1 :::"Real_Sequence":::); :: original: :::"Partial_Sums"::: redefine func :::"Partial_Sums"::: "s" -> ($#m1_subset_1 :::"Real_Sequence":::); end; definitionlet "s" be ($#m1_subset_1 :::"Real_Sequence":::); attr "s" is :::"summable"::: means :: SERIES_1:def 2 (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) "s") "is" ($#v2_comseq_2 :::"convergent"::: ) ); func :::"Sum"::: "s" -> ($#m1_subset_1 :::"Real":::) equals :: SERIES_1:def 3 (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) "s" ")" )); end; :: deftheorem defines :::"summable"::: SERIES_1:def 2 : (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) "iff" (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) ")" )); :: deftheorem defines :::"Sum"::: SERIES_1:def 3 : (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" )))); theorem :: SERIES_1:4 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: SERIES_1:5 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_membered :::"complex-membered"::: ) ($#v7_membered :::"add-closed"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k2_series_1 :::"Partial_Sums"::: ) (Set (Var "s1")) ")" ) ($#k1_valued_1 :::"+"::: ) (Set "(" ($#k2_series_1 :::"Partial_Sums"::: ) (Set (Var "s2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_series_1 :::"Partial_Sums"::: ) (Set "(" (Set (Var "s1")) ($#k1_series_1 :::"+"::: ) (Set (Var "s2")) ")" ))))) ; theorem :: SERIES_1:6 (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s1")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s2")) ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" (Set (Var "s1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "s2")) ")" )))) ; theorem :: SERIES_1:7 (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s1")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set (Var "s2")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "s1")) ($#k1_series_1 :::"+"::: ) (Set (Var "s2"))) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "s1")) ($#k1_series_1 :::"+"::: ) (Set (Var "s2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "s1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "s2")) ")" ))) ")" )) ; theorem :: SERIES_1:8 (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s1")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set (Var "s2")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "s1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "s2"))) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "s1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "s2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "s1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "s2")) ")" ))) ")" )) ; theorem :: SERIES_1:9 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "s")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ))))) ; theorem :: SERIES_1:10 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "s"))) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set (Var "s")) ")" ))) ")" ))) ; theorem :: SERIES_1:11 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" (Set (Var "s")) ($#k1_valued_0 :::"^\"::: ) (Num 1) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k1_valued_0 :::"^\"::: ) (Num 1) ")" ) ($#k47_valued_1 :::"-"::: ) (Set (Var "s1"))))) ; theorem :: SERIES_1:12 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "n"))) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:13 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "s")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "n"))) "is" ($#v1_series_1 :::"summable"::: ) ))) "holds" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) ; theorem :: SERIES_1:14 (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s1")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s2")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))))) ; theorem :: SERIES_1:15 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "s")) ($#k1_valued_0 :::"^\"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))))) ; theorem :: SERIES_1:16 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s"))) "is" bbbadV7_VALUED_0())) ; theorem :: SERIES_1:17 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool "(" (Bool (Set ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s"))) "is" ($#v1_seq_2 :::"bounded_above"::: ) ) "iff" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) ")" )) ; theorem :: SERIES_1:18 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s"))))) ; theorem :: SERIES_1:19 (Bool "for" (Set (Var "s2")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool (Set (Var "s1")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "s2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))))))) "holds" (Bool (Set (Var "s2")) "is" ($#v1_series_1 :::"summable"::: ) )) ; theorem :: SERIES_1:20 (Bool "for" (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s2")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" ) ")" ) & (Bool (Set (Var "s2")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool "(" (Bool (Set (Var "s1")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s2")))) ")" )) ; theorem :: SERIES_1:21 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) "iff" (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))))) ")" )) ; theorem :: SERIES_1:22 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" (Set (Var "a")) ($#k1_prepower :::"GeoSeq"::: ) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ))))) ; theorem :: SERIES_1:23 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" )))))) ; theorem :: SERIES_1:24 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k1_prepower :::"GeoSeq"::: ) ) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set "(" (Set (Var "a")) ($#k1_prepower :::"GeoSeq"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ))) ")" )) ; theorem :: SERIES_1:25 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) & (Bool (Set ($#k4_series_1 :::"Sum"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "a")) ")" ))) ")" ))) ; theorem :: SERIES_1:26 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) ; theorem :: SERIES_1:27 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:28 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) ; theorem :: SERIES_1:29 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:30 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; registrationlet "k", "n" be ($#m1_hidden :::"Nat":::); cluster (Set "k" ($#k3_power :::"to_power"::: ) "n") -> ($#v7_ordinal1 :::"natural"::: ) ; end; definitionlet "k", "n" be ($#m1_hidden :::"Nat":::); :: original: :::"to_power"::: redefine func "k" :::"to_power"::: "n" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: SERIES_1:31 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" bbbadV8_VALUED_0()) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ")" ))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ) "iff" (Bool (Set (Var "s1")) "is" ($#v1_series_1 :::"summable"::: ) ) ")" )) ; theorem :: SERIES_1:32 (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_power :::"to_power"::: ) (Set (Var "p")) ")" ))) ")" )) "holds" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:33 (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_power :::"to_power"::: ) (Set (Var "p")) ")" ))) ")" )) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )))) ; definitionlet "s" be ($#m1_subset_1 :::"Real_Sequence":::); attr "s" is :::"absolutely_summable"::: means :: SERIES_1:def 4 (Bool (Set ($#k56_valued_1 :::"abs"::: ) "s") "is" ($#v1_series_1 :::"summable"::: ) ); end; :: deftheorem defines :::"absolutely_summable"::: SERIES_1:def 4 : (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#v2_series_1 :::"absolutely_summable"::: ) ) "iff" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "s"))) "is" ($#v1_series_1 :::"summable"::: ) ) ")" )); theorem :: SERIES_1:34 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ")" ))))) ; registration cluster ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCT_2((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) )) ($#v2_series_1 :::"absolutely_summable"::: ) -> ($#v1_series_1 :::"summable"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; theorem :: SERIES_1:35 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "s")) "is" ($#v2_series_1 :::"absolutely_summable"::: ) )) "holds" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) ; theorem :: SERIES_1:36 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) )) "holds" (Bool (Set (Var "s")) "is" ($#v2_series_1 :::"absolutely_summable"::: ) )) ; theorem :: SERIES_1:37 (Bool "for" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Var "s")) "is" ($#v2_series_1 :::"absolutely_summable"::: ) )) ; theorem :: SERIES_1:38 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))))) & (Bool (Set (Var "s")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: SERIES_1:39 (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:40 (Bool "for" (Set (Var "s1")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool (Set (Var "s")) "is" ($#v2_series_1 :::"absolutely_summable"::: ) )) ; theorem :: SERIES_1:41 (Bool "for" (Set (Var "s1")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) & (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; theorem :: SERIES_1:42 (Bool "for" (Set (Var "s1")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_power :::"-root"::: ) (Set "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "s")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool "not" (Bool (Set (Var "s")) "is" ($#v1_series_1 :::"summable"::: ) ))) ; begin definitionlet "s" be ($#m1_subset_1 :::"Real_Sequence":::); let "n" be ($#m1_hidden :::"Nat":::); func :::"Sum"::: "(" "s" "," "n" ")" -> ($#m1_subset_1 :::"Real":::) equals :: SERIES_1:def 5 (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) "s" ")" ) ($#k1_seq_1 :::"."::: ) "n"); end; :: deftheorem defines :::"Sum"::: SERIES_1:def 5 : (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "s")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_series_1 :::"Partial_Sums"::: ) (Set (Var "s")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))); definitionlet "s" be ($#m1_subset_1 :::"Real_Sequence":::); let "n", "m" be ($#m1_hidden :::"Nat":::); func :::"Sum"::: "(" "s" "," "n" "," "m" ")" -> ($#m1_subset_1 :::"Real":::) equals :: SERIES_1:def 6 (Set (Set "(" ($#k6_series_1 :::"Sum"::: ) "(" "s" "," "n" ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_series_1 :::"Sum"::: ) "(" "s" "," "m" ")" ")" )); end; :: deftheorem defines :::"Sum"::: SERIES_1:def 6 : (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "s")) "," (Set (Var "n")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "s")) "," (Set (Var "n")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "s")) "," (Set (Var "m")) ")" ")" )))));