:: LIMFUNC2 semantic presentation begin theorem :: LIMFUNC2:1 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k")))))))) ; theorem :: LIMFUNC2:2 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))))) ; theorem :: LIMFUNC2:3 (Bool "for" (Set (Var "r2")) "," (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" ) "," (Set (Var "r1")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: LIMFUNC2:4 (Bool "for" (Set (Var "r2")) "," (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "r1")) "," (Set "(" (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "r2")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: LIMFUNC2:5 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" ))) ")" )))) ; theorem :: LIMFUNC2:6 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) & (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" ))) ")" )))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Real":::); pred "f" :::"is_left_convergent_in"::: "x0" means :: LIMFUNC2:def 1 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) "x0")) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) "x0") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) "x0" ")" )))) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) ")" ); pred "f" :::"is_left_divergent_to+infty_in"::: "x0" means :: LIMFUNC2:def 2 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) "x0")) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) "x0") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) "x0" ")" )))) "holds" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) ) ")" ) ")" ); pred "f" :::"is_left_divergent_to-infty_in"::: "x0" means :: LIMFUNC2:def 3 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) "x0")) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) "x0") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) "x0" ")" )))) "holds" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_limfunc1 :::"divergent_to-infty"::: ) ) ")" ) ")" ); pred "f" :::"is_right_convergent_in"::: "x0" means :: LIMFUNC2:def 4 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) "x0" ")" )))) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) ")" ); pred "f" :::"is_right_divergent_to+infty_in"::: "x0" means :: LIMFUNC2:def 5 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) "x0" ")" )))) "holds" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) ) ")" ) ")" ); pred "f" :::"is_right_divergent_to-infty_in"::: "x0" means :: LIMFUNC2:def 6 (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "x0" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) "x0" ")" )))) "holds" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_limfunc1 :::"divergent_to-infty"::: ) ) ")" ) ")" ); end; :: deftheorem defines :::"is_left_convergent_in"::: LIMFUNC2:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"is_left_divergent_to+infty_in"::: LIMFUNC2:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) ) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_left_divergent_to-infty_in"::: LIMFUNC2:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_limfunc1 :::"divergent_to-infty"::: ) ) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_right_convergent_in"::: LIMFUNC2:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"is_right_divergent_to+infty_in"::: LIMFUNC2:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) ) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_right_divergent_to-infty_in"::: LIMFUNC2:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_limfunc1 :::"divergent_to-infty"::: ) ) ")" ) ")" ) ")" ))); theorem :: LIMFUNC2:7 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" )))) ")" ) ")" ))) ; theorem :: LIMFUNC2:8 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")))) ")" ) ")" )) ")" ) ")" ) ")" ))) ; theorem :: LIMFUNC2:9 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" )) ")" ) ")" ) ")" ))) ; theorem :: LIMFUNC2:10 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" )))) ")" ) ")" ))) ; theorem :: LIMFUNC2:11 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")))) ")" ) ")" )) ")" ) ")" ) ")" ))) ; theorem :: LIMFUNC2:12 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" )) ")" ) ")" ) ")" ))) ; theorem :: LIMFUNC2:13 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: LIMFUNC2:14 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: LIMFUNC2:15 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: LIMFUNC2:16 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: LIMFUNC2:17 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) ) ")" ))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:18 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "," (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:19 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) ) ")" ))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:20 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "," (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:21 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ")" ))) ; theorem :: LIMFUNC2:22 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" & "(" (Bool (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) ")" ")" ))) ; theorem :: LIMFUNC2:23 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) "or" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" )) "holds" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:24 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) "or" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" )) "holds" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:25 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v7_valued_0 :::"non-decreasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:26 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:27 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v8_valued_0 :::"non-increasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:28 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:29 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v8_valued_0 :::"non-increasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:30 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_seq_2 :::"bounded_above"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:31 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v7_valued_0 :::"non-decreasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:32 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v5_valued_0 :::"increasing"::: ) ) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_seq_2 :::"bounded_below"::: ) )) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:33 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:34 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:35 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:36 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:37 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:38 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:39 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:40 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Real":::); assume (Bool (Set (Const "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Const "x0"))) ; func :::"lim_left"::: "(" "f" "," "x0" ")" -> ($#m1_subset_1 :::"Real":::) means :: LIMFUNC2:def 7 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) "x0" ")" )))) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) it) ")" )); end; :: deftheorem defines :::"lim_left"::: LIMFUNC2:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) "iff" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "b3"))) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Real":::); assume (Bool (Set (Const "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Const "x0"))) ; func :::"lim_right"::: "(" "f" "," "x0" ")" -> ($#m1_subset_1 :::"Real":::) means :: LIMFUNC2:def 8 (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) "x0") & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) "x0" ")" )))) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) it) ")" )); end; :: deftheorem defines :::"lim_right"::: LIMFUNC2:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) "iff" (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "b3"))) ")" )) ")" )))); theorem :: LIMFUNC2:41 (Bool "for" (Set (Var "x0")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "iff" (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" ))) ")" ))) ; theorem :: LIMFUNC2:42 (Bool "for" (Set (Var "x0")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "iff" (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) ")" ) ")" ))) ")" ))) ; theorem :: LIMFUNC2:43 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:44 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:45 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:46 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:47 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k2_real_1 :::"""::: ) )) ")" ))) ; theorem :: LIMFUNC2:48 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:49 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k2_real_1 :::"""::: ) )) ")" ))) ; theorem :: LIMFUNC2:50 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:51 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:52 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:53 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:54 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:55 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:56 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k2_real_1 :::"""::: ) )) ")" ))) ; theorem :: LIMFUNC2:57 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k56_valued_1 :::"abs"::: ) (Set (Var "f"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" ($#k56_valued_1 :::"abs"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_complex1 :::"abs"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:58 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k2_real_1 :::"""::: ) )) ")" ))) ; theorem :: LIMFUNC2:59 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:60 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: LIMFUNC2:61 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: LIMFUNC2:62 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: LIMFUNC2:63 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) & (Bool "(" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) ")" ) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ))) ; theorem :: LIMFUNC2:64 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ))) ; theorem :: LIMFUNC2:65 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) & (Bool "(" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) ")" ) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ))) ; theorem :: LIMFUNC2:66 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ))) ; theorem :: LIMFUNC2:67 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) ")" ) ")" ))) "holds" (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )))) ; theorem :: LIMFUNC2:68 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ) ")" ) ")" ))) "holds" (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" )))) ; theorem :: LIMFUNC2:69 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" (Bool (Set (Var "f")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) "or" (Bool (Set (Var "f")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: LIMFUNC2:70 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" (Bool (Set (Var "f")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) "or" (Bool (Set (Var "f")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: LIMFUNC2:71 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:72 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:73 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:74 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:75 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:76 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:77 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC2:78 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ;