:: RFUNCT_4 semantic presentation begin theorem :: RFUNCT_4:1 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ))) ; theorem :: RFUNCT_4:2 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R1")) "," (Set (Var "R2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set "(" (Set (Var "R1")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r1")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) "," (Set "(" (Set (Var "R2")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r2")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_rvsum_1 :::"mlt"::: ) "(" (Set (Var "R1")) "," (Set (Var "R2")) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "r1")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r2")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: RFUNCT_4:3 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "st" (Bool (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "R"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "R"))))) "holds" (Bool (Set (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: RFUNCT_4:4 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "R"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: RFUNCT_4:5 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "holds" (Bool (Set ($#k15_rvsum_1 :::"mlt"::: ) "(" (Set "(" (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; begin definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_strictly_convex_on"::: "X" means :: RFUNCT_4:def 1 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set (Var "s")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )))) ")" ) ")" ); end; :: deftheorem defines :::"is_strictly_convex_on"::: RFUNCT_4:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )))) ")" ) ")" ) ")" ))); theorem :: RFUNCT_4:6 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:7 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )) "iff" (Bool "(" (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )))) ")" ) ")" ) ")" ))) ; theorem :: RFUNCT_4:8 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "c")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "b")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "c")) ")" ) ")" ))) ")" ) ")" ) ")" ))) ; theorem :: RFUNCT_4:9 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "c")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "b")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "c")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "c")) ")" ) ")" ))) ")" ) ")" ) ")" ))) ; theorem :: RFUNCT_4:10 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "Y"))))) ; theorem :: RFUNCT_4:11 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Set (Var "f")) ($#k15_valued_1 :::"-"::: ) (Set (Var "r"))) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: RFUNCT_4:12 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: RFUNCT_4:13 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "g")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g"))) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:14 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "g")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g"))) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:15 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "g")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) & (Bool "(" (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" )) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X")))))) ; theorem :: RFUNCT_4:16 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "r")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ))) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "r")) ")" ))) ")" ) ")" ) ")" ) ")" ))) ; theorem :: RFUNCT_4:17 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "r")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" ))) & (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "a")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "a")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "b")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ")" ) ($#k12_binop_2 :::"/"::: ) (Set "(" (Set (Var "b")) ($#k10_binop_2 :::"-"::: ) (Set (Var "r")) ")" ))) ")" ) ")" ) ")" ) ")" ))) ; theorem :: RFUNCT_4:18 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "P")) "," (Set (Var "E")) "," (Set (Var "F")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "st" (Bool (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "P"))))) "holds" (Bool "(" (Bool (Set (Set (Var "P")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "F")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "E")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ))) ")" ) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k15_rvsum_1 :::"mlt"::: ) "(" (Set (Var "P")) "," (Set (Var "E")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k15_rvsum_1 :::"mlt"::: ) "(" (Set (Var "P")) "," (Set (Var "F")) ")" ")" )))) ")" ) "iff" (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ")" )) ; theorem :: RFUNCT_4:19 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Interval":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "x1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x2"))) ")" )) & (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "I")))) "holds" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "a")))))) ; begin definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_quasiconvex_on"::: "X" means :: RFUNCT_4:def 2 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"is_quasiconvex_on"::: RFUNCT_4:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_rfunct_4 :::"is_quasiconvex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_strictly_quasiconvex_on"::: "X" means :: RFUNCT_4:def 3 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "s"))))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"is_strictly_quasiconvex_on"::: RFUNCT_4:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_rfunct_4 :::"is_strictly_quasiconvex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_strongly_quasiconvex_on"::: "X" means :: RFUNCT_4:def 4 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set (Var "s")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"is_strongly_quasiconvex_on"::: RFUNCT_4:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_rfunct_4 :::"is_strongly_quasiconvex_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k11_binop_2 :::"*"::: ) (Set (Var "r")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_upper_semicontinuous_in"::: "x0" means :: RFUNCT_4:def 5 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) "x0" ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set "(" "f" ($#k1_seq_1 :::"."::: ) "x0" ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_upper_semicontinuous_in"::: RFUNCT_4:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_rfunct_4 :::"is_upper_semicontinuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_upper_semicontinuous_on"::: "X" means :: RFUNCT_4:def 6 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r5_rfunct_4 :::"is_upper_semicontinuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_upper_semicontinuous_on"::: RFUNCT_4:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_rfunct_4 :::"is_upper_semicontinuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r5_rfunct_4 :::"is_upper_semicontinuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_lower_semicontinuous_in"::: "x0" means :: RFUNCT_4:def 7 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) "x0" ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) "x0" ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_lower_semicontinuous_in"::: RFUNCT_4:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_rfunct_4 :::"is_lower_semicontinuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_lower_semicontinuous_on"::: "X" means :: RFUNCT_4:def 8 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r7_rfunct_4 :::"is_lower_semicontinuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_lower_semicontinuous_on"::: RFUNCT_4:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r8_rfunct_4 :::"is_lower_semicontinuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r7_rfunct_4 :::"is_lower_semicontinuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" ))); theorem :: RFUNCT_4:20 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "f")) ($#r5_rfunct_4 :::"is_upper_semicontinuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f")) ($#r7_rfunct_4 :::"is_lower_semicontinuous_in"::: ) (Set (Var "x0"))) ")" ) "iff" (Bool (Set (Var "f")) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ))) ; theorem :: RFUNCT_4:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "f")) ($#r6_rfunct_4 :::"is_upper_semicontinuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r8_rfunct_4 :::"is_lower_semicontinuous_on"::: ) (Set (Var "X"))) ")" ) "iff" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) ")" ))) ; theorem :: RFUNCT_4:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_rfunct_4 :::"is_strictly_convex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r4_rfunct_4 :::"is_strongly_quasiconvex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:23 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_rfunct_4 :::"is_strongly_quasiconvex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_rfunct_4 :::"is_quasiconvex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:24 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_rfunct_3 :::"is_convex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_rfunct_4 :::"is_strictly_quasiconvex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:25 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_rfunct_4 :::"is_strongly_quasiconvex_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_rfunct_4 :::"is_strictly_quasiconvex_on"::: ) (Set (Var "X"))))) ; theorem :: RFUNCT_4:26 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r3_rfunct_4 :::"is_strictly_quasiconvex_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Var "f")) ($#r4_rfunct_4 :::"is_strongly_quasiconvex_on"::: ) (Set (Var "X"))))) ;