:: RINFSUP1 semantic presentation begin theorem :: RINFSUP1:1 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool "(" (Bool (Set (Set (Var "s")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t"))) & (Bool (Set (Set (Var "s")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "t"))) ")" ) "iff" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "t")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" )) ; definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"upper_bound"::: "seq" -> ($#m1_subset_1 :::"Real":::) equals :: RINFSUP1:def 1 (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) "seq" ")" )); end; :: deftheorem defines :::"upper_bound"::: RINFSUP1:def 1 : (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")) ")" )))); definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"lower_bound"::: "seq" -> ($#m1_subset_1 :::"Real":::) equals :: RINFSUP1:def 2 (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) "seq" ")" )); end; :: deftheorem defines :::"lower_bound"::: RINFSUP1:def 2 : (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")) ")" )))); theorem :: RINFSUP1:2 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set (Var "seq2"))) ($#r2_funct_2 :::"="::: ) (Set (Var "seq1")))) ; theorem :: RINFSUP1:3 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")))) "iff" (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ))) ")" ))) ; theorem :: RINFSUP1:4 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_member_1 :::"--"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")) ")" )))) ; theorem :: RINFSUP1:5 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2()) "iff" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) ) ")" )) ; theorem :: RINFSUP1:6 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2()) "iff" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) "is" ($#v3_xxreal_2 :::"bounded_below"::: ) ) ")" )) ; theorem :: RINFSUP1:7 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) & (Bool "(" "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))))) ")" ) ")" ) ")" ))) ; theorem :: RINFSUP1:8 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool "(" "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s"))))) ")" ) ")" ) ")" ))) ; theorem :: RINFSUP1:9 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) "iff" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2()) & (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) ")" ))) ; theorem :: RINFSUP1:10 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) "iff" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq")))) ")" ) ")" ))) ; theorem :: RINFSUP1:11 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2()) "iff" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "seq"))) "is" bbbadV2_SEQ_2()) ")" )) ; theorem :: RINFSUP1:12 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2()) "iff" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "seq"))) "is" bbbadV1_SEQ_2()) ")" )) ; theorem :: RINFSUP1:13 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" )))) ; theorem :: RINFSUP1:14 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" )))) ; theorem :: RINFSUP1:15 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq2")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq2")) ")" )))) ; theorem :: RINFSUP1:16 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq2")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq2")) ")" )))) ; notationlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); synonym :::"nonnegative"::: "f" for :::"nonnegative-yielding"::: ; end; definitionlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); redefine attr "f" is :::"nonnegative-yielding"::: means :: RINFSUP1:def 3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))); end; :: deftheorem defines :::"nonnegative"::: RINFSUP1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )); theorem :: RINFSUP1:17 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" ($#v4_partfun3 :::"nonnegative"::: ) ))) ; theorem :: RINFSUP1:18 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: RINFSUP1:19 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: RINFSUP1:20 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2"))) "is" bbbadV2_SEQ_2()) & (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq1")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq2")) ")" ))) ")" )) ; theorem :: RINFSUP1:21 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2"))) "is" bbbadV1_SEQ_2()) & (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq1")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq2")) ")" ))) ")" )) ; theorem :: RINFSUP1:22 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ; theorem :: RINFSUP1:23 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ; theorem :: RINFSUP1:24 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:25 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:26 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" bbbadV1_SEQ_2()))) ; theorem :: RINFSUP1:27 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" bbbadV2_SEQ_2()))) ; theorem :: RINFSUP1:28 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ))) ; theorem :: RINFSUP1:29 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )))) ; theorem :: RINFSUP1:30 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) "}" ))) ; theorem :: RINFSUP1:31 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set (Var "R")) "is" ($#v4_xxreal_2 :::"bounded_above"::: ) )))) ; theorem :: RINFSUP1:32 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set (Var "R")) "is" ($#v3_xxreal_2 :::"bounded_below"::: ) )))) ; theorem :: RINFSUP1:33 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set (Var "R")) "is" ($#v5_xxreal_2 :::"real-bounded"::: ) )))) ; theorem :: RINFSUP1:34 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0())) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))))))) ; theorem :: RINFSUP1:35 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0())) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))))))) ; definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"inferior_realsequence"::: "seq" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: RINFSUP1:def 4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set "(" "seq" ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "Y")))))); end; :: deftheorem defines :::"inferior_realsequence"::: RINFSUP1:def 4 : (Bool "for" (Set (Var "seq")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "Y")))))) ")" )); definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"superior_realsequence"::: "seq" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: RINFSUP1:def 5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set "(" "seq" ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "Y")))))); end; :: deftheorem defines :::"superior_realsequence"::: RINFSUP1:def 5 : (Bool "for" (Set (Var "seq")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) "}" )) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "Y")))))) ")" )); theorem :: RINFSUP1:36 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:37 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" (Set (Var "seq")) ($#k1_valued_0 :::"^\"::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:38 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:39 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:40 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s"))))) ")" ) ")" ) ")" )))) ; theorem :: RINFSUP1:41 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) & (Bool "(" "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )))) ")" ) ")" ) ")" )))) ; theorem :: RINFSUP1:42 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "(" (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ))) ")" ) "iff" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )))) ; theorem :: RINFSUP1:43 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "(" (Bool "(" "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "m")))) ")" ) "iff" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )))) ; theorem :: RINFSUP1:44 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "(" (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) "iff" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" )))) ; theorem :: RINFSUP1:45 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "(" (Bool "(" "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) "iff" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" )))) ; theorem :: RINFSUP1:46 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_0 :::"min"::: ) "(" (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )))) ; theorem :: RINFSUP1:47 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set "(" (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )))) ; theorem :: RINFSUP1:48 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: RINFSUP1:49 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; theorem :: RINFSUP1:50 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) "is" bbbadV7_VALUED_0())) ; theorem :: RINFSUP1:51 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) "is" bbbadV8_VALUED_0())) ; theorem :: RINFSUP1:52 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; theorem :: RINFSUP1:53 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ))))) ; theorem :: RINFSUP1:54 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; theorem :: RINFSUP1:55 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" )))) ; theorem :: RINFSUP1:56 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )) ; theorem :: RINFSUP1:57 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ))) ")" )) ; theorem :: RINFSUP1:58 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ))) ")" )) ; theorem :: RINFSUP1:59 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:60 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:61 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" )))) ; theorem :: RINFSUP1:62 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" ) ")" )))) ; theorem :: RINFSUP1:63 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0())) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: RINFSUP1:64 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0())) "holds" (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Var "seq")))) ; theorem :: RINFSUP1:65 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: RINFSUP1:66 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: RINFSUP1:67 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq")))) & (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ))) ; theorem :: RINFSUP1:68 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV7_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:69 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; theorem :: RINFSUP1:70 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0())) "holds" (Bool (Set ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Var "seq")))) ; theorem :: RINFSUP1:71 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; theorem :: RINFSUP1:72 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: RINFSUP1:73 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq")))) & (Bool (Set ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq"))) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ))) ; theorem :: RINFSUP1:74 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" bbbadV8_VALUED_0()) & (Bool (Set (Var "seq")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:75 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) ")" )) ; theorem :: RINFSUP1:76 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq2")) "is" bbbadV2_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:77 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq2")) "is" bbbadV1_SEQ_2())) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:78 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:79 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" bbbadV2_SEQ_2()) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; theorem :: RINFSUP1:80 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" bbbadV1_SEQ_2()) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool (Set (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))))) ; definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"lim_sup"::: "seq" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: RINFSUP1:def 6 (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) "seq" ")" )); end; :: deftheorem defines :::"lim_sup"::: RINFSUP1:def 6 : (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k2_rinfsup1 :::"lower_bound"::: ) (Set "(" ($#k4_rinfsup1 :::"superior_realsequence"::: ) (Set (Var "seq")) ")" )))); definitionlet "seq" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"lim_inf"::: "seq" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: RINFSUP1:def 7 (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) "seq" ")" )); end; :: deftheorem defines :::"lim_inf"::: RINFSUP1:def 7 : (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rinfsup1 :::"upper_bound"::: ) (Set "(" ($#k3_rinfsup1 :::"inferior_realsequence"::: ) (Set (Var "seq")) ")" )))); theorem :: RINFSUP1:81 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s"))))))) ")" ))) ; theorem :: RINFSUP1:82 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )))))) ")" ))) ; theorem :: RINFSUP1:83 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s"))))) ")" ) & (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ))))) ")" )) ")" ))) ; theorem :: RINFSUP1:84 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))))))) ")" ))) ; theorem :: RINFSUP1:85 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s"))))))) ")" ))) ; theorem :: RINFSUP1:86 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq")))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "r")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s"))))) ")" ) & (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "r")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")))))) ")" )) ")" ))) ; theorem :: RINFSUP1:87 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq"))))) ; theorem :: RINFSUP1:88 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq")))) ")" ) "iff" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) ")" )) ; theorem :: RINFSUP1:89 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool "(" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq")))) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq")))) ")" )) ; theorem :: RINFSUP1:90 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq")) ")" ))) ")" )) ; theorem :: RINFSUP1:91 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "seq1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "seq2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool "(" (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")))) & (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")))) ")" )) ; theorem :: RINFSUP1:92 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")) ")" ))) & "(" (Bool (Bool "(" (Bool (Set (Var "seq1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) "or" (Bool (Set (Var "seq2")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) ")" )) "implies" (Bool "(" (Bool (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" (Set (Var "seq1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "seq2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")) ")" ))) ")" ) ")" ")" )) ; theorem :: RINFSUP1:93 (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq1")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq1")) "is" ($#v4_partfun3 :::"nonnegative"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "seq2")) "is" ($#v4_partfun3 :::"nonnegative"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq1")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_rinfsup1 :::"lim_inf"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_rinfsup1 :::"lim_inf"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" ))) & (Bool (Set ($#k5_rinfsup1 :::"lim_sup"::: ) (Set "(" (Set (Var "seq1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "seq2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq1")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k5_rinfsup1 :::"lim_sup"::: ) (Set (Var "seq2")) ")" ))) ")" )) ;