:: INTEGR16 semantic presentation begin theorem :: INTEGR16:1 (Bool "for" (Set (Var "z")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set ($#k3_complex1 :::"Re"::: ) (Set "(" (Set (Var "r")) ($#k5_binop_2 :::"*"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k3_complex1 :::"Re"::: ) (Set (Var "z")) ")" ))) & (Bool (Set ($#k4_complex1 :::"Im"::: ) (Set "(" (Set (Var "r")) ($#k5_binop_2 :::"*"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k4_complex1 :::"Im"::: ) (Set (Var "z")) ")" ))) ")" ))) ; registrationlet "S" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ); cluster (Set ($#k3_comseq_3 :::"Re"::: ) "S") -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; cluster (Set ($#k4_comseq_3 :::"Im"::: ) "S") -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; definitionlet "S" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ); :: original: :::"Re"::: redefine func :::"Re"::: "S" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); :: original: :::"Im"::: redefine func :::"Im"::: "S" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); end; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); mode :::"middle_volume"::: "of" "f" "," "D" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) means :: INTEGR16:def 1 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "D")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "D"))) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" "f" ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ))) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_binop_2 :::"*"::: ) (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" "D" "," (Set (Var "i")) ")" ")" ) ")" ))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"middle_volume"::: INTEGR16:def 1 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "D")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "D"))))) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ))) & (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_binop_2 :::"*"::: ) (Set "(" ($#k3_integra1 :::"vol"::: ) (Set "(" ($#k2_integra1 :::"divset"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" ) ")" ))) ")" )) ")" ) ")" ) ")" ))))); definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "D" be ($#m1_integra1 :::"Division"::: ) "of" (Set (Const "A")); let "F" be ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Const "f")) "," (Set (Const "D")); func :::"middle_sum"::: "(" "f" "," "F" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) equals :: INTEGR16:def 2 (Set ($#k17_rvsum_1 :::"Sum"::: ) "F"); end; :: deftheorem defines :::"middle_sum"::: INTEGR16:def 2 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "holds" (Bool (Set ($#k3_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k17_rvsum_1 :::"Sum"::: ) (Set (Var "F")))))))); definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); mode :::"middle_volume_Sequence"::: "of" "f" "," "T" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set ($#k2_numbers :::"COMPLEX"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) means :: INTEGR16:def 3 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) "is" ($#m1_integr16 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"middle_volume_Sequence"::: INTEGR16:def 3 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" (Set ($#k2_numbers :::"COMPLEX"::: ) ) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) "is" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Set (Var "T")) ($#k2_integra2 :::"."::: ) (Set (Var "k"))))) ")" ))))); definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); let "k" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"."::: redefine func "S" :::"."::: "k" -> ($#m1_integr16 :::"middle_volume"::: ) "of" "f" "," (Set "T" ($#k2_integra2 :::"."::: ) "k"); end; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "T" be ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Const "A")); let "S" be ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Const "f")) "," (Set (Const "T")); func :::"middle_sum"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"Complex_Sequence":::) means :: INTEGR16:def 4 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_integr16 :::"middle_sum"::: ) "(" "f" "," (Set "(" "S" ($#k4_integr16 :::"."::: ) (Set (Var "i")) ")" ) ")" ))); end; :: deftheorem defines :::"middle_sum"::: INTEGR16:def 4 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Complex_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k5_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "S")) ($#k4_integr16 :::"."::: ) (Set (Var "i")) ")" ) ")" ))) ")" )))))); begin theorem :: INTEGR16:2 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")))))) ; theorem :: INTEGR16:3 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")))))) ; registrationlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); cluster (Set ($#k3_comseq_3 :::"Re"::: ) "f") -> ($#v1_funct_2 :::"quasi_total"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" "A" "," (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set ($#k4_comseq_3 :::"Im"::: ) "f") -> ($#v1_funct_2 :::"quasi_total"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" "A" "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: INTEGR16:4 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k1_integr16 :::"Re"::: ) (Set (Var "S"))) "is" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) "," (Set (Var "D"))) & (Bool (Set ($#k2_integr16 :::"Im"::: ) (Set (Var "S"))) "is" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) "," (Set (Var "D"))) ")" ))))) ; theorem :: INTEGR16:5 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k1_integr16 :::"Re"::: ) (Set "(" (Set (Var "F")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_integr16 :::"Re"::: ) (Set (Var "F")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k3_complex1 :::"Re"::: ) (Set (Var "x")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: INTEGR16:6 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k2_integr16 :::"Im"::: ) (Set "(" (Set (Var "F")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_integr16 :::"Im"::: ) (Set (Var "F")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k4_complex1 :::"Im"::: ) (Set (Var "x")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: INTEGR16:7 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Fr")) ($#r1_hidden :::"="::: ) (Set ($#k1_integr16 :::"Re"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "Fr"))) ($#r1_hidden :::"="::: ) (Set ($#k3_complex1 :::"Re"::: ) (Set "(" ($#k17_rvsum_1 :::"Sum"::: ) (Set (Var "F")) ")" ))))) ; theorem :: INTEGR16:8 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Fi")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Fi")) ($#r1_hidden :::"="::: ) (Set ($#k2_integr16 :::"Im"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "Fi"))) ($#r1_hidden :::"="::: ) (Set ($#k4_complex1 :::"Im"::: ) (Set "(" ($#k17_rvsum_1 :::"Sum"::: ) (Set (Var "F")) ")" ))))) ; theorem :: INTEGR16:9 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) (Bool "for" (Set (Var "Fr")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "Fr")) ($#r1_hidden :::"="::: ) (Set ($#k1_integr16 :::"Re"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k3_complex1 :::"Re"::: ) (Set "(" ($#k3_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) "," (Set (Var "Fr")) ")" ))))))) ; theorem :: INTEGR16:10 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_integra1 :::"Division"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "F")) "being" ($#m1_integr16 :::"middle_volume"::: ) "of" (Set (Var "f")) "," (Set (Var "D")) (Bool "for" (Set (Var "Fi")) "being" ($#m1_integr15 :::"middle_volume"::: ) "of" (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "Fi")) ($#r1_hidden :::"="::: ) (Set ($#k2_integr16 :::"Im"::: ) (Set (Var "F"))))) "holds" (Bool (Set ($#k4_complex1 :::"Im"::: ) (Set "(" ($#k3_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "F")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_integr15 :::"middle_sum"::: ) "(" (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) "," (Set (Var "Fi")) ")" ))))))) ; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); attr "f" is :::"integrable"::: means :: INTEGR16:def 5 (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) "f") "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) "f") "is" ($#v3_integra1 :::"integrable"::: ) ) ")" ); end; :: deftheorem defines :::"integrable"::: INTEGR16:def 5 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_integr16 :::"integrable"::: ) ) "iff" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) "is" ($#v3_integra1 :::"integrable"::: ) ) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) "is" ($#v3_integra1 :::"integrable"::: ) ) ")" ) ")" ))); theorem :: INTEGR16:11 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) "iff" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" )) ; theorem :: INTEGR16:12 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "g")))) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "g")))) ")" )))) ; theorem :: INTEGR16:13 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) "iff" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) ")" ))) ; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func :::"integral"::: "f" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) equals :: INTEGR16:def 6 (Set (Set "(" ($#k12_integra1 :::"integral"::: ) (Set "(" ($#k5_comseq_3 :::"Re"::: ) "f" ")" ) ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k12_integra1 :::"integral"::: ) (Set "(" ($#k6_comseq_3 :::"Im"::: ) "f" ")" ) ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" )); end; :: deftheorem defines :::"integral"::: INTEGR16:def 6 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k6_integr16 :::"integral"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_integra1 :::"integral"::: ) (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k12_integra1 :::"integral"::: ) (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" ))))); theorem :: INTEGR16:14 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_integr16 :::"integrable"::: ) ) & (Bool (Set ($#k2_integra3 :::"delta"::: ) (Set (Var "T"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k2_integra3 :::"delta"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k5_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" ($#k5_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_integr16 :::"integral"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: INTEGR16:15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_integr16 :::"integrable"::: ) ) "iff" (Bool "ex" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"DivSequence":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "S")) "being" ($#m2_integr16 :::"middle_volume_Sequence"::: ) "of" (Set (Var "f")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#k2_integra3 :::"delta"::: ) (Set (Var "T"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" ($#k2_integra3 :::"delta"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k5_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_comseq_2 :::"lim"::: ) (Set "(" ($#k5_integr16 :::"middle_sum"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))) ")" ))) ; definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); pred "f" :::"is_integrable_on"::: "A" means :: INTEGR16:def 7 (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) "f") ($#r1_integra5 :::"is_integrable_on"::: ) "A") & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) "f") ($#r1_integra5 :::"is_integrable_on"::: ) "A") ")" ); end; :: deftheorem defines :::"is_integrable_on"::: INTEGR16:def 7 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f"))) ($#r1_integra5 :::"is_integrable_on"::: ) (Set (Var "A"))) ")" ) ")" ))); definitionlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func :::"integral"::: "(" "f" "," "A" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) equals :: INTEGR16:def 8 (Set (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k5_comseq_3 :::"Re"::: ) "f" ")" ) "," "A" ")" ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k6_comseq_3 :::"Im"::: ) "f" ")" ) "," "A" ")" ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" )); end; :: deftheorem defines :::"integral"::: INTEGR16:def 8 : (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k2_integra5 :::"integral"::: ) "(" (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" ))))); theorem :: INTEGR16:16 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) "iff" (Bool (Set (Var "g")) "is" ($#v1_integr16 :::"integrable"::: ) ) ")" )))) ; theorem :: INTEGR16:17 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_integr16 :::"integral"::: ) (Set (Var "g"))))))) ; definitionlet "a", "b" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func :::"integral"::: "(" "f" "," "a" "," "b" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) equals :: INTEGR16:def 9 (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k5_comseq_3 :::"Re"::: ) "f" ")" ) "," "a" "," "b" ")" ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k6_comseq_3 :::"Im"::: ) "f" ")" ) "," "a" "," "b" ")" ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" )); end; :: deftheorem defines :::"integral"::: INTEGR16:def 9 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k8_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" (Set "(" ($#k4_integra5 :::"integral"::: ) "(" (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) "," (Set (Var "a")) "," (Set (Var "b")) ")" ")" ) ($#k5_binop_2 :::"*"::: ) (Set ($#k7_complex1 :::""::: ) ) ")" ))))); begin theorem :: INTEGR16:18 (Bool "for" (Set (Var "c")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k5_comseq_3 :::"Re"::: ) (Set "(" (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k3_complex1 :::"Re"::: ) (Set (Var "c")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k4_complex1 :::"Im"::: ) (Set (Var "c")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) ")" ))) & (Bool (Set ($#k6_comseq_3 :::"Im"::: ) (Set "(" (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k3_complex1 :::"Re"::: ) (Set (Var "c")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k6_comseq_3 :::"Im"::: ) (Set (Var "f")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k4_complex1 :::"Im"::: ) (Set (Var "c")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k5_comseq_3 :::"Re"::: ) (Set (Var "f")) ")" ) ")" ))) ")" ))) ; theorem :: INTEGR16:19 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k3_binop_2 :::"+"::: ) (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) & (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f1")) "," (Set (Var "A")) ")" ")" ) ($#k4_binop_2 :::"-"::: ) (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f2")) "," (Set (Var "A")) ")" ")" ))) ")" ))) ; theorem :: INTEGR16:20 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k5_binop_2 :::"*"::: ) (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; theorem :: INTEGR16:21 (Bool "for" (Set (Var "c")) "being" ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_integr16 :::"is_integrable_on"::: ) (Set (Var "A"))) & (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set "(" (Set (Var "c")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_binop_2 :::"*"::: ) (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" ))) ")" )))) ; theorem :: INTEGR16:22 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ))))) ; theorem :: INTEGR16:23 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_measure5 :::"closed_interval"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "b")) "," (Set (Var "a")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k1_binop_2 :::"-"::: ) (Set "(" ($#k7_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_integr16 :::"integral"::: ) "(" (Set (Var "f")) "," (Set (Var "a")) "," (Set (Var "b")) ")" ))))) ;