:: LIMFUNC4 semantic presentation begin theorem :: LIMFUNC4:1 (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set (Var "X"))))) "holds" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set (Var "X")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) ")" )))) ; theorem :: LIMFUNC4:2 (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "X"))))) "holds" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "X")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) ")" )))) ; theorem :: LIMFUNC4:3 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:4 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:5 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:6 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:7 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:8 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:9 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:10 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:11 (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")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:12 (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")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:13 (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")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:14 (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")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:15 (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")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:16 (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")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:17 (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")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:18 (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")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:19 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:20 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:21 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:22 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:23 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:24 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:25 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:26 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:27 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:28 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:29 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:30 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:31 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:32 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:33 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:34 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:35 (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")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:36 (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")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:37 (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")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:38 (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")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:39 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:40 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:41 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:42 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:43 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:44 (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")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:45 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:46 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:47 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) )) ; theorem :: LIMFUNC4:48 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (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 "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) )) ; theorem :: LIMFUNC4:49 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:50 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:51 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:52 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))))) ; theorem :: LIMFUNC4:53 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v4_limfunc1 :::"divergent_in+infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f2")))) ")" )) ; theorem :: LIMFUNC4:54 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v5_limfunc1 :::"divergent_in+infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f2")))) ")" )) ; theorem :: LIMFUNC4:55 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v7_limfunc1 :::"divergent_in-infty_to+infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f2")))) ")" )) ; theorem :: LIMFUNC4:56 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v8_limfunc1 :::"divergent_in-infty_to-infty"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f2")))) ")" )) ; theorem :: LIMFUNC4:57 (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")) ($#r2_limfunc2 :::"is_left_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:58 (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")) ($#r3_limfunc2 :::"is_left_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:59 (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")) ($#r5_limfunc2 :::"is_right_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:60 (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")) ($#r6_limfunc2 :::"is_right_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:61 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:62 (Bool "for" (Set (Var "x0")) "being" ($#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 (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:63 (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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r")) "being" ($#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 "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#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 "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:64 (Bool "for" (Set (Var "x0")) "being" ($#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")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:65 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:66 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:67 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:68 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:69 (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")) ($#r2_limfunc3 :::"is_divergent_to+infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:70 (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")) ($#r3_limfunc3 :::"is_divergent_to-infty_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f2")))) ")" ))) ; theorem :: LIMFUNC4:71 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set (Var "f2")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_limfunc1 :::"convergent_in+infty"::: ) ) & (Bool (Set ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k4_limfunc1 :::"lim_in+infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:72 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set (Var "f2")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1"))))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v6_limfunc1 :::"convergent_in-infty"::: ) ) & (Bool (Set ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k5_limfunc1 :::"lim_in-infty"::: ) (Set (Var "f1")) ")" ) ")" )) ")" )) ; theorem :: LIMFUNC4:73 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:74 (Bool "for" (Set (Var "x0")) "being" ($#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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc2 :::"is_left_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc2 :::"lim_left"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:75 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")))) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:76 (Bool "for" (Set (Var "x0")) "being" ($#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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r4_limfunc2 :::"is_right_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k2_limfunc2 :::"lim_right"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ; theorem :: LIMFUNC4:77 (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")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2")))) "holds" (Bool "ex" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g1"))) & (Bool (Set (Var "g1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x0"))) & (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "g2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "x0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g2"))) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) ")" )) ")" ) & (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ($#k2_rcomp_1 :::".["::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "g")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ")" )))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" )) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_limfunc3 :::"is_convergent_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f2")) "," (Set "(" ($#k1_limfunc3 :::"lim"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ")" )) ")" ))) ;